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Full text of "The Mechanical Properties Of Fluids"

Applied Physics Series 



THE MECHANICAL PROPERTIES 
OF FLUIDS 



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BLACKIK & SON LIMITKD 

66 Chando* Pface, lONtX)N 

i/ tatatili(n htitct, Gl.AMjOW 

ni.ACKlE & SON (IND!A) LIMiTKil 
Warwick Huuw, I-urt Httu't, 



BLACKIK & SON (CANADA) UMtH I) 
TOHUNHI 



THE MECHANICAL 
PROPERTIES OF 

FLUIDS 

A Collective ffiork by 
C. V. Drysdale F. R. W. Hunt 

,-^^^:==;-.JteJBaB 1. ^ 

D Sc.(Lond ) M A,, B Sc 

Allan Ferguson Horace Lamb 

M A , D St (Lond ) LL D , Sc D , F R S 

A. E. M. Geddes A. G. M. Michell 

O B E , M A , D Sc M C E , F R S 

A. H. Gibson Sir Geoffrey Taylor 

D Sc , M Inst C E. M A , I R S. 

Engineer Vice-Admiral 

Sir George Goodwin 
K.C B , LL D. 



SECOND EDITION 



BLACKIE & SON LIMITED 

LONDON AND GLASGOW 







BOOK 

PRODUCTION 

WAR ECONOMY 

STANDARD 



THE PAPER AND- BINDING OF THIS BOOK 

CONFORM TO THE AUTHORIZED ECONOMY 

STANDARDS 



Fiist ^ssue, 1923 

jKejjnnled 1025 

Second edition, tevwd and cnlaiged, 1036 

Repmtcd, with comclions, 1937 

Mepnnted Wd4, 1946 



Printed m Great Britain by 
Blackie & Son, Limited, Glasgow 



PUBLISHERS' NOTE 



In recent years a great many researches have been made 
ito the mechanical properties of fluids by physicists and 
igineers. These researches are of the utmost practical 
iportance to engineers and others, but it is not unusual 

find that the people who are called upon to apply the 
suits m industry have considerable difficulty in finding 
mnected accounts of the work It is hoped that this 
Election of essays, ^many of which are written by men 
10 are the actual pioneers, will prove of use in making 
e recent discoveries in the mechanical properties of fluids 
are generally known The mathematical notation has 
en made uniform, and the different chapters have been 
Hated as far as possible. 



CONTENTS 



Introduction 

By ENGINEER VIOE-ADMIEAL SIE GEOEGE GOODWIN, 
KO.B., LL.D. 



itioduction 



CHAPTER I 

Liquids and Gases 
By ALLAN FEEGUSON, MA, DSc(Lond) 

efinitiona Density Compressibility Surface Tension Viscosity 
Equations of State Osmotic Pressure 1 

CHAPTER II 

Mathematical Theory of Fluid Motion 
By PEOFESSOR HORACE LAMB, LL.D., ScD, F.R.S. 

ream-line Motion Vortex Motion Wave Motion Viscosity - - 56 

CHAPTER III 

Viscosity and Lubrication 

By A G. M. MIOHELL, M.O.E., F.E.S. 

VISCOSITY. Laminar Motion Coefficient of Viscosity Relative 
Velocities Conditions at the Bounding Surfaces of Fluids Motion 
Parallel to Bounding Surfaces Viscous Flow in Tubes Use of 
Capillary Tubes as Viscometers Secondary or Commercial Viscometers 
Coefficients of Viscosity of Various Fluids Variation of Viscosity 

Vli 



viii CONTENTS 

Page 

with Pressure Viscous Plow between Paiallel Planes Mow between 
Parallel Planes having Eelative Motion Cup-and-ball Viacometer - 102 

B LUBRICATION The Connection between Lubrication and Viscosity 
Essential Condition of Viscous Lubrication Inclined Planes Un- 
limited in One Direction Applications to Actual Bearings Self- 
adjustment of the Positions of Bearing Surfaces Self-adjustment 
in Journal Bearings Exact Calculation of Cylindrical Journal and 
Beaimg Approximate Calculation of Cylindrical Bearings Plane 
Bearings of Finite Width Cylindrical Bearings of Finite Width- 
Experimental Eesults Types of Pivotal Bearings Flexible Bearings 
Limitations of the Theory Bibliography 128 



CHAPTER IV 

Stream-line and Turbulent Flow- 
By PROFESSOR A H GIBSON, D Sc. 

Stream-line Motion Stability of Stream-line Motion Hele Shaw's 
Experiments Critical Velocity Critical Velocity in Converging 
Tubes The Measurement of the Velocity of Mow in Fluids The 
Ventun Meter Measurement of Flow by Diaphragm in Pipe Line 
The Pitot Tube The Effect of Fluid Motion on Heat Transmission- 
Application of the Principle of Dimensional Homogeneity to Problems 
involving Heat Transmission - ..... 160 

CHAPTER V 

Hydrodynamical Resistance 
By PROFESSOR A H GIBSON, DSc. 

Dimensional Homogeneity and Dynamical Similarity Eesistance to the 
Uniform Flow of a Fluid through a Pipe Skin Friction Besistance 
of Wholly Submerged Bodies Eesistance of Partially Submerged 
Bodies Model Experiments on Eesistance of Ships Scale Effects 
Eesiatance of Plane Surfaces, of Wires and Cylinders, of Strut 
Sections Resistance of Smooth Wires and Cylinders ... 19] 

CHAPTER VI 

Phenomena due to the Elasticity of a Fluid 
By PROFESSOR A. H GIBSON, D.Sc 

Compressibility Sudden Stoppage of Motion . Tdeal Case Effect of 
Friction m the Pipe Line Magnitude of Eise in Pressure, following 
Sudden Closure Effect of Elasticity of Pipe Line Valve Shut 



CONTENTS 

Suddenly but not InstantaneouslySudden Stoppage of Motion la 
a Pipe Line of non-Uniform Section Sudden Initiation of Motion- 
Wave Transmission of Energy Theory of Wave Transmission of 

Energy 



CHAPTER VII 

The Determination of Stresses by Means 
of Soap Films 

By Sir GEOFFREY TAYLOR, M.A., FJt , 

Prandtl'a Analogy Experimental Methods Accuracy of tlio Mothod 
Example of the Uses of the Method Comparison of Soap Film 
Results with those obtained in Direct Torsion ExperimentsTorsion 
of Hollow Shafts Example of the Application of the Soap film 
Method to Hollow Shafts 237 



CHAPTER VIII 
Wind Structure 

By A E. M. GEDDES, QBE, MA, I) So. 
Wind Structure 200 

CHAPTER IX 

Submarine Signalling and the Transmission 
of Sound through Water 

By V DRYSDALE, D So (Loml ) 

Fundamental Scientific Principles Velocity of Propagation Wave 
length Transmission of Sound through Various SubMlances- - 
Pressure and Displacement Receivers Directional Transmission and 
Reception PRACTICAL UNDERWATER TRANSMITTERS AND HKOBIVKUS , 
SUBMARINE TRANSMITTERS OR SOURCES OF SOUND Electromagnetic 
Transmitters Submarine Sirens RECEIVERS OR HTDROPHONKS - 
The C Tube Magnetophones PRACTICAL CONSTIUJOTIQN on 1 HYDRO- 
PHONES DIRECTIONAL DEVICES Sound Banging Leader Q oar- 
Acoustic Depth Sounding Ecbo Detection of Ships and Obstacles 
ACOUSTIC TRANSMISSION os 1 POWER Developments in Echo Depth- 
sounding Gear 208 



x CONTENTS 

CHAPTER X 

The Reaction of the Air to Artillery Projectiles 
By F. E. W. HUNT, M.A 

i'age 

Introduction THE DRAG Early Experiments. The Ballistic Pendulum 
The Bashforth Chronograph Later Experiments Krupp's 1912 
Experiments Cranz's Ballistic Kmeinatograph Experiments with 
High- velocity Air Stream The Drag at Zero Yaw The Drag at 
Low Velocities The Drag at High Velocities The Scale Effect 
Dependence of the Drag on Density (The Function f(v/a, vdjv 
Shape of Projectile The Base The Pressure Distribution on the 
Head of a Projectile The Effect of Yaw on the Drag The Drag 
Coefficient Concluding Eeniarks EEACTION TO A YAWING, SPIN- 
NING PROJECTILE The Principal Eeactions The Damping Eeactions 345 

INDEX 377 



INTRODUCTION 

BY ENGINEER VICE-ADMIRAL 
SIR GEORGE GOODWIN, K.C.B., LL.D. 

(Late Engineer-in-Chief of the Fleet) 



To those engaged in the practice of engineering, and 
ble and willing to utilize the information that mathematical 
ad physical science can offer them, it is of great assistance 
) have such information readily available in direct and 
ilevant connection with the problems with which they 
e confronted. 

The collective work of this book, issuing as it does 
om authors highly qualified and esteemed in their respec- 
ve fields, whose views and statements will be accepted as 
ithontative, supplies concisely and consistently valuable 
formation respecting the mechanical properties of fluids, 
id elucidates the evolution of many successful practical 
>plications from first considerations. 

Much has been done in this direction in regard to solids, 
id this has been assimilated and usefully applied by 
any; but much less has been produced on the subject of 
lids, especially in compact form, and this collective work 
ill doubtless on this account be very welcome. 

The necessities of the war brought us face to face with 
any new problems, a large number of which required 
>t only prompt application of the knowledge available, 



xii INTRODUCTION 

but intensive research and rapid development in order to 
comply with the constantly Increasing standards of quality 
that were demanded. Most of the results are well known: 
the principles by which they were reached, especially in 
regard to fluids, are perhaps only vaguely understood, 
except by a few. The results are certainly appreciated, 
but further application is probably hindered in many direc- 
tions for want of this knowledge. 

The several contributions to this work enunciate clearly 
the principles involved, and indicate that a wide field is open 
for the application of these principles to those who are 
engaged in industrial avocations and pursuits, as well as to 
others whose duties continue to be confined exclusively to 
preparation for war. 

Chapters dealing mainly with theoretical considerations 
form a prelude, clarifying ideas of the physical properties of 
fluids and providing a sketch of the mathematical theory 
of fluid motion, with indicators to practical utility. 

The sections devoted to practical applications are de- 
veloped from the underlying theoretical and mathematical 
considerations. The retention of this method of treatment 
of the several subjects throughout the work is a valuable 
feature. These sections will interest a variety of readers, 
some parts being of particular value to specialists such as 
the gunnery expert, the naval architect, and the aeronautical 
engineer, but by far the greater portion of the book will 
be of general interest to the large body of engineers who 
have to deal with or use fluids for many purposes in theii 
everyday work. 

The chapter on viscosity and lubrication should point 
the way to the better appreciation and further application 
of the correct principles of lubrication. The best known 
present application, that of enabling propeller thrust loads 
of high intensity to be taken on a single collar, has been highly 
successful, and it is gratifying to observe that other develop- 



INTRODUCTION xiri 

nents are already In contemplation, and that some are well 
dvanced. 

The description of the determination of stress by means 
f soap films is fascinating and deeply interesting; and it is 
heering to know that certain forms of stress in members 
f irregular form under load, not amenable to calculation, 
nd hitherto not determinable and therefore provided for by 
factor of safety, can now be closely approximated to by 
scperimental means, and it may be hoped that this or some 
ther experimental process can be extended practically in the 
ear future to determine other forms of stress. Success in 
lis direction would be directly attended with economy of 
taterial and would facilitate design. 

The chapter on submarine signalling indicates the 
arch of progress in a new branch of engineering, and 
e author makes the important and significant remark that 
e science of acoustics shows signs of developing into the 
igmeermg stage, a statement worthy of the careful con- 
ieration of all thoughtful engineers. 

The section dealing with the wave tiansmission of energy 

mes opportunely in view of the laige number of practical 

^plications of this form of power transmission that are 

ing developed, and of those that have matured The same 

ction gives information respecting the principles govern- 

2j the various foims of flow-meters, and should prove 

eful to engineers associated with high-powei installations 

10 are, by reason of the magnitude of the individual in- 

illations, being forced to use flow measurements in lieu 

the definite bulk measurements hitherto favoured by 

my, and should give a greater confidence in the use and 

curacy of flow-meters designed on a sound basis. 

The preceding cases are merely mentioned as examples; 

sry chapter contains a great deal of matter of practical 

)ical engineering interest connected with the mechanical 

)perties of fluids. I have selected these examples as some 



xiv INTRODUCTION 

of those familiar to me in which I have personally felt the 
want of some preparatory and explanatory information, such 
as that given in this book; and it is my recollection that 
such information was more difficult or inconvenient to obtain 
in regard to fluids than for solids. My own experiences 
must, I feel, be those of many others. 

The whole series of articles has been to me most interest- 
ing, and they show clearly that engineering in the present 
day requires a great deal of help from pure and experimental 
science, and is adapting itself to the utilization of branches of 
science with which it has hitherto not been closely associated. 
Engineering practice to be worthy of the name must keep 
itself abreast of and well in touch with those sciences and 
the developments and discoveries connected with them. 
This is an onerous task and can only be effected collectively; 
it is too big for one individual; but works such as this will 
tend to ease the burden, and convert the task into a pleasing 
duty. 



LIST OF SYMBOLS USED 



p or P, pressure. 

v or V, volume; velocity. 

N, modulus of iigidity. 

E, Young's modulus. 

t, temperature C. (or F.), time. 

p, density. 

s, specific giavity. 

K, bulk modulus; eddy conductivity 

T, absolute temperature in C. 

T, absolute temperature in F. 

C^, specific heat at constant volume. 

C p , specific heat at constant pressure 

7, surface tension. 

jS, compressibility. 

M, molecular weight. 

m, mass. 

/*, viscosity. 

j>, kinematic viscosity z= . 

P 
u, v, w, component velocities; displacements 

N, Avogadro's constant 

tu, specific volume of water; cross section; angular velocity. 

S, shearing stress. 

5E, twist. 

T fl , torque 

A, wave-length; film energy. 

R, acoustic resistance. 

, frequency (cycles per second). 



THE MECHANICAL 
PROPERTIES OF FLUIDS 

CHAPTER I 
liquids and Gases 

Definitions 

We propose to discuss in this chapter some of the moie important 
leial piopeities of fluids Common knowledge enables us to 
ociate with the teims fluid, liquid, vapour, gas, certain properties 
ich we regaid as fundamental, and which serve to difteientiate 
'se forms of existence from the form which we know as solid A 
id is, etymologically and physically, that which flows, and the 
uid or the gaseous state is a special case of the fluid foim of exist- 
:e A liquid, in geneial, is only slightly compiessible and pos- 
ses one free bounding surface when contained in an open vessel. 
gas, on the other hand, is easily compressible under ordmaiy 
:umstances, and always fills the vessel which contains it * The 
st elementary observation forces upon our notice distinctions 
h as these just mentioned, but it still remains to be seen whether 
se can be made the basis of a satisfactory classification 
Indeed, it is doubtful whether we can make a classification which 
I conveniently pigeon-hole the different states of matter, for, as 
shall see in the sequel, these different states shade over, under 
cial circumstances, one into the other, without the slightest 
ach of continuity. 
Ordinarily the change from solid to liquid as when ice becomes 

* But compare the quotation on p. 2. 



2 THE MECHANICAL PROPERTIES OF FLUIDS 

water or fiom liquid to vapour as when watei boils is quite shai p, 

and the propeities of any one substance m the three states aie cleaily 

marked off. But substances such as pitch or sealing-wax behaving 

under some circumstances as solids, under others as liquids aie 

distinctly troublesome to the enthusiast for classification. Thus 

a bell or tuning fork, cast from pitch, will emit a note perfectly clear 

and distinct as that given by a bell of metal. Nevertheless a block 

of pitch, left to itself, will in time flow like any ordinary liquid 

Steel balls placed on the top of pitch contained in a vessel slowly 

sink to the bottom, and corks placed at the bottom of the vessel will 

in time appear at the upper surface of the pitch. Such anomalies 

serve to emphasize the difficulties attendant on any attempt at a 

rigorous classification. Indeed it is sometimes held that the diffeience 

between the solid and liquid states is one of degree, and that all solids 

in some measuie show the properties of liquids Howevei this may 

be, it is enough to note now that the diffeiences between the solid, 

liquid, and gaseous states are sufficiently pronounced to make it 

convenient to attempt a classification which shall emphasize these 

differences We shall therefore discuss certain properties of matter 

which serve to define ideal solids, liquids, and gases. We shall find 

that no substances in nature confoini to our ideal, which will therefoie 

be but a first approximation to the truth, and later we shall find that 

small corrections, applied to the equations of state which are the 

expression of our fundamental definitions, will serve to make the 

equations represent with considerable accuracy the behavioui of 

actual substances. This process, involved though it may appear, is 

both historically correct and physically convenient 

Thus the reader may remember that m 1662 the Honourable 
Robert Boyle took a long glass tube " which by a dexterous hand and 
the help of a lamp was in such a manner crooked at the bottom that 
the part turned up was almost parallel to the rest of the tube, and the 
orifice of this shorter leg . . . being hermetically sealed, the length 
of it was divided into inches . . . Then putting in as much quick- 
silver as served to fill the arch, . . we took care, by fiequently 
inclining the tube, so that the air might freely pass from one leg into 
the other, . . . (we took, I say, care) that the air at last included m 
the shorter cylinder should be of the same laxity with the rest of the 
air about it. This done, we began to pour quicksilver into the 
longer leg, . . till the air m the shorter leg was by condensation 
reduced to take up but half the space it possessed (I say possessed not 
filled} before; we cast our eyes upon the longer leg of the glass, . . 



LIQUIDS AND GASES 3 

and we observed, not without delight and satisfaction, that the quick- 
silver in that longer part of the tube was 29 in. higher than the other."* 
The pressure and volume of a gas at constant temperature are 
therefore in reciprocal propoition; that is, at constant temperature 
L he equation of state of a gas is given by 

pv = k. 

Succeeding experiments emphasized the truth of this result, and it 
vas not until instrumental methods had advanced considerably that 
imall deviations from this law were shown to exist under ordinary 
onditions It was then proved that an equation of the type 

= k 

nore closely represented the behaviour of even the more permanent 
;ases, later work has shown this equation does not represent with 
ufficient accuracy the icsults of expenment, and vaiious other 
quations of state have, from time to time, been proposed To these 
quations we shall latei have occasion to icier 

Again, the physical convenience of such a method of appioach 
may be illustrated by lesults deduced from the principles ol rigid 
ynamics No body in naluie is peifcctly ngid that is, is such that 
line joining any two pai tides of the body remains invariable in 
ingth dm ing the motion of the body but considerable simplification 
f the equations of motion results if we make this assumption, and 
le results obtained are in many cases of as high an 01 der of accuracy 
s is required. We can, if necessary, obtain a closer approximation 
D the truth by consideimg the actual defoimation suffered by the 
ody the problem then becoming one in the theory of elasticity, 
uppose, for example, that it is oui object to deduce the acceleiation 
ue to giavity fiom observation of the period of a compound pendu- 
im It would be possible to attack the problem, taking into account 
b initio such effects as are due to, say, deformation of the pendulum 
i its swing, yielding of the supports and the like, afterwards ne 
iecting such effects as expenence has shown to be very small. But 
ich a method would lender the problem almost unbearably complex, 
esides tending to distract attention from the essentials, s and it is 
3th moie convenient and moie philosophic to focus one's mind on 
ie more impoitant issues, to solve the problem first for an ideal 

* Boyle's works (Birch's edition, Vol. I, p. 156, 1743). 



4 THE MECHANICAL PROPERTIES OF FLUIDS 

rigid body, and afterwards to introduce as small corrections effects 
due to elasticity, viscosity, and so forth. 

This, then, is the course which we shall follow in discussing the 
properties of fluids, and we shall seek, using elastic properties as a 
guide, definitions which will emphasize those differences which 
undoubtedly exist between the solid, liquid, and gaseous states of 
existence. 

If we wish to describe completely the elastic behaviour of a 
crystalline substance, we find that in the most general case twenty-one 
coefficients are required. For isotropic substances, fortunately, the 
problem is much simpler, and the coefficients reduce to two, the bulk 
modulus (K) and the rigidity modulus (N) These coefficients are 





B h 



(b) 



(c) 



Fig i 



easily specified. Thus, if a cube of unit edge be subjected to a 
uniform hydrostatic pressure P, so that its volume deci eases by an 
amount Sv, the sides of the cube deci easing by an amount e, then, 
$v being the change in volume per initial unit volume, the ratio of 
stress to strain, which is the measure of the bulk modulus, is given 

P 

by = K, and, to the first order of small quantities, $v = 32. 
ov 

Suppose now that our unit cube is strained in such a way that 
in one direction the sides are elongated by an amount e, in a per- 
pendicular dnection are contracted by an amount e (fig i a), the sides 
perpendicular to the plane of the drawing being unaltered in length 
Such a strain is called a shearing strain, and may be supposed to be 
produced by stresses (P) acting as shown. Considering the rect- 
angular prism BCD, which is in equilibrium under the stresses 
acting normally over the faces EC and CD, and the forces due to 
the action on BCD of the portion ABD of the cube, we see that the 
resultant of the two forces P is a force Pv 2 acting along BD. The 



LIQUIDS AND GASES 5 

>rce due to the action of ABD on BCD must be equal and opposite 
> this. But the area of the face BD is \/2 units, and there is there- 
ire a tangential stress of P units, in the sense DB, acting over the 
agonal area of the cube, due to the action on the prism BCD of 
ie matter in the prism ABD, and called into being by the elastic 
splacements Thus the shearing stress, which produces the shearing 
r atn, may be measured by the stress on the areas of purely normal or 
purely tangential stress. 

If we suppose the directions of the pimcipal axes of shear to be 

mg the diagonals AC and BD, so that these diagonals are conti acted 

d elongated respectively by an amount e (per unit length), then it 

n easily be shown that, assuming the strains to be small, the side of 

e square, the area of the square, and the perpendicular distance 

tween its sides are, to the fiist order of small quantities, unaltered 

the strain. Hence (fig. i b) the square ABCD strains into the 

Dmbus abed, and by rotating the rhombus through the angle cEC 

iich rotation does not involve the introduction of any elastic forces 

arrive at the state shown in fig. i c. Hence, rotation neglected, 

shearing stiam may be icgarded as being due to the sliding of 

*allel planes of the solid through horizontal distances which are 

>portional to their vertical distances from a fixed plane DC, the 

ling being brought about by a tangential stress P applied to the 

ne AB. The angle Q is taken as a measure of the strain, and the 

idity modulus (N) is given by the equation 



It is to be remembered that an elastic modulus such as Young's 
dulus (E) is not independent of K and N, but is connected with 
m, as can readily be proved, by the relation 

E = 9* N * 
3/c + N' 

We are now in a position to define formally the terms " solid " 
" fluid ". 

A solid possesses both rigidity and bulk moduli. If subjected to 
iring stress or to hydrostatic pressure it takes up a new position 
quilibrium such that the forces called into existence by the elastic 
dacements form, with the external applied forces, a system in 
ilibrium. 

ait, Properties of Matter, p 155, or Morley, Strength of Materials (1931), p. iz. 



T> THE MECHANICAL PROPERTIES OF FLUIDS 

A fluid possesses bulk elasticity, but no rigidity. It follows, 
there! 01 e, that a fluid cannot permanently resist a tangential sh ess, and 
that, however small the stress may be, the fluid will, in time, sensibly 
yield to it. In a solid, the stress on an element-plane may have any 
dheetion with reference to that plane. It may be purely normal, 
as on the plane BC (fig. i a), or purely tangential, as on the plane BD. 
In a fluid at test the stress on an element-plane must be normal to 
that plane And it follows at once from this normality, as is proved 
m all elementary treatises on hydrostatics, that the pressure (p) at 
a point in a fluid at rest under the action of any forces is independent 
of the orientation of the element-plane at that point. Thus if #, y, z 
aie the co-ordinates of the point in question, 



In upofett fluid, no tangential stresses exist, whether the fluid be 
at icst, or whether its different parts be in motion relative to each 
othci. In all fluids known in nature, tangential stresses tending to 
damp out this relative motion do exist, persisting as long as the 
relative motion persists The fluid may be looked on as yielding to 
these stresses, different fluids yielding at very diffeient time-rates, 
the late oi yield depends on the propeity known as viscosity 

A perfect liquid may be defined as an incompiessiblc peifect fluid, 
No fluid in nature is completely incompiessiblc, and the quantitative 
study ol the bulk moduli of liquids and their i elation to othci constants 
oi the liquid substance is a matter ol gieat theoietical and piactical 
importance 

Jt must be icmcmbeicd that the magnitude of the bulk modulus 
depends on the conditions under which the compicssion is earned 
out Two moduli aic of primary importance that in which the 
tcnipeialuic of the substance icmains constant, and that m which 
the compicssion is aduibatic, so that heat ncithci cntcis noi 
leaves the substance under compiession Remembcnng this, 
we may define a perfect gas as a substance whose bulk modulus 
ol isothermal elasticity is numeiically equal to its pressure. 
this we have at once by definition 




or pdv -[- vdp = o. 



LIQUIDS AND GASES 7 

Whence, integrating, 

pv k, 

nd our perfect gas follows Boyle's Law. 

Density 

Having obtained woiking definitions of the substances with which 
^e have to deal, we pioceed to discuss in order certain of their more 
indamental properties and constants One of the most important 
f these constants is the density of the fluid, defined as the mass of 
nit volume of the fluid. The density of a liquid is accepted, in a 
lemical laboratory, as one of the tests for its identification, and the 
aportance in industry of the " gravity " test needs no emphasis, 
fe shall therefore detail one or two methods for the measurement 
the density of a liquid methods for the measurement of gaseous 
vapoui densities aie peihaps moie appropriately discussed in a 
eatise on heat. 

To determine the density of a substance we have to measure 

ther (a) the mass of a known volume or (b) the volume of a known 

ass Fluids must be weighed in some soit of containing vessel, 

d if we know the volume of the containing vessel, the measuiement 

the mass of fluid which fills it at a given temperature at once 

/es us the density of the fluid. The most convenient way of 

librating a containing vessel is by finding the weight o$ some 

uid of known density which fills the vessel at a known temperature 

us assumes, of course, that the density of the standard liquid 

lally water or meicury has been determined by some inde- 

ident method, and much labonous research has been done on the 

asuiement of the vanation of density with tempeiature of these 

} fluids. 

Thus, Halstrom * measured carefully the linear expansion of 
lass rod, the relation between length and temperature being ex- 
ssed by the formula 

L = L (i + at + bt z ). 

A piece of this rod of volume V was taken and weighed in water 
iifferent temperatures, the loss in weight on water at f being 
;n by 

W = W (i + lt + mt* -f nt*). 

*Ann. Clum. Phys , 28, p. 56. 



8 THE MECHANICAL PROPERTIES OF FLUIDS 

The quantities a, Z>, /, m, n are determined by experiment, and it is 
clear that the volume at t of the portion used is 

V = V (i + at -|- fo 2 ) 3 

Now since the loss of weight in water is, by Archimedes' principle, 
equal to the weight of water displaced, and since the volume of this 
displaced water is equal to the volume of the glass sinker, we have for 
the density of water at t 

p =.__ = ,_. ^ + at + 
or 

where a, /?, y are known in terms of /, m, n, a, b. The figures obtained 
in an actual experiment are quoted below: 

a 0-000001690 



b = 0-000000105 
/ = 0-000058815 
m = 0-0000062168 
n = 0-00000001443 



a 0-000052939 
whence -J/8 = 00000065322 
\y = o 00000001445 



It may be noted in passing that the temperature of maximum density 
of water may be determined from these results with consideiable 

accuracy. For when p is stationary we have J- o, and hence 
3y2 2 -|- 2/fe + a = O 

This equation is a quadiatic in t, one of the roots is outside the 
lange ot the expeiimental figures, the other is 4-108 C 

From expeiiments of which this may be quoted as a type, 
Table I (p. 9) has been drawn up 

It will be seen that, if water be used as the calibrating liquid, 
the determination of the density of a liquid becomes identical with 
the opeiation of determining its specific gravity that is, we find by 
experiment the ratio of the weight of a certain volume of liquid to 
that of an equal volume of water at the same temperature. The 
magnitude of this ratio is conveniently denoted by the symbol s* t , 
and may be reduced to density mass per cubic centimetre 
by means of Table I on p. 9. It is more usual to compare the 



LIQUIDS AND GASES 9 

TABLE I 
DENSITY OF WATER IN GM /c.c AT VARIOUS CENTIGRADE TEMPERATURES 

Temperature. Density. Temperature. Density. 



Degrees Degrees. 

O 0-99987 42 0-99147 

2 o 99997 44 o 99066 

4 i ooooo 46 o 98982 

6 o 99997 48 0-98896 

8 o 99988 50 o 98807 

10 099973 52 0-98715 

12 099953 54 098621 

14 099927 56 098525 

1 6 o 99897 58 0-98425 

18 0-99862 60 0-98324 

20 o 99823 62 o 98220 

22 099780 64 098113 

24 o 99732 66 o 98005 

26 o 99681 68 o 97894 

28 o 99626 70 o 97781 

30 099567 75 097489 

32 099505 80 097183 

34 0-99440 85 o 96865 

36 099371 90 096534 

38 0-99299 95 096192 

40 o 99224 100 o 95838 



eight of the liquid with that of an equal volume of water at 4 C., 

id this value, s* 4 , may also be deduced from the experimental figures 

y means of the table. It should be noted that the specific gravities 

and s* 4 are often doubtful in meaning, for they refer sometimes 

the ratio of true weights, sometimes to the ratio of apparent 

eights, no correction being made for displaced air. In experimental 

oik of high accuracy, it is well both to make this correction, and 

indicate that it has been made. 

For ordinary work, the common specific-gravity bottle may be 
jed, but for precision measurements some form of pyknometer is 
icessary The pyknometer is usually a U-tube of small cubic content , 
5 ends terminating in capillary tubes. Three forms are outlined in 
r.2 . (i) is the original Sprengel type, (ii), a modification introduced 
1 Perkin, possesses several advantages. The instrument, filled by 
iction, is placed in an inclined position in a thermostat, and excess 

(D312) 



io THE MECHANICAL PROPERTIES OF FLUIDS 

liquid is withdrawn from a by means of filter paper, until the level 
in the othei limb falls to b. The tube is now removed and restored 
to the veitical position, when the liquid recedes from a. If now 
expansion takes place before weighing, the bulb above b acts as a 
safety space, and all danger of loss by overflow is obviated. The 
form shown in (id) was introduced by Stanford, and reduces to a 
minimum those paits of the vessel not contained in the thermostat, 
whilst its shape does away with the necessity for suspending wires, 
as the bottle can be weighed standing upon the balance pan. 

In technological practice much specific-gravity work is carried 
out by means ol vaiiable-immersion hydrometers Hydrometer piac- 
tice and methods can hardly be said to be in a satisfactory state. 



il/c 




FIB 



Not only has one to plough through a jungle of aibitiaiy scales, but 
the reduction of these scale leadings to specific giavities, defined 
accurately as we have defined them alteady, is no easy mattei All 
hydrometers should cany, maiked permanently on their surfaces, 
some indication of the pimciple ol their graduation, so that then- 
readings may be reduced to s|, s*, or some othei definitely known 
standaid For rough woik, of course, the aibituuy giaduations 
suffice, and a workman soon learns to associate a reading of, say, 
* degrees Twaddell with some definite pioperty ol the liquid with 
which he is working. But with moic delicate hydromcteis an 
absence of exact reference to some definite standard is distinctly 
unsatisfactory. 

Thus the common hydrometer is graduated so that a reading of 
1035 corresponds to a specific gravity of 1-035 the standard of 
reference being very often doubtful and the Twaddell hydrometer 
is so constructed that the specific gravity s is given by 

s = i-ooo ~|~ 0-005 r, 
where T is the reading in Twaddell degrees. Clearly on the common 



LIQUIDS AND GASES 11 

lydrometer the " water-point " is 1000, on the Twaddell hydrometer 
,ero, and, unless the hydrometer carries some reference to the tem- 
>erature at which its water-point is determined, it becomes impossible 
atisfactorily to compare the performances of two different hydro- 
neters. 

Confusion is woise confounded when we introduce Baume" 
eadmgs. In the original Baume* hydrometer water gave the zero 
3omt, and a 15 per cent solution of sodium chloride gave the 15 
nark. This for liquids heavier than water. For liquids lighter 
han water a 15 per cent solution of salt marked the 
zero point, the water-point being at 10. Now it is usual ^ 
to mark the point to which the hydrometei sinks in sul- 
phunc acid of density 1-842 as 66 B We can easily 
work out a foimula of i eduction giving the specific 
gravity in terms of these fixed points. Thus, let __o" 

V be the total volume of hydiometer up to o, 

p l5 the density of water, 

v, the volume of hydrometer between o and w, - n 

p, the density of liquid m which hydiometei floats 

at mark n, and 
a, the cioss-sectional aiea of neck 

By Aichimedes' pimciple the mass of the hydiometer 
is given by the two expiessions 

VPJL and (V v)p 

Hence Fig 3 

VftL = (V - v) P = (V - an} P - 

and if 5 is the specific gravity of the liquid, 



Pi 
Hence we have 

ns __ V __ , 



s i a 

If we put 5 = 1-842 when n = 66, we find that k = 144-3, and 
theiefore for any other liquid giving a reading x, 

s = I44 ' 3 
144-3 - *' 

It is obvious that we do not know where we are unless the densities 



u 'TUH MKCHANICAL PROPERTIES OF FLUIDS 

used in eahbiatum aie sharply defined, and the clouds are not 
appiet uhly lightened by the piactices of Dutch and Ameiican 
Iiydiometei inakeis, who take the constant k as 144 and 145 respec- 
tivcl} pitsumably ioi the convenience of dealing in integial numbers. 
IMohr's balance is exceedingly convenient for use in those techno- 
logical laboratories m winch a laige number of determinations of 
density are made. As fig. 4 shows, it is a balance of special form, 
one arm being divided into ten equal parts and canying, suspended 
horn A hook by a silk fibre, a glass thermometer which also serves as 
a sinker The weights piovided are m the form of riders, the two 





FIB i 

I digest being equal, the other two being o i and o-oi 
respectively ol the largest weights The hook is so E^Ir^ 
uljiisted that a body suspended from it is in position EI~^:~z 
it the tenth maik ~ ~7 ~ 

The othci arm of the balance canies a counter- - - _- 
ioit,e \Mth a pointed end, which point, when the H~~ 
ulanee is in ec[uihbinnn, is exactly opposite to a 
iducial nuuk on the fixed suppoit Suppose now 
hat the balance is levelled and is in equilibrium, the sinker 
)eing in an Place the tmiket in water at 15. // will be found that 
we o/ the ]i<'(r<<i("<f wciqhh, \mf>em1ed from tJic hook, will restore cqmh- 
ntuin A little thought, based on a knowledge oi the law ol moments, 
ihould convince the stmlcnt that the specific gravity of a liquid, 
sou eel to ooor, tan he lead oil at once liom the positions of the 
'aumi.s lideis when (he balance is in equilibrium, the sinker being 
mmcised in the luimcl. Thus the specific gravity of the liquid, 
he ndeis being disposed as m fig 4, is 1-374 rci erred to water at 
5", and may be expressed as a density by means ol Table I, p 9. 

It happens on occaiuon that a detci mination o specific gravity 
s called Tot, and that no suitable instruments are at hand. It is 
vorth while knowing that an accurate icsult may be obtained with 
10 moie elaborate apparatus than a wooden rod, which need not be 
inifoim, but should be uniformly graduated, and a few counter- 
Kusea of unknown weight, made 01 material unacted on by the liquid 
mder test. Suppose a knitting needle, or a piece of a small triangulai 



LIQUIDS AND GASES 13 

ile, to be fixed in the rod to form a fulcrum (fig. 5) Suspend the 
veights W and Wx from the rod by loops of thread, and move Wj. 




Fig.s 



mtil the lod is level. If w be the weight of the rod acting at a 
hstance y from the fulcrum, we have 

WY + toy = 



f now, without disturbing W, we allow W x to hang in a beaker of 
rater at a temperature f, we have, if the point of balance be now 
hifted to #2, and W 2 be the apparent weight of W 1$ 

WY + wy = W 2 # 2 
These equations give 

VV i VV 2 ^2 ~~" "~ *^i 

f a beaker of the liquid under test at a temperature t be substituted 
or the water and the balance point be now at x 3 , we have by similar 
easoning 



or w.-w, 



lence 

* = 



W 



nd the specific gravity, which may as before be reduced to density 
y means of Table I, p. 9, is given accurately in terms of lengths 
leasured along the rod 

The variation of density with temperature has been the subject 
f many investigations; most of the equations proposed to represent 
lis variation (under certain specified conditions) are applicable, 
r ith great exactness, over limited ranges only. A formula has, how- 
ler, been put forward recently, which gives the relation between 
rthobaric density and temperature with very considerable accuracy, 
ver the whole range of existence of the liquid phase. It is developed 



xus: 



14 THE MECHANICAL PROPERTIES OF 

We shall see later that, for unassociated liquids, the relation 
between surface tension and reduced temperature (-m) is given by 



where n varies slightly from liquid to liquid, but does not deviate 
greatly from the value 1-2. Further, for any one such liquid the 
relation between surface tension and the densities of the liquid and 

vapour phases is 
* F V 



where p does not deviate greatly from the value 4. Eliminating y 
between these two equations, we find 



where B stands for the p tn root of y /C. If we assume that, at the 
absolute zero, the density of the super-cooled liquid is about four 
times the critical density, and that of the vapour is negligible, we 

have , , 

P* P = 4Pc(i -m) 03 , 

assuming constant values for n and for p. But it is well known 
that, if we take the mean of the orthobaric densities of liquid and 
vapour at any temperatuie, and plot these mean values against tem- 
perature, the result is, to a high degree of approximation, a straight 
line inclined at an obtuse angle to the temperature axis. That is, 

Pe H- Pv = P Q^2- 

The condition that, at the critical point (yn = i) we have p e = p v = 
p c , combined with the condition previously mentioned that for 
wt = Q we have p e = 4p c and p v = o, gives us 



Taking these equations for (p e -f- p v ) and (p B p v ), eliminating 
p v and dropping the subscript /, we find 

p = 2p e [(i - m) 03 + (i 



which is a reduced equation between density and temperature 
applicable to all unassociated substances. The equation may be 
tested by writing it in the form 



(i 
po 
or Y = X s , 



LIQUIDS AND GASES 15 

vhere s may or may not be equal to 0-3 A logarithmic plot of Y 
gainst X shows, in general, very good straight lines, whose slope 
leviates very little from the value 0-3. But the lines do not pass 
hrough the origin. 
It follows then that 

Y=GX" 

p = 2 Pc [G(i - vrif* + (i - o sw)], 

/here G is a constant, whose value vanes from liquid to liquid. 
[lie variation is not great, and a mean value of G is about 0-91 
This general form gives very satisfactory results, but, if very close 
grcement with the experimental figures is necessaiy, the values of 
j and of ^ special to the particular liquid must be chosen 

If we put wi = o in this equation, and take the value of G as 
91, we see that the (absolute) zero density of the supercooled 
quid is about 3 82/3 c , probably a better approximation than the 
sual value 4p c 

Again, the equation enables us to compute a reasonably good 
alue for the critical density if the density at any one reduced 
smperature is known This, of course, involves a knowledge of 
tie critical temperature If the critical temperature is not known 
/e may make use of Guldberg's rule, that for unassociated liquids 
lie boiling-point under noimal pressure is very approximately 
wo-thirds of the cntical temperature Putting, therefore, m = f , 
fc have 

Pfc = 2/40 91(1- !) 03 -h(i-o 5 x*)], 



geneial relation between the density at the normal boilmg-pomt 
nd the critical density 

By eliminating p e instead of p v a similar formula may be derived 
3 show the march with temperature of the density of the saturated 
apour. 

Compressibility 

We have seen that a perfect liquid is, as a matter of definition, 
icompressible that is, its bulk modulus is infinite. Liquids m 
ature are under ordinary circumstances very slightly compressible,* 
nd the determination of their compressibilities is, in effect, a de- 

* Constantmesco, p. 222,, 



if. THE MECHANICAL PROPERTIES OF FLUIDS 

teimination of then bulk moduli which, at any given pressure and 
tempeiature, is defined by the equation 



8w /dp\ 

, = v(* \ 

v vWr 



where 8/> is the additional stiess (i e pressure per unit aiea) causing 
a da icase in volume Bv of a substance whose initial volume is v t 

and (^ j stands foi the uite of decrease oi piessure with volume 
\fa>A 



under isolheimal conditions (T constant). The compressibility, at 
any given pressure and tempeiatme, may be defined as the recipiocal 
of the bulk modulus, i e. the latio 



V \9/> 

Another definition of compicssibility is sometimes used, namely? 

/"i \ 

( v \ ; in this case v is the volume of unit mass of the liquid. 

\9/>/T 

We shall heie confine out selves to a discussion of the com- 
pressibilities oi liquids those of gases and vapouis will be tieatcd 
later. It is clear that a complete study of the compressibility ot a 
liquid lesolvcs itsell into the drawing of a p, v, T suiface for the 
.substance in question, so that the volume of i gm of the substance 
is known at any piessuic and tcmpciatuic The impoitancc oi this 
knowledge can luudly be ovei estimated When we have diawn the 
/>, v, T surface for any liquid we aie in a position completely to de- 
teunme its most impoitant thei modynamic piopertics In this 
connection the iccent woik of Bridgman f is pic-eminent in value, 
and we shall heic give a discussion, as brief as may be, of his woik, 
leaving the icadei to study details oi the oldei cxpcuments, il he be 
so minded, in other books The punciplcs involved are simple, 
but it must be icmembeied that the experimental difficulties, when 
the pleasures aic pushed up to the older ol 20,000 Kgm per 
square centimetre, aie very gicat. 

The substance under test is placed in a stiong chiome-vanadiurn 
steel cylinder, and the pressure is produced by the advance ^of a 
piston of known cross-section, the amount of advance of the piston 

* This notation means that /> is icRatdcd as n function of v and T, and T is kent 
constant m finding the clenvative '^ 

f Proc, Amcr, Acad Set,, 48, 309 (1912). 



LIQUIDS AND GASES 17 

giving the change in volume. It would make the story too long were 
we to discuss in detail the method of packing of the piston to ensure 
freedom from leakage, and the manner of correction for the change 
in volume of the cylinder, but it may be of interest to iote that the 
pressure was measured by the change of electrical resistance of a 
coil of mangamn. The resistance of the coil was about 100 ohms, 
and it was constructed of wire, seasoned under pressure, of resistance 
30 ohms per metre. For high-pressure measurements this forms a 
very simple and convenient form of gauge. It must, of couise, be 
calibrated, and Bndgman performed this by making, once for all, 
a series of measuiements of the change of electrical resistance oi the 
wiie with piessure, measuring the piessure by means of a specially 
constructed absolute gauge It was found that the change of resistance 
with pressure was so accurately linear up to pressures of 12,000 
Kgm. per square centimetie that the readings could be extra- 
polated with confidence up to 20,000 Kgm. The changes of 
icsistance were measured on a specially constructed Caiey Foster 
bridge 

The whole apparatus was immersed in a theimostat, and series 
of pressure-volume readings were taken at diffeient temperatures 
From these readings Table II was drawn up, exhibiting the be- 
haviour of water up to 12,500 Kgm per square centimetre pressure, 
and 80 C. 

A caieful study of this table will show that we can extiact from 
it data which give veiy complete details of the thermodynamic pro- 
peities of water within the range considered. The reader is stiongly 
recommended to work out a few of these results, by so doing he will 
learn in an hour or two moie of the principles of thermodynamics 
and of the properties of water than he would gam from a week's 
reading of books where everything is painstakingly explained for 
him The hints given below should suffice to set him going, and 
should he have access to Biidgman's papers, it would be well to com- 
pare his results with the curves given by Bndgman * 

/) \ /) \ 

( i ) Calculate the compressibility ( ) 01 - ( ) , and plot a curve 

\op/i v \9p/T 

between this quantity and p at any one temperature Repeat for 
vanous tempei atures f 

* Loc cit 

| The vauous thcimodynarmcal iclations given in (i) to (10) will be found in 
treatises on theunodynamics, e R Dictionary of Applied Physics, article "Thermo- 
dynamics ", Remember that T here stands foi absolute temperatuie. 



i8 THE MECHANICAL PROPERTIES OF FLUIDS 

(Ci \ / l \ 

- ) 01 - ( = ) , and plot it as 
Ol/p V \0i/p 

a function of the pressuie at various temperatures 

(3) The mechanical work done by the external pressure in com- 
pressing the liquid at constant temperature is given by 



between given limits, and is obtained by mechanical integialion 
(planimeter, square counting, or the like) of the cuives showing the 
relation between p and v at constant temperature 

(4) The total heat given out, Q, during an isothermal com 
pression is similarly derived by mechanical integration fiom 

3T. 

using the results of (2) to plot the desired curve. 

(5) Knowing the mechanical work and the heat liberated in com- 
pression, we can find the drffeience between these, thus giving the 
change of internal eneigy along an isothermal, and can plot this 
against the pressure. 

(6) The pressure coefficient is given by 



idv\ 

= -W 



dT 



It can thus be determined with the aid of the results of (i) and (2), 
and can be plotted against the pi essure 

(7) The specific heat at constant pi essure may be obtained by 
mechanical integration from the equation 



This, of course, involves working out the second derivative from the 

(o \ 
) in the same manner as the fiist deiivative is 
Oi/P 

worked out fiom the original tables. Values of the specific heat as 
a function of the tempeiature at atmospheric pressure may be taken, 
as Bridgman took them, from the steam tables of Marks and Davis, 



1ATURE 



Pressure, 












Kgm 


60 


65 


70 


75. 


So". 


per cm a 












O 


IOI68 


1-0195 


i 0224 


10255 


I -0287 


500 


99 6 5 


9992 


I 'OO2O 


I 0049 


1-0075 


I,OOO 


9791 


9816 


9842 


9869 


9896 


1,500 


9632 


9657 


9682 


9707 


9732 


2,OOO 


9489 


95*3 


'9537 


9561 


9585 


2,500 


9363 


9386 


9409 


9433 


9457 


3,000 


9247 


9269 


9292 


'93 H 


9337 


3,5 


9138 


9160 


9182 


9204 


9226 


4,000 ( 


937 


9058 


9080 


9101 


9123 


4,500 


8945 


8965 


8986 


9008 


9028 


5,000 


i 8858 


8879 


8899 


8920 


8940 


5>5oo 


8777 


8798 


8818 


8838 


8858 


6,000 


8702 


8722 


8742 


8762 


8781 


6,500 


8631 


8650 


8670 


8689 


-8709 


7,000 


8564 


8583 


8602 


8621 


8640 


7,500 


8499 


8519 


8538 


8557 


8575 


8,000 


8438 


8457 


'8477 


8495 


8513 


8,500 


8381 


8400 


8419 


8437 


8455 


9,000 


8327 


8346 


8364 


8383 


8401 


9,500 


8275 


8294 


8313 


833i 


8349 


10,000 


8226 


8245 


8264 i 


8282 


8300 


10,500 


8179 


8198 


8216 


8235 


8252 


11,000 


8i33 


8152 


8170 


8188 


8206 


11,500 


8088 


8107 


8125 


8143 


8160 


12,000 


8043 


8062 


8080 


8098 


8115 


12,500 


7999 


8017 


8036 


8054 


8071 


[Facing p 18 



LIQUIDS AND GASES tg 

(8) Knowing C^, we can determine C u from the equation 

'80V 
A9T/, 




(9) The rise in temperature accompanying an adiabatic change of 
ressure of i Kgm per square centimetre may be deduced with the 
^lp of the equation 

= Z ( dv ' 
Cp\i 

I the quantities on the light-hand side of the equation being 
lown from the icsults of previous sections. <f> lefers to the 
itropy 

Finally (10): The difference between the adiabatic and isothermal 
impressibilities is given by 

f dv\ {dv\ __ T 

d may theiefore be calculated 

Bndgman has added to the value of this work by making similar 
idies oi twelve organic liquids. For details the student should 
nsult his original papei s * 

Intel estmg iclations exist between the compressibility of a liquid 
d ceitam other of its physical constants, these we shall discuss 
er undei other heads. Meantime we pass on to a consideiation of. 

Surface Tension 

We may take it as an expeimiental fact easily deduced fiom the 
)st ordinary obseivations that the surface of a liquid is in a state 
tension and is the seat of energy. The spherical shape of small 
ndrops or of small globules of mercuiy shows that the liquid 
face tends to become as small as possible in the circumstances, 

a sphere is that surface which for a given content has the smallest 
Derficial area. Again, the fact that the surface is the seat of eneigy 
illustrated by a simple experiment suggested by Cleik Maxwell, 
agme a large jar containing a mixtuie of oil and water well shaken 

* Proc. Amer. Acad, Set., 49, 3 (1913)- 



ao THE MECHANICAL PROPERTIES OF FLUIDS 

up, so that the oil is dispersed through the water in small globules. 
If the system be left for some time it will be found that the oil has 
" settled out ", and it is clear that the settlmg-out process has involved 
the motion of considerable masses of matter that is, a definite 
amount of work has been done. The only difference between the 
two states of the system is that before the settling out the surface-aiea 
of the oil-watei interface was considerably greater than the area of 
the interface in the final state. We conclude that the surface pos- 
sesses energy, and it will be seen shortly that an important i elation 
exists between surface energy and surface tension 

We assume, then, that across any line of length ds drawn in the 
surface of a liquid there is exerted a tension yds, the dnection of this 
tension being normal to the element ds and in the tangent plane to 
the surface. The quantity y is called the surface tension of the 
liquid, its dimensions aie clearly those of force length, and a surface 
tension is reckoned, m C.G.S. units, in dynes per centimetre, 01 
in grammes per second per second. This tension differs fiom 
the tension in a sheet of stretched india-rubber, with which it is 
commonly compared, in that it is, within wide limits, independent 
of the area of the surface. It is constant at constant tempera- 
ture, but varies with the temperature, and the calculation of 
its temperature coefficient is a matter of great theoretical im- 
portance. Textbook writers usually give the relation between 
surface tension and temperature in the form 



a result of little value, holding good over a very limited range The 
small value attaching to the formula can be shown at once, if we 
remember that at the critical temperature the suiface tension vanishes, 

so that we must have t c = -. But values of t c calculated in this 

a 

way are wildly wrong, showing that the range of the foimula is 
exceedingly restricted. It can be shown that, for liquids which 
do not show molecular complexity, the relation between suiface 
tension and tempeiature is given by 

y = y (i - bt)\ 

where n varies from liquid to liquid, but in general has a value not 
differing very greatly from 1-2. This equation holds good from 
freezing point to critical temperature, and its accuracy may be tested 



LIQUIDS AND GASES 21 

jy comparing the value of t c obtained fiom direct experiment with 

hat obtained from the relation * c = 7- The test is shown in 
Table III below. b 



Substance 

Ether 

Benzene 

Caibon tetrachlonde 

Methyl formate 

Propyl formate 

Propyl acetate 



TABLE III 
b 



I 


248 


0005155 


I 


218 


o 003472 


I 


206 


0-003553 


I 


210 


0-004695 


I 


2 3 I 


o 003774 


I 


294 


o 003623 



t c Calcu- t c Ob- Differ- 
lated served ence 



+ 02 
-OS 

-16 
i o 

+ 01 

-02 



Cent 
Degrees. 
194 


Cent 
Degrees 

I 93 S 


288 


2885 


281 5 


2831 


213 


214-0 


265 


264 9 


276 


276 2 



/e have previously leferred to the relation between suiface tension 
id surface energy The assumption that the suiface tension 
F a surface is equal to its suiface , *,, >,., 

lergy (per unit aiea) is anothei 
>mmon error. The "pi oof" given 
Biially follows these lines Imagine 
soap film stt etched over the veitical 
lie frame shown in fig 6, the bar CD 
nng movable If the bar be pulled 
Dwnwards through a distance 8x, the c 
oik done against the suiface forces 
2yLSx (remember that the film has 
ro sui laces) But if A be the eneigy 
:i unit area, since the increase of 
rface is 2lSx, the inciease of super- 
ial eneigy is A.2/Sx Hence, equating these quantities, we have 

y = A. 

it this argument overlooks the important fact that surface tension 
mmishes with increasing temperature. Hence it follows from 
ermodynamic principles that, in oider to stretch the film isother- 
illy, heat must flow into the film to keep its temperature constant, 



Fig 6 



22 



THE MECHANICAL PROPERTIES OF FLUIDS 



and this heat goes to increase the surface energy. A simple thermo- 
dynamic argument shows that the relation between y and A is given by 



7 ~ ^ A 8T' 

where T stands for absolute temperature, and only if the tempeiature 
coefficient of surface tension were zero would the simpler equation 
hold. 

Since we know the relation between surface tension and tem- 
peiature for unassociated substances we can easily work out, by 

substituting for y and 

9y . , . . 

-~ in the equation just 

given, the relation be- 
tween surface energy 
and temperatuie. This 
relation is shown in 
fig 7, the dotted curve 
showing the variation 
of surface tension, the 
full curve the vaiiation 
of surface eneigy with 
temperature. The two 
""--,. curves intersect each 

~~~~ TEMP other and the axis of 

Fjg ? temperature at the 

critical temperature, 

showing that at that point both surface tension and surface energy 
vanish. But for lower temperatures the two quantities are in 
general very different in numerical magnitude, surface energy 
increasing much faster than surface tension with falling tem- 
perature. This important fact should carefully be borne m mind 
Many relations, empirical and otherwise, have been suggested 
connecting surface tension with other physical constants Thus, 
Macleod * has recently found that for any one liquid at different 
temperatures 

y = 



* Trans Faraday Soc , 1923. The present writer has also shown (Trans 
Faiaday Soc,, July, 1923) that the constant C may be expressed m the form 
C = AT C / M$pc*P, where M is the molecular weight, p c the critical density, and 
A a constant independent of the nature of the liquid 



LIQUIDS AND GASES 23 

wheie C is a constant independent of the temperature, p t the density 
of the liquid, and p v that of the saturated vapour of the liquid. 

We should naturally expect surface tension and compressibility 
(f?) to stand in intimate relation, and expeiiment shows that liquids 
of high compiessibility have low surface tensions and conversely. 
Richards and Matthews * have examined the quantitative relation 
between these two constants, and find that, for a large number of 

unassociated substances, the product y^ is a constant quantity. 

A most important equation connecting surface tension, density, 
and temperature, is that proposed by E6tvos,f 

- T - 8), 

>vhere M is the molecular weight of the liquid, p its density, and 
5 and K are constants for unassociated liquids, 8 being about 6 
md K 2 12 The equation shows that a knowledge of the tempera- 
ui e variation of y enables us to calculate the molecular weight of 
he liquid under examination, and hence to determine whether its 
nolecules aie or are not associated 

In iccent yeais this test of association has been slightly altered. 

[nstead of examining the vanation of y( ) with temperature, the 

, , /M\ 1 , . , j , i 

variation of A( has been studied, wheie, as we have seen, 

\P/ 

= 7- IgFp 

3ennett and Mitchell J have shown that for unassociated liquids 
his quantity, which we may call the total molecular surface energy, 
s constant over a fanly wide lange of temperatuie, and have used 
his constancy as a test of non-association 

We now tuin to the discussion of a problem of fundamental 
mportance that of the i elation between the pressure-excess (posi- 
ive or negative) on one side of a curved suiface and the tension in 
he suiface It is fanly cleai that the piessure just inside a curved 
iurface such as that of a spherical bubble is greater than the pressure 
ust outside the suiface, and the manner in which pressure- excess 
s connected with surface tension may be calculated as follows. 

* Zeit Phys Chem , 61, 49 (1908) 

| See Nernst, Theoretical Chemutry, p. 270 (1904). 

[ Zeit Phys Chem , 84, 475 (1913). 



24 THE MECHANICAL PROPERTIES OF FLUIDS 

Imagine a cylm ducal suiiace whose axis is peipendicular to the 
plane of the paper, pait of the trace of the surface by the plane of 
the paper being the curve AB (fig. 8). Consider the equilibiium 
of a poition of this cylindrical surface of unit length perpendicular 




Fig. 8 

to the plane of the paper, and of length ds in the plane ot the paper. 
If II and II -f p be the pressures at CD on the two sides of the 
cylinder, we have, resolving normally, 

n 

2y sin- -)- lids -- (II -j- p)ds, 
2 

when 6 is the radian measure of the angle indicated m fig. 8, 
or, since 9 is small, .,0 _ p^ s 

ds 
But 9 ~ where R is the radius of curvature at C, and theiefoie 

ft - Z 
P ~ R* 

If the surface is one of double curvature, the effects aie additive and 
we have 



where R x and R 2 are the principal radii of curvature at the point in 
question. Thus for a spherical drop, or a spherical air-bubble in a 
liquid, we have 2 y 

P = P 



LIQUIDS AND GASES 21 

where R is the radius of the drop or bubble. For a spherical soap- 
bubble, which has two surfaces, we should have 



p = 



R 



The use of the pressure-excess equation, combined with a know- 
ledge of the fact that a liquid meets a solid at a definite angle called 
the contact-angle, will suffice to solve many important surface-tension 
problems. Thus the rise or fall of liquids m capillary tubes is 
readily explained. Water, for example, meets glass at a zero contact 





Fig 9 

angle, hence the suiface of water in a capillary tube must be sharply 
curved, and the nanower the tube the sharper must be the curvature 
m order that the liquid may meet the glass at the proper angle The 
state of affairs shown in fig 9 (i) is impossible, for the pressuie at 
A being atmosphenc, the pressuie at B must be less than atmosphenc 

2v 

by -, where R is the radius of curvature of the meniscus at B But 

the pressure at C in a liquid at rest must be equal to that at B, and 
the pressure at C is clearly atmospheiic. Hence the liquid must rise 
in the tube until the additional pressure due to the head h just brings 
the pressure at B up to atmospheric value. We must have therefore 

- *. 



26 THE MECHANICAL PROPERTIES OF FLUIDS 

If the tube be very narrow and the ciitenon of nanowness that is 

r - shall be small compared with unity the meniscus will be a segment 

of a sphere, and the contact angle being zeio we may put R = r, 
the radius of the tube, giving the well-known equation 

y = $gprh 
If -, though small, be not negligible compared with unity the meniscus 

will be flattened; a very close approximation to the truth may be 
obtained by treating the meridional curve as the outline of a semi- 
ellipse. Suppose the semi-axes of the ellipse to be r and b (fig 9 n). 
If we take the contact angle as zero, theie will be an upwaid pull of 
2-nry on the liquid m the tube all round the line of contact of the 
liquid with the glass. Equating this to the weight of liquid raised 
(including the weight of that in the meniscus) we have 

ZTrry Trr^hpg -f- lvr z bpg 
or 20? = rh -f- ^rb, 

if for brevity we wiite a 2 for . But if R be the radius of curvatuie 

SP 
at O, we have accurately 

Piessuie-excess = gph = , 
or za 2 = Rh. 

Now R, the radius of curvature at the end of the semi-axis minor of 

r a 
an ellipse, is equal to . Hence 



2,0 

V fl 

and therefore 2a z r h -f- lr. , 

2<2 2 

or i2 4 6rhaP r^h = o. 



Solving this as a quadratic in aP and expanding the surd, we obtain 

O 



T 

In all practical cases - is small compared with unity, and the above 



LIQUIDS AND GASES 27 

juation gives values for a z (and therefore for y) in close numerical 
agreement with those obtained from the equation 

3 2 = ^/! _)_ ^ _ 0-1288^ + 

\ il (v 

btained by the late Lord Rayleigh * as the result of a lather complex 
id difficult analysis. 

The problem of the measurement of interfacial tensions has 
jcently assumed great technological importance, mainly on account 
" the rapid development of colloid chemistry and physics. Tanning, 
yeing, dairy chemistry, the chemistry of paints, oils, and varnishes, 
" gums and of gelatine are all concerned deeply with the properties 
* colloidal systems in which one phase is dispersed in very small 
irticles through the substance of another phase. There is conse- 
aently a relatively great extent of surface developed between the 
ro phases, and the interfacial tension at the surface of separation of 
ie phases may play an important, not to say decisive, part in deter- 
iining the behaviour of the system 

This tension may be measured by a modification of the capillary 
ibe experiment desciibed above For example, the tension at a 
mzene-water interface has been measured by surrounding the 
ipillary with a wider tube, and filling with benzene the space 
reviously occupied by air 

Exact determinations can conveniently be made by the di op- 
eight method, wherein a drop of liquid is formed at the end of 
id detached slowly fiom a veitical thick- walled capillary tube 
timeised in the second (and lighter) liquid The method can, of 
)urse, be used to deteimme liquid-au tensions So many erroneous 
atements have been made concerning the practice and theory of 
us method that it is woith while consideimg it in some detail 
or example, a common practice in physico-chemical works is to 
juate the weight of the detached diop to zirry, a procedure which, 
Lit for the fact that the di op-weight method is often used as a com- 
uative one, would give results about 100 per cent in error. Those, 
jam, who are alive to the error of writing 

mg = zirry 

3t infrequently tell us that the constant ZTT must be replaced by 

lother constant of value 3 8, for no very apparent or adequate 

sason. Let us then investigate the problem as exactly as may be 

* Proc. Roy. Soc. 92 (A), 184 (1915). 



2 8 THE MECHANICAL PROPERTIES OF FLUIDS 

m an elementary manner, and see if some justification exists for this 
procedure. Suppose for the moment that the drop is formed in air. 
If we assume that the diop is cylindrical at the level AB (fig. 10), then, 
II being the atmospheric pressure, the pressure at any point in the 

plane AB is II -f -. Consequently, resolving vertically for the 
forces acting on the portion of the drop below AB, we have 



mg -f- 




?" 



2-rrry 



II. 



.77T 



leading to 



mg 



Fig 10 



exactly half the value of the 
weight of a drop as given by 
most of the textbooks. 

But the detachment of a drop 
is essentially a dynamical pheno- 
menon, and no statical treatment 
B can be complete. We can, how- 
ever, obtain some assistance from 
the theory of dimensions. As- 
sume that the mass of a detached 
drop depends on the surface 
tension and the density of the 
liquid, the radius of the tube, 
and the acceleration due to 
gravity. We may thus write 



m = 



Dimensionally 

[M] = [MT- 2 ]*[LT~ 2 f [ 

leading to 

x -j- ss = i, x + y = o, y 3* + > o. 

Solving for w, y, and # in terms of x we find 



T , yr 
m = K-- 

8 



or 



m ' 



LIQUIDS AND GASES 29 

inhere F is some arbitrary function of the variable -. The late 

gpr* 

jord Rayleigh determined the weight of drops of water let fall slowly 
rom tubes of various external diameters. Knowing the surface 
snsion of water, he was enabled to tabulate the variation of the 

unction F with that of the independent variable -; for, as we see, 
hie function F is given by 

F = ^ 
yr' 

n this way the following table was drawn up. 

TABLE IV 



\ yr 



2-58 4-13 

1-16 397 

o 708 3 80 

o-44i 3 73 

0-277 3-78 

o 220 3 90 

o 169 4 06 



* 

will be seen that foi a considerable variation of the variable ~- 

gpr* 

-and this means a considerable variation of r the function F does not 
actuate seriously, and foi most purposes it is permissible to assume 
lat F is constant and equal to 3-8. Hence the reason for the 

luation o 

1 mg = 3 8ry. 

The argument for mterfacial tensions follows identical lines, and 
ie reader should have no difficulty in working it out for himself, 
membering that the drop of density p, say, is now supposed to be 
sndent in a lighter liquid of density p v 

If we assume that the liquids with which we are dealing obey the 
3wer law for the variation of surface tension with temperature, we 
ive ! 



sir < . i i 4 j* i * ' .a,% 

" |T) I *"** ^ " 

? * I \vV ^~~~*^"' 



30 THE MECHANICAL PROPERTIES OF FLUIDS 

where, for convenience, we express temperatures in the reduced 
form. The total surface energy A is given by 



= y m~ 9 

am 
and, with this form for y, is 

A = y (i <m} n -' L {i + (n i)-m}, 

and we see that, contrary to some statements, there is no indication 
of a maximum value for A, the march of A with temperature following 
the curve shown in fig 7. 

We have seen that a reduced equation may be developed between 
orthobaric density and temperature which, in its simplest form, 
may be written 

p = 2p c [(i w) 03 -j- i 



It follows then that free molecular surface energy (e) defined as 
y(M/p)t and total molecular surface energy (E) defined as A(M//>)i, 
have their vanation with temperature at once determined on sub- 
stituting m these expressions the appropriate expressions for y, A 
and p 

The deduction of the equations showing how e and E vary with 
the temperature is left to the reader, but it may be noted that e is 
not a linear function of the temperature, nor is E independent of 
the temperature, although the variation at fairly low temperatures 
is very small, and the assumption of constancy over ordinal y ranges 
of temperature need lead to no serious error. Neveitheless it is 
worthy of note that, considering the whole range vn = o to wi = i, 
the quantity E rises very slowly to a not very pronounced maximum 
at a temperature about f of the critical value, thereafter falling 
rapidly to zero at the critical point It is interesting to see that this 
slight maximum is shown in the experimental figures, but was 
overlooked, as workers in the subject were looking rather for 
constancy than variation with temperature. 

Some time ago Katayama remarked that very considerable 
simplification resulted if the difference of the liquid and vapour 
densities were substituted for the liquid density in the definitions 
of E and e. We thus have 

/ M \l . ,, ,/ M \f 
e = y [ ) and E = A ( - 1 , 

\Pc PJ \Pe PJ 



LIQUIDS AND GASES 31 

and Katayama points out that, in these circumstances, e and E are 
linear functions of the temperature given by 

e = <? (i m], E = E (i + o'2iri). 

As the reader may easily convince himself, these results depend 
on the power law being followed with n equal to 1-2, and Macleod's 
law being obeyed with the index equal to 4 

If we write this latter law in the more general form 

y = C( Pe - Pv y, 

and do not assume any special value for n in the expression for the 
power law, we readily find that E = E (i m} x {i + (n i)m}, 
where for brevity x is written for (n i znfep) If n 1-2 
and p = 4, we have x = o, and Katayama's value for E results In 
no instances that we have examined is this exactly true The index 
x is small but positive, and the result is that E climbs by an almost 
linear ascent to a definite maximum, thereafter falling very rapidly 
to zero at the cutical point behaviour much more consonant with 
our usual conception of surface energy than that given by Kata- 
yama's equation, which gives E its highest value at the critical point 

To establish these results is not difficult If we have, for a sub- 
stance whose critical temperature is known, a series of values of 
surface tensions determined over a wide temperature range, a 
logarithmic plot of (i m) and y serves to test the power law, 
and to determine the value of n where the power law is followed 
The values of A, e and E at different temperatures may then readily 
be computed E 0) the zero value of the total molecular surface 
energy, is readily deduced, and it may be remarked that this 
quantity varies in very mteiestmg and icgular fashion with varia- 
tion in chemical constitution 

The quantity My i /(p e />) (where M stands for molecular 
weight) has been named the parachor. It provides us with a 
number which measures the molecular volume of a liquid at a 
temperature at which the surface tension is unity, and therefore gives 
a most valuable means of comparing molecular volumes under 
corresponding conditions. 

Viscosity 

We have seen that a perfect fluid is one in which tangential stresses 
do not exist, whether the fluid be at rest, or whether its different 
portions be m motion relative to each other. Such stresses do, 



32 THE MECHANICAL PROPERTIES OF FLUIDS 

howevei, appear in all known fluids when lelative motion exists, 
and the fluid may be looked upon as yielding under the stress, different 
fluids yielding at very different rates. 

The most obvious effect of the existence of such tangential 
stresses between different parts of the fluid is the tendency to damp 
out relative mption. Thus, if we have a layer of liquid flowing over 
a plane solid surface, the flow taking place in parallel horizontal layers, 
the layer of liquid m contact with the surface will be at rest, and 
there will be a steady increase, with increase of height above the 
solid suiface, in the horizontal velocity of the successive layers. 
Considering the surface of separation between any two layers, the 
tangential stress existing there will tend to retard the faster moving 
upper layer, and to accelerate the slower moving lower layer. The 
magnitude of the tangential stiess may be written down if we assume, 
following Newton, that the tangential stress is propoitional to the 
velocity gradient, so that, if the horizontal velocity is v at a vertical 
distance y from the fixed surface, we have 

dv , dv 

S oc , i e. equals p -, 

ay ay 

where //, is a constant called the coefficient of viscosity of the fluid. 

If - is unity, then S = ju. Hence we are led to Maxwell's well- 
ay 

known definition of p " The viscosity of a substance is measured 
by the tangential force on unit area of either of two horizontal planes 
of indefinite extent at unit distance apart, one of which is fixed, 
while the other moves with unit velocity, the space between being 
filled with the viscous substance " 

The dimensions of // are those of stress divided by velocity 
gradient; this works out to 

M = [ML-'T-'J, 

so that a coefficient of viscosity in C.G.S. units is correctly given as 
x gm. per centimetre per second. 

If in fig. i (c), p 4, we put Aa dx, AD = dy, we see that the 
ligidity modulus (N) is given by 

S = N^. 
dy 

d"V 

Comparing this with S = ft , it is clear that the dimensions of 

iy 



LIQUIDS AND GASES 33 

.cosity differ from those of rigidity by the time unit, in the same 
y as the dimensions of length differ from those of velocity. In 
t, the rigidity modulus of a solid determines the amount of the 
ain set up by a given tangential stress, and the viscosity modulus 
a fluid determines the rate at which the fluid yields to the stress. 

The fall of a sphere through a viscous fluid aptly illustrates 
r eral interesting physical phenomena; we shall therefore study 
5 problem in some little detail. Considerable assistance is given 

an application of the theory of dimensions. Suppose that the 
istance (R) experienced by the sphere depends on its radius (a), 
velocity (v), and the density (/>) and viscosity (/u,) of the surrounding 
id, we then have 

R = ka x p y ^v M , 

I as the right-hand side must have the dimensions of a force, we 
ain, by equating the exponents of M, L, and T, 

x zv, y = w -- i, z = 2 w, 

so that R = fof--v 





\ p> I P 

low velocities we may assume that the resistance is proportional 
he velocity. Putting therefore zu = i, we find 



R = 

lore complex analysis, originally given by Stokes,* shows that 
R = 



tiowever, we assume that for high velocities the resistance varies 
be square of the velocity, we have, putting zv = 2, 

R = kv*a z p, 

viscosity does not enter into the question energy is expended, 
in overcoming viscous resistance, but in producing turbulent 
ion in the liquid. 

Returning to the problem of low velocities, let us write down the 
ition of motion of a sphere falling vertically through an infinite 
n of fluid. The forces acting are the weight (W) of the spheie 

* Lamb, Hydrodynamics, p. 532 (1895). 
(D312) 3 



34 11IK MECHANICAL PROPERTIES OF FLUIDS 

(downwards), the lesistancc (R), and the buoyancy (B) of the displaced 
lluul (upwaids). This gives 

W - (B + R) = mf, 

whcic m is the mass and /the downward acceleration of the sphere 
Hut us the velocity of the spheie incieases, R increases pan passu 

so that the acceleration steadily 

^_ diminishes, until when R has 

increased to such an extent 

that 

W - B = R, 

/ becomes zero, and the sphere 
henceforward falls with a con- 
stant velocity known as the 
" terminal velocity ". Calling 
this velocity V, the density and 
radius of the sphere p and a 
respectively, and the density of 
the fluid p > we have 

jiTtt s g(p PQ) = 
leading to 



UDC 




I'Jtf I 



r 9 V ' 

Clear ly, measurement of the 
terminal velocity V enables us 
to determine the viscosity of 
a liquid The method is 
peculiarly suited for the mea- 
surement of the viscosities of 
very viscous hquids such as 
heavy oils or syrups, and has 
been much used ol late yeais. The simple apparatus required is 
shown m %. n The outci cylinder icpiesents a thermostat; the 
inner cylinder contains the liquid under expeiiment. 

The sphere steel ball bearings 0-15 cm in diameter are suitable 
An liquids having viscosities comparable with that of castor oil is 
diopped centrally through the tube AB, and its velocity is measured 
over the suifacc CD, which icpiesents one-third of the total depth 
of the liquid. 



LIQUIDS AND GASES 35 

Two important corrections are necessary one for " wall-effect ", 
e for " end-effect "; for it must be remembered that the simple 
sory given above applies only to slow motion through an infinite 
'an of fluid. 

These corrections have been investigated by Ladenburg,* who 
s shown that in order to correct for wall-effect we must write 



- V 



., 



iere V is the observed velocity, V^ the corresponding velocity 
an infinite medium, and - the ratio of the radius of the sphere to 

it of the cylinder containing the liquid. 
Similarly for the end-effect 



Lere h represents the total height of the liquid which is supposed 
be divided into three equal poitions, V representing the mean 
ocity over the middle third. Introducing these corrections into 
}kes's formula, we obtain 



4- r 

^ O 7 

, 

The method has been much used dining the war period for the 
asurement oi /x for liquids of high viscosity, and is fully desciibed 
a papei by Gibson and Jacobs f 

A commercial viscometcr has iccently come into use, which is of 
;eedmgly simple type, and gives fairly ichable icsults A steel 
1 m. in diameter is placed inside a hemisphencal steel cup of 
jhtly larger dimensions. The cup cairies on its internal surface 
ee small pi ejections in length about 0-002 in. A little of the oil 
der examination is poured into the cup, and the ball placed in 
sition inside the cup. The ball is pressed down on to a table, 
; cup being uppermost, and at a given instant cup and ball are 
ed clear of the table. The time taken for the ball to detach itself 
measured, and this gives a measure of the viscosity of the oil.| 

* Ann der Physik (IV), 23, 9 and 447 (1907) 

f>r. Chem Soc , 117, 473 (1930). 

j For fuller description see Chapter III, p. 119. 



36 THE MECHANICAL PROPERTIES OF FLUIDS 

Comparative measurements only can be made, and the instru- 
ment must be standardized by means of a liquid of known 
viscosity 

Viscometeis for use with oidmaiy liquids usually depend on 
measurements of the flow of a liquid through a horizontal or vertical 
capillary tube. The solution of the pioblem for a horizontal capillary 
affords an interesting application of the general equations of hydro- 
dynamics, and we shall attack the problem from that side. The 
reader may or may not be able to follow the arguments by which 
these equations are established he may study them at leisure in the 
tieatises of Lamb, of Bassett, or of Webster what is important is 
that he should see clearly their physical significance and obtain piac- 
tice in handling them. This is best done by a caieful study of one 
or two of theii applications. 

The equations of motion of an incompressible fluid aie: * 

Du d , 



, D a , a , a , 9 

where = -f u -j- v -f- w~, 

ut ot ox ay oz 

v* = !L 4- !L + *. 

dx* dy z a* 2 ' 

p is the pressure at any point, X, Y, Z the components of the external 
force per unit mass, u,v,w the velocity components. 

To apply these equations to the steady flow of a liquid thiough 
a horizontal capillary tube, we take the axis of the tube as ^-axis, 
and assume the flow everywhere parallel to this axis Then u = 
v = o, and from (a) and (J3) we have 

dp dp^ 

dx dy 

so that the mean pressure over any section of the tube is uniform. 
* See Chapter II, p. 83. 



LIQUIDS AND GASES 

Also from the equation of continuity, 

du . dv . dw 

_|_ _ _j = o, 

dx dy dz 

, dw 

we have = o. 

9# 

fence (a) and (/3) vanish, and (y) becomes, assuming no extraneous 
>rces, 



dp 
ince w varies only with t, x, and y, and p only with #, then 

= constant = c (say), and therefore 
f, i, i j_ 



/e can now obtain the equation known as Poiseuille's equation, for 

the motion is unaccelerated - = o, and 

dt 



"ransforming to polar co-ordinates,* 

id z w i dw _, i 
A 1 \ h o i ~ ^r~ ~r ~^ 



r, w being independent of 9, 

d z w 

'o~T 

or z r or 
""his may be written 

,/ 9 2 2y . 9zw\ a 9 

q r + ) = C r- = c. 

r\ or* or/ r or \ or 



\ 
)= 

/ 

ntegrating, we have 



dw c r 2 . . 

V _ . - . _ _ _|_ f\* 

' ^5^ i^ - tl 

9r /u, 2 
* See p. 53 et seq. 



3 8 THE MECHANICAL PROPERTIES OF FLUIDS 
and integrating a second time 

10 = I : - r z 4- A losr r 4- B, 

*JL JLC ' O ' ' 

where A and B are constants of integration When r = o (on the 
axis of the tube), w is finite. Consequently A must be zeio, and 

w i"[I" r + 

Also, if there is no slipping of the liquid at the walls of the tube, 
when r = a,w = o, and consequently 



If V is the volume of liquid which escapes from the tube in a time T 
the volume issuing in unit time is given by 

Y = f 
T I'* 



*" / / > P\ J TT U 

= c (a r*} rar - c 
4ft V ; 8 /* 

Ifpi and /> 2 are the pressures at the entrance to and exit horn the 

dp 
pipe, then remembeiing that stands for the late oi inucase of p 

with z, we have 

whei e / is the length of the pipe. Hence 



8 V / 

If the liquid is supplied to the tube under a constant head //, and 
escapes into the air at a low velocity, we have 

__ TT a*T gph 
p _ _._ T . 

This equation is known as Poiseuille's equation. All the quantities 
on the light-hand side may be determined expei mien tally, and 
hence ju may be evaluated. 



LIQUIDS AND GASES 39 

Comparative measurements by this method are usually made using 
twald's viscometer (fig. 12). The bulb C is filled with the liquid 
der examination, which is then drawn up by suction until it fills 
5 bulb D. The pressure is then released, and the time of transit 
L ween the marks A and B observed. The pressure head is varying 
oughout the fall, and clearly we cannot apply Poiseuille's equation 
it stands. But, noting that for liquids of equal densities and 
Ferent viscosities the times will be proportional to the viscosities, 
i that for liquids of equal viscosities and different densities the 
les will be inversely proportional to the densities, 

have in general, 




t = K or \i = cot, 



ere c is a constant for the apparatus to be deter- 
tied by using a liquid of known viscosity. 
A viscometer of dimensions suitable for the detei- 
nation of the viscosity of water is not suited for use 
h heavy oils But if we have a series A, B, C, 
viscometers of gradually increasing bore, calibrate 
by using water, and then use the most viscous 
aid suitable for A m order to calibrate B, continuing 
s process as far as necessary, we are provided with 
ham of viscometers which can be used over a veiy 
ie lange. 

Since the viscosity of liquids decreases rapidly 
h increase of temperature, it is of vital necessity 
t the apparatus be enclosed in some form of thei- Fig J2 
stat and that the temperature of experiment be 
en and recorded. This rapid change of viscosity with temperature 
kes it very difficult to obtain relations between the viscosities of 
Ferent chemically related substances, as it is by no means easy to 
tie the temperature of comparison. It has been found, however, 
t consistent results may be obtained if viscosities are compaied 

temperatures of equal slope that is at temperatures for which ~ 

CIL 

he same. Using this standard it has been shown, for example, 
t the molecular viscosities * of a homologous series increase by a 
istant amount for each addition of CH 2 

* The molecular viscosity of a liquid is defined as /J.(Mv)$, where M is the 
ecular weight and v the specific volume of the fluid concerned, 




40 THE MECHANICAL PROPERTIES OF FLUIDS 

There are many important piactical problems which depend foi 
their solution on a knowledge of friction in fluids The viscosities 
of mixtures of liquids, the viscosities of gases, the theoiy of lubiication 
the discussion of turbulent motion, to mention but a few, piesen 
important and most inteiestmg aspects. These matters are full) 
discussed in Chapter III 

One interesting problem may be mentioned m passing the 
suspension of clouds in air, where we have the appaient paiadox oi 
a fluid of specific gravity unity suspended in a fluid of specific 
gravity 0-0013. The paradox is cleared up by an application oi 
Stokes' formula, 

V = - (^ZJ ^ 3 . 
9 P- 

Taking the viscosity of air as 0-00017 m C.G S units, the leader is 
recommended to calculate the teiminal velocities of spheres of water 
say o-i, o-oi, . . . cm in radius The terminal velocities ol minute 
drops will be found to be surprisingly small 

The kinetic theory of liquid viscosity has not icceived a great 
deal of serious attention, and formulse developed to show, for example, 
the dependence of liquid viscosity on tempeiature have usually a 
purely empirical basis Of these, one proposed by Porter may be 
specially noted. Suppose that the variation of viscosity with tem- 
perature has been experimentally investigated foi two liquids 
Take a temperature T x at which one liquid has a viscosity ?j , Find 
the temperature T 2 at which the second liquid has the viscosity 77 1 
Repeat foi different values of T x . Then Tj/Tg is a hneai function 
ofTj. 

Recently Andrade has put forward a kinetic theory in which he 
assumes " that the viscosity is due to a communication of momen- 
tum from layer to layer, as in Maxwell's theory of gaseous viscosity 
but that this communication of momentum is not effected to uny 
appreciable extent by a movement of the equilibrium position oJ 
molecules from one layer to another, but by a tempoiary union al 
the periphery of molecules in adjacent layeis, due to their large 
amplitudes of vibration." 

This assumption leads to a formula connecting viscosity and 
temperature of the form 



r, = 
a formula which had been put forward previously on an empirical 



LIQUIDS AND GASES 41 

isis. Porter's relation, as the reader may verify, follows at once 
om this equation. 

In the deduction of this equation, variation of volume with 
tnperature has been neglected and, taking this factor into account, 
idrade deduces a second formula, 



iere v is the specific volume. For a great many organic liquids 
is formula gives a very good fit, though, as was to be expected, 
iter and the tertiary alcohols show abnormalities. 

Equations of State 

Much labour has been expended on the problem of devising 
uations which shall represent accurately the pressure- volume- 
nperature relations of a substance in its liquid and its gaseous 
ases It may be said at once that it is impossible to devise an 
nation which shall be accurate over such a range without being 
possibly cumbrous Nevertheless the simpler equations have, 

we shall see, considerable value in giving a fairly adequate 
)resentation of the general behaviour of a homogeneous fluid, 



IL 




FIR 13 



A very simple type of such a fluid is a gas, considered as an 
emblage of material points which are in rapid and random 
ition, and which do not exert any attractive forces on each other 
nsider a given quantity of such a gas enclosed m a cube of side 
md let the component velocities of any one particle be u, v, w 
shown (fig. 13). 

The pressures on the faces of the cube are due to the impacts of 
particles thereon. At any impact on, say, the face A the velocity 
nponent normal to that face will be reversed, and the change of 
menturn of the particle consequent on the rebound will be 

mu ( mu} = zmu. 

(D312) 3* 



42 THE MECHANICAL PROPERTIES OF FLUIDS 

2/ 

The time of tiavel fiom A to B and back is sec.; the frequency 

u u 

of the impacts on the face A is ,; hence for any one particle 

the change of momentum at A per second will be 

u mu z 
zmu x - = =-, 
2/ / 

and similarly for the other particles 

The force on the face A due to molecular bombardment is, 

therefore, -~u*. 

V 

Ifp is the pressure on the face A, 



^ o v po 

p = _ _ = -Zir = -Z,r, 

where V is the volume of the cube. Similarly, foi the pressures on 
the faces perpendicular to A, we have the expressions 

~Zv*, 1 ~Sw\ 

But these pressures are equal, and therefore 

p = 



^(a -I- 
where U 2 = 2 -f v z + ro 2 

Now let us define a mean velocity U by the relation 

f? _ jff 
U - 2, N , 

N being the total number of particles in the cube. We then have 



and wN being the mass of the gas, we have, if /> be its density, 



, 
p = and /> = 

Hence Boyle's Law. 

If we assume that U 2 is pioportional to the absolute temperature, 



LIQUIDS AND GASES 43 

i have Charles's law, and can write as the characteristic equation of 
ir " perfect " gas 

pV = RT. 

V stands for the volume of unit mass of the gas, R will be different 
r different substances. A simple deduction from our fundamental 
uation shows, however, that the gramme-molecular volume* is the 
me for all gases. Consider two different gases for which 

and 



the pressures and volumes are the same, 



the temperatures are equal, then, assuming that the mean kinetic 
jrgies are the same, 



that Nj = N 2 

at is, equal volumes of two gases, under the same conditions of 
iperature and pressure, contain the same number of molecules 
This is the formal statement of Avogadro's hypothesis. It 
ows, therefore, that the weights of these equal volumes are pro- 
tional to the molecular weights of the gases, and hence that the 
mme-molecular volumes of all gases, measured under the same 
iditions of tempeiature and pressure, are the same. 
The giamme-moleculai volume, measured at o C and 760 mm. 
Hg, is 22-38 litres. If then V stands for this volume, the constant 
vill be the same for all gases. Its value should be calculated by 
reader. 

But no gas behaves in this simple manner, although for moderate 
ssures and high temperatures the equation is accurate enough for 
inary computations, as far as the more permanent gases are 
cerned. 

Suppose that we study experimentally the p - v relations of 
erent fluids, drawing the isothermals for various different tem- 
itures (fig. 14). Starting at a sufficiently low temperature we find 
the volume steadily diminishes with increase of pressure up to a 
ain point at which the fluid separates into two phases liquid and 
;ous. The pressure then remains constant until the gaseous 

I e the volume occupied by M gm of a gas at normal temperature and 
sure, where M is the molecular weight. 



PRESS 



44 THE MECHANICAL PROPERTIES OF FLUIDS 

phase has completely disappeared, when further mciease of pressure 
causes but small diminution in volume. If we now repeat the 
experiment at a higher tempeiature, we find that the hoiizontal 
poition AB of the curve, representing the period of tiansition from 
the gaseous to the liquid phase, is shoitei, and shortens steadily with 
increasing temperature until the isothermal for a certain temperature 
exhibits a point of in- 
flexion with a hori- 
zontal tangent, run- 
ning for a moment 
parallel to the volume- 
axis, and then turning 
upwards again. The 
tempeiature for which 
this isothermal is 
drawn is called the 
critical temperature, 
and the point G the 
critical point * Above 
this temperature no 
amount of pressure 
causes a separation 
into two distinct 
phases 

The experimental 
determination of these 
curves, over a wide 
range of pressure and 
temperature, is a 
matter of no small 

difficulty. Once a suitable pleasure gauge has been devised we 
have seen that the change of electrical lesistance of manganin may 
be utilized obseivations are fairly straightfoiwaid, but the cali- 
bration of such a gauge demands experimental work on a hcioic 
scale Amagat, for example, peiformed a Boyle's Law experiment 
in which nitrogen was compressed in the closed (shorter) limb of a 
U-tube, the open lirnb being installed on the side of a shaft 327 m. 
deep. The p-v relation for nitrogen being known, this gas may 
then be used as a standard m studying the behaviour of othci 

* Foi a description of the physical state of the fluid at the cntical point, consult 
any of the standaid treatises on heat, e g Poyntmg and Thomson, or Pieston. 




_ 

VOL 



Fig 14 



LIQUIDS AND GASES 4S 

ases, or in calibiating a different type of piessure gauge. 

Let us now examine briefly the character of the curves obtained 
om experiments o this type. It is convenient to plot pv against 
, as this procedure exhibits very clearly the departure of the gas 
mcemed from the " perfect " state. Some of the results obtained 
e shown in fig. 15 

The reason for the difference between the isothermals for an 
'eal and for an imperfect fluid is not far to seek. The equation 

pV = RT 

kes no account of the forces of attraction between the molecules, 
)r of the volume occupied by the molecules themselves. It makes 

'} pv 




"Perfect Gas 



Carbon. DuDxUie 



Fig 15 



zeio when p becomes indefinitely great, and it is cleaily more in 
cordance with the properties of fluids to put 

p(V - b) = RT, 

that asp increases indefinitely, V tends to the limit b, b represents, 
eiefoie, the smallest volume into which the molecules can be 
eked. 

Further, the mutual atti action of the molecules will result m the 
oduction of a capillary pressure at the fluid surface, the intensity 
the molecular bombardment will be diminished, and the pressure 

the surface of the containing vessel correspondingly decreased, 
it, theiefore, 

(p + w)(V - b) = RT. 

ithout discussing the matter very closely, we can determine the 
lue of cu from consideration of the fact that the attraction between 
o elementary portions of the fluid is jointly proportional to their 



46 THE MECHANICAL PROPERTIES OF FLUIDS 

masses that is, in a homogeneous fluid, to the square of the density, 
or inversely as the square of the volume. We see, then, reasons for 

writing co = -, and the equation of state becomes 



the form originally pioposed by van der Waals 

This equation is a cubic m v, and if the isothermals are plotted 
for different values of 0, we obtain cmves whose general shape is 
that of the curve HAEDCBK of fig. 14. It will be observed that for 
temperatures below the ciitical tempeiature a hoiizontal constant 
pressure line cuts any given isothermal either in one point or in three 
points corresponding to the roots of the van der Waals cubic 
Taking an isothermal nearer to the critical tempeiatuie, we see that 
the three real roots are more nearly coincident, and at G, the ciitical 
point itself, the roots coincide. Above the critical point, a hoiizontal 
line cuts any given isothermal in one point only two ol the loots of 
the cubic are imaginary. If we write down the condition that the 
three roots shall be coincident, we easily ainve at values ol the ciitical 
constants in terms of the constants of van dei Waals' equation 
These are 

rn 

I 
* c 

But it is preferable to write down the condition that at the ciitical 
point the isothermal has a point of inflexion with a horizontal tangent 
If we therefore differentiate van der Waals' equation with icspcct 

dp 3 2 /> 
to v, put ?j- and -^ equal to zero, the resulting equations, combined 

with the original equation of state, serve to deteimine p c , v c , and T c 
The work is left as an exercise to the reader. 

This method is pieferable, since it is perfectly general and may 
be applied to characteristic equations which are not cubics in v, and 
to which, therefore, the " equal-root " method, beloved ol wiitcis 
on physical chemistiy, is not applicable 

It will be observed that the equation tells us nothing concerning 
the straight line AB 3 which lepresents the actual passage observed m 
nature from the vapour to the liquid phase. The position of this 
line on any given isothermal can, however, be obtained from the 
simple consideration that the areas DBCD and AEDA must be 



LIQUIDS AND GASES 47 

qual,* and the line must be drawn to fulfil this condition. 

Fig 1 5 shows that the pv-p curves in general exhibit a minimum 

alue for pv, and that the locus of these points lies on a definite curve. 

'he equation to this curve may be obtained by writing pv as y and 

as x in the characteristic equation, and expressing the condition 

lat y should have a minimum value. 

Of the other characteristic equations that have from time to time 
sen proposed we may cite, naming them by their authors: 

Clausms (a): 



Clausius (b). 

Dieterici (a): 

L + * \ (V - b) = RT, 

Dieterici (6). 

/>(V b) = RTe 

he deduction of the cutical constants from these equations is left 

the reader. 

The value of a chaiactenstic equation which shall closely repiesent 
e pressure- volume relations of a fluid over a wide range of pressuies 
id temperatuies, is obvious We have seen, in the section on 
impossibility, that many important physical constants may be 
pressed in terms of theimodynamic equations involving certain 
ffeiential coefficients and integrals The values of these physical 
instants may be worked out by substituting, in the appropriate 
ermodynamic equations, the values of the differential coefficients 

* Foi in any reversible cycle ( /) -~ o If the cycle be isothermal, 

(M9 = 1 (f) dQ, and therefore (l)dQ = o. 

it for any cycle 

( /) (dQ -I- dW) = o, 

)cfW = o So that, if we take 

it mass of the substance leversibly round the cycle AEDCBDA (fig. 14), the 
>rk done, lepresented by the sum of the positive and negative areas AEDA and 
3BD, must be zero. Hence the two areas are equal (For critical remarks on 
is pioof see Preston, Theory of Heat, 479 (1904), or Jeans, Dynamical Theory oj 
uses, p. 159.) 



48 THE MECHANICAL PROPERTIES OF FLUIDS 

obtained from the differentiation of the equation of state Un- 
fortunately, no equation yet pioposed covers the whole giound 
satisfactorily. An equation which fits the experimental figuies at one 
end of the scale is usually unsatisfactory at the other end, and con- 
versely. 

A few simple tests may be suggested by which the fitness of any 
given equation may be roughly examined. The expenments oi 

RT 
Piofessor Young show that, while the value of the latio - c vanes 

PPc 

slightly from substance to substance, its mean value may be taken 
as about 3-75 Now the equation of state of an ideal gas gives unity 
for this ratio, and is cleaily veiy far out of it. Van der Waals' 
equation gives 

RT C / Sa \ / a \ , 

____ c = (R x O ( : X 36) = 267, 
p e v e \ 2 7 R/;/ \2 7 & 2 6 I h 

and is not a veiy good approximation to the truth 

Similarly the (a) and (b) equations of Clausms give for this ratio 
the values 2-67 and 3 oo respectively, and the corresponding equations 
of Dieterici give 3-75 and 3-69, if in the (a) equation we take the 
value of k as -* 

Another test may be derived from the expenmental fact that the 
critical specific volume v c is about four times the liquid volume 
Now the constant b, which represents the least volume into which 
the molecules can be packed, cannot be seiiously drfleient from the 
liquid volume. Accordingly we find, on woikmg out the values lor 
v c) that foi the (a) and (b} equations of Clausms the values ol v c ai e 
3& and 4/; i espectively,* whrle the corresponding equations ol 
Dretencr give the values $ and 2&. Foi van dei Waals' equation 
the value is, as we have seen, 36. 

But the subject may be studied from a different point of view 
Instead of attempting to devise an equation which shall leprescnt 
the properties of a substance over a wide lange a process which 
usually lesults in a cumbrous foimula we may tiy to anive at an 
equation which shall be simple and manageable in foim, so that 
the vanous physical constants of the fluid may be leadily worked out 
from the corresponding thermodynamic relations, while at the same 
time the equation shall represent a very close approximation to the 
tiuth over a limited lange, the range chosen being one of practical 
impoitance. Whether such a formula can, 01 cannot, be extrapolated 

* If m the (6) equation we put zC = b. 



LIQUIDS AND GASES 49 

yond the limits of the range is a mattei of secondary interest what 

important is that the formula should be as exact as may be within 

2se limits. 

Let us then investigate the form which such an equation as the 
equation of Clausms assumes for moderate pressures. Re- 

iting the equation as 



see that at model ate pressures, when the volume is large, we shall 
t be seriously in error if we write 

(v+V = ^ ( a pp roximatel y)- 

1 thus have 

p(V - b) = RT - a 



TV 

I again, putting, m the small term, 

V - RT 

T' 

find, on rearranging the equation, 

V = *I-^ + a, 

p T 2 ' 

2re c is put for . 

If we replace T 2 by the more general form T", wheie n varies from 
stance to substance, we have 



- , 

p T" 

ch is the form known as Calendar's equation 
This equation has been applied very successfully to elucidate 
pioperties of steam over a lange of pressure from o to 34 atmos- 
res; the value of n appropriate to steam is --3- Space will not 
nit us to discuss at length this important equation Indeed, the 
ussion lies within the piovince of thermodynamics, and the 
ler desirous of further information should consult the articles 
hermodynamics " and " Vaporization " in the Encyclopedia 
'anmca, or a textbook such as Ewing's Thermodynamics for 
ineers, 



50 THE MECHANICAL PROPERTIES OF FLUIDS 

Osmotic Pressure 

If we throw a handful of currants or raisins into water and leave 
them for a while, we find that the fiuits, originally shiunken and 
wrinkled, have swelled out and become smooth.* Water has passed 
through the skin of the fruit, while the dissolved substances inside 
cannot pass out or at least do not stream out so freely as the water 
streams in. This unilateral passage of a substance thiough a mem- 
brane is termed osmosis.^ 

In the limiting case, when we have a solution on one side of a 
membrane and pure solvent on the other, the membrane is called 
a semi-permeable membrane if it freely admits of the passage of the 
solvent, but is strictly impervious to the dissolved substance tt 
has been asserted that no such membranes exist in nature, but, as 
far as experiment can show, a membrane of copper ferrocyanide 
forms a true semi-permeable membrane to a solution of sugar in 
water 

Suppose such a membrane, prepared with due precautions J 
and it is not so easy as one would imagine to prepare a thoroughly 
resistant membrane to be deposited on the inside of a cylindncal 
porous pot. The pot is filled with a sugar solution, closed, and 
attached to a suitable manometer which shall measuie the pressuie 
inside the pot It is then placed in a vessel containing pine water 
We shall find that the pressure in the pot uses, finally i caching a 
maximum stable value The maximum value of this pressure, 
assuming that the membrane is truly semi-permeable, is called the 
osmotic pressuie of the solution. 

It will thus be seen that osmotic pressure is defined in teims of 
a semi-permeable membrane. It is only in very loose phiaseology 
that one can speak of the osmotic pressure of a solution without 
reference to the existence of a semi-permeable membiane. A solu- 
tion, qua solution, has no osmotic pressure. 

If this definition be consistently followed, a great deal of vague 
and loose reasoning of the sopoiific-power-of-opium variety will be 
swept away. It is all the more needful to emphasize this point as 
there has arisen, in biological (and even in engineering) circles, a 
tendency to ascribe to " osmotic pressure " a power and potency 

* Imbibition of water by the dried tissues will also play a part m the smoothing 
process 

f From iSoy*6s, a raie Greek noun meaning " thrusting " or " pushing through ". 
t Morse, Jour. Amer Chem, Soc., 45, 91 (1911). 



LIQUIDS AND GASES 51 

ich is almost proportional to the vagueness with which the 
chanism of that pressure is conceived. 

Consider, for example, the common remark that osmotic pressure 
cts the wrong way " that is, causes motion from a region of lower 
lottc pressure to a region of higher osmotic pressure. It only 
uires a little consideration of the definition of osmotic pressure 
y to realize that the argument involves a varrepov irporepov, 
it is clear that it is osmosts which produces osmotic pressure, not 
lotic pressure which produces osmosis. 

The quantitative laws of osmotic pressure were first studied by 
ffer, whose figures show that for dilute solutions the pressure, 
constant temperature, is proportional to the concentration, and 
at constant concentration the pressure is proportional to the 
)lute temperature. We may therefore write 

PV = KT, 

it has been shown by van't Hoff that the constant K has the 
e value as the gas constant R Hence it follows that the osmotic 
ssure of the solution is the same as that which would be exerted 
the dissolved substance were it dispersed in the form of a gas 
>ugh a volume equal to that occupied by the solution, 
[f we desire to correct this simple gas law, we find it necessary 
ook at the matter from a different angle. We define an ideal 
tion as containing two completely miscible unassociated com- 
ents, of such a nature that there occurs no change of volume on 
ing, and that the heat of dilution is negligible 
For such a solution it can be shown that the osmotic pressure P 

^^ P^ 2 RT f 1 ( 

P + 0- = -^-| ~ log a (l - 

re /? denotes the compressibility of the solvent, V its molecular 
me, and x the ratio of the numbei of molecules of the dissolved 
stance to the total number present. If we neglect /?, which is 
illy small, and expand the logarithmic term, we have the con- 
entform RT/r ^ ^ v 

P = -^r-U-h- + -+ ) 

V \ 2 3 / 

3 equation holds good for any concentration. 

Despite a large amount of criticism, the Idnetic theory of osmotic 

sure still holds the field as the only one which gives values of the 



52 THE MECHANICAL PROPERTIES OF FLUIDS 

pressure calculated from theory.* The properties of the membiane, 
which play a large part in some theories, whilst of great inteiest and 
value, are distinctly of secondary importance in the kinetic theory. 
It is the thermal agitation of the molecules of the solute which is 
effective in producing osmotic pressure, and the magnitude of the 
pressure calculated from the agitation of the molecules is equal to 
the value obtained by experiment, " Any other theoiy put forwaid 
to account for osmosis must fulfil, then, a double duty, not only must 
it be competent to explain osmosis, but it must also explain away the 
effects that we have the light to expect from the molecular agitation 
of the solute."! 



NOTE ON TRANSFORMATION OF CO-ORDINATES 

Perhaps the simplest way of passing from one foirn to the other is 
to consider the concentiation of fluid, reckoned pei unit volume per 
second, at a point 

If u, v, w are the component velocities at a point P, the fluid leaving 
the elemental volume SxSySst m time 8t is readily found, by the method 
given below for a more difficult case, to exceed that which enters it, by 

an amount ,~ ~ 

(mi . ov 
/>(;r- + 
\ox 

where p is the density, i.e the concentration, icckoned in mass pei unit 



volume per second, is 

r 



. 

OU 



Q- 
oy 



Taking now the element of volume shown m fig. 16, 




rS6 




Fig. 1 6 



* Porter, Trans Far Soc , 13, 10 (1917) The reader desiring more information 
on the subject should consult this valuable discussion f Porter, loc, cit,, p. 8, 



LIQUIDS AND GASES 53 

ind letting u' be the ladial velocity and / the tangential velocity at P-, 



ve sret 

u'rBdSzBt for the radial flow in, and 

u' + ~-o>) (r80 -f l(rS0)8fW* for the radial flow out. 
\ or / \ or / 



The latter exceeds the former by 

OU , / 

r -- L u ' 
or 



The tangential flow in is 
nd the tangential flow out, 



r hich exceeds the former expression by 



e the total flow out is 

, . du' . 3w'\ s /j<j r> o. 
u + r- + -^ )808r88f, 

9r 30 / 

e the total flow //o the element is this expiession taken with the 
unus sign 

The elemental volume is rWSzftr , 

cnce the concentration is 

in' , du' . i 

. . f\\ L _L 

P\-~ I -5" r ~ 
\r or r 

The concentration mass per unit volume per second must be the 
mie whatever co-ordinates we use, hence 

du , dv u' du' . i d^ ,-, 

^.^ nzn 4 *i lil 

/-\ I -\ I *-i I O/l ^ ' 

9^; oy r or r ou 

If the fluid motion has a velocity potential, the component velocity 
i any direction is the gradient of potential in that direction, i e 

3c4 3<i 

' 11 = - L. 

rv J ^ o > 

3^ 9y 

, 8(4 / 1 3(4 

M' I rri' : 7" 

"X" ' 'vi' 

3r r 30 



54 THE MECHANICAL PROPERTIES OF FLUIDS 

for the element of length perpendicular to r, i.e. in the tangentia 
direction, is rd&\ hence 



by substituting the values of w, w', w, e;', in equation (i). 

We have proved the transformation when the dependent variable is 
the velocity potential <, but as the transformation is a purely analytical 
one in its nature, the form must be equally true whatever be the 
physical nature of (j>; for instance, if (f> = w, the # component of 

2 



the form which actually occurs (p. 33), and, in geneial, 



at every point in a plane where U is a single-valued function of the 
co-ordinates of the point and possesses finite derivatives up to those of 
the second order there. 



CHAPTER II 
Mathematical Theory of Fluid Motion 

It is assumed throughout this chaptei that the fluid with which 
we deal may be regarded as incompressible. This only means that 
changes of pressuie aie propagated in it instantaneously, instead of 
with the (very great) velocity of sound. Since the velocity of sound 
in air is only about four times less than in water, it is cleai that 
many of our lesults will be equally applicable to gaseous fluids, 
the influence of compressibility being negligible except in the case 
of veiy rapid differential motions. 

It is further assumed in the first instance that the fluid isfnction- 
less, i e. that it presses perpendicularly on any surface with which 
it is in contact, whether it be the surface of an adjacent poition 
of fluid, or of a solid boundary This hypothesis of the absence 
of all tangential stress is not in accordance with fact, but it 
greatly simplifies the mathematics of the subject, and there are, 
moreover, many cases of motion in which the influence of friction 
is only secondary It follows from this assumption that the state of 
stiess at any point P of the fluid may be specified by a single quan- 
tity p, called the " pressuie-intensity " or simply the " piessme ", 
which measures the force per unit area excited on any suiface through 
P, whatever its aspect. This is in fact the cardinal pioposition ol 
hydrostatics. 

It is convenient here to prove, once ior all, that the icsultant 
of the pressures exerted on the boundary of any small volume Q 
of fluid is a force whose component in the direction of any line element 
s is 



where dpfis is the gradient of p in the direction of 8$. Take first the 
case of a columnar portion of fluid whose length $x is parallel to 



MATHEMATICAL THEORY OF FLUID MOTION 57 

he axis of #, and suppose that the dimensions of the cross-section 
w) are small compared with Bx. The pressure-intensities at the 
wo ends may then be 
lenoted by pa} _ + Q ;)+( P <.$BS X)Q) 

p and p + /Sx, * Flg , 

ox 

o that the component parallel to the length of the pressure on the 
olumn is 



he pressures on the sides being at right angles to Bx Since 
5 the volume of the column, the formula (i) is m this case verified. 
iince, moreover, any small volume Q may be conceived as built up 
f columnar portions of the above kind, and since there is nothing 
pecial to the dn action Ox, the result is seen to be general. 

Stream -line Motion 

i Bernoulli's Equation 

A state of steady 01 stream-line motion is one in which the stream- 
nes, i e the actual paths of the particles, preseive their confirmation 
nchanged The most obvious examples are where a stream flows 
ast a stationary solid, and the designation is naturally extended 
) cases where a solid moves umfoimly m a straight line, without 
station, through a surrounding fluid, piovided the superposition of 
uniform velocity equal and opposite to that of the solid reduces 
ic case to one of steady motion m the formei sense This super- 
osed velocity does not of couise make any difleience to the dynamics 
f the question 

The stream-lines drawn through the contoui of any small area 
ill mark out a tube, which we may call a stream-tube. Since the 
ime volume of fluid must ti averse each cioss-section in the same 
me, we have 



heie co!, o> 2 are the areas of any two cross-sections at P L , P 2 , and 
L , # 2 the corresponding velocities in the direction (say) from P x 
> P 2 . Now consider the region included between these two sections. 
i a short time dt a volume Q = oj^dt will have entered it at 



58 THE MECHANICAL PROPERTIES OF FLUIDS 

P 1} and an equal volume Q = o> z q 2 dt will have left it at P 2 The 
work done by hydrostatic pressure in this time on the mass of fluid 

which originally occupied the space 
PjPg will be Pi^iqidt or piQi, at P lf 
and p z a) z q z dt or p z Q at P 2 . The 
same mass will have gained kinetic 
energy of amount ^pQ(q^ #i 2 ), where 

j"/"^ ^X/^ P ' 1S tne density, i.e. the mass per 

unit volume. If V denotes the poten- 
tial energy of unit mass, the gam of 
potential energy will be pQ(V 2 Vj) 
Hence, equating the woik done on 
the mass to the total increment of energy we have 




or Pi 

Hence along any stream-line 

P + W + /v = c, ............. (2) 

where C is a constant for that particular line, but may vaiy from 
one stream-line to another. This equation is due to D. Bernoulli 
(1738) and was proved by him substantially in the above manner 
The formula has many applications For instance, in the case 
of water issuing from a small orifice in the wall of an open vessel 
we have at the upper surface p = p (the atmospheric piessure), 
and q = o, approximately Again, the value of V at the upper 
surface exceeds that at the orifice by gsi, where a is the diflcrence ol 
level and g the acceleiation due to gravity Hence if q be the velocity 
at the surface of the issuing jet, 

Po -I- gpz = A + lpq z , 



^ r (? 2 = 2gZ, ........... (3) 

r a foimula due to Tomcelli (1643). If S' be 

the section of the jet at the " vena contracta ", 

where it is sensibly paiallel, the piessure over S' 

will be p Q The velocity will therefore be given 

Jg 3 by the above value of q, and the discharge per 

unit time will be pqS' The ratio of S' to the 

area S of the orifice is called the " coefficient of contraction ". 

It is not easy to determine this coefficient theoretically, but a 



MATHEMATICAL THEORY OF FLUID MOTION S9 

r ery simple argument shows that in the case of an orifice in a 

hm wall it must exceed f. Take, for instance, an orifice in a 

r ertical wall. In every second a mass of pqS' escapes with the 

elocity q, and carries with it a momentum pq z S f . This represents 

he horizontal force exerted by the vessel on the fluid. There must 

e a contrary reaction of this amount on the vessel. On the opposite 

rail of the vessel, where the velocity is 

isignificant, the pressure has sensibly the 

tatical value due to the depth, and if this 

/ere also the case on the wall containing c" 

tie orifice there would be an unbalanced > 

3rce gpzS urging the vessel backwards r. 

Actually, owing to the appreciable 

elocity, the pressure near the orifice will 

e somewhat less, so that the reaction ex- Flff 4 

2eds gpzS . Hence pq z S' > gpzS , or (since 

* = 2 gs) S' > |S In a particular case, where the fluid escapes 

y a tube projecting inwards in the manner shown in the figure, the 
atical pressure obtains practically over the walls, and S' = -|S 

sactly. This arrangement is known as " Borda's mouthpiece " 
Another application of (3), much used in aeronautics and engi- 

eering, is to the measurement of the velocity of a stream, e g of 

ic relative wind in an 

iioplme. The quantities / ' i i ^ 

and p -\- Ipq* are mea- > 
ired independently, and 
icir diffeience detei mines 

A fine tube, closed at ( 
le end and connected with 

pressure-gauge at the F ' s 

her, points up the stieam 

) as to interfere as little as possible with the motion, and contains 
few minute holes in its side, at a little distance from the closed 
id; the gauge therefore gives the value of p On the other hand, 
i open tube diawn out at the end almost to a point, and 
>nnected to a second gauge, will give the value of p -f %pq z at a 
tort distance ahead of the vertex. For if p' be the pressure at the 
*tex itself, where the velocity is arrested, we have p + %pq z 
p', for points on the same stream-line. The two contrivances 
e often united (as in the figure) in a single appliance known as a 
Pitot and static pressure tube". 



60 THE MECHANICAL PROPERTIES OP FLUIDS 

2. Two-dimensional Motion, Stream-function 

There are two types oi stream-line motion which ate specially 
simple and important. We take first the two-dimensional type, 
where the motion is m a system of parallel planes, and the velocity 
has the same magnitude and direction at all points of any common 
normal. It is sufficient then to confine oui attention to what takes 
place in one of these planes. Any line drawn in it may be taken to 
represent the portion of the cylindrical surface, of which it is a 
cross-section, included between this plane and a parallel plane at 
unit distance from it. By the " flux " across the line we understand 
the volume of fluid which in unit time crosses the surface thus defined 
Now taking an arbitrarily fixed point A and a vaiiable point P, the 
flux (say from right to left) will be the same across any two lines 
drawn from A to P, provided the space be- 
tween them is wholly occupied by fluid 
This is m virtue of the assumed constancy of 
volume The flux will theiefoie be a function 
only of the position of P, it is usually denoted 
Fig o by the lettei ?//. It is evident at once from 

the definition that the value of ^ will not altei 
as the point P describes a stream-line, and theiefore that the equation 

i/r = constant . . (4) 

will define a sti earn- line For this icason ;/ is called the stt cam- 
function. If P' be any point adjacent to P, the flux across AP' will 
differ from that across AP by the flux acioss PP', whence, wilting 
PP' = 8s, we have 8i/r q n Ss, where q n is the component velocity 
normal to PP', to the left. Thus 



where dfi/ds is the gradient of ^ in the direction of 8s. This leads 
to expressions for the component velocities w, v parallel to rectangulai 
co-ordinate axes. If we take 8s parallel to Oy we have q n = w, 
whilst if it be taken parallel to Ox we have q n = v. Thus 




%? 5 

These satisfy the relation 

du . dv 



MATHEMATICAL THEORY OF FLUID MOTION 61 

which is called the equation of continuity. It may be derived other- 
wise by expressing that the total flux across the boundary of an 
elementary area BxBy is zero. Again, if we use polar co-ordinates 
', 6, and take Bs (= Br) along the radius vector, we have q n = v, 
where v denotes the transverse velocity; whilst if Ss (== rS6) be at 
ight angles to r, q n = w, the radial velocity. Hence 

difj di[t /o\ 

* = ~m* v " fr ( } 

t follows that 

f v 0V f \ 

"f\" V "/ 1^ /"v/i ~~"~~ > * * i**o** \;7/ 

9r 30 

vhich is another form of the equation of continuity. 

It is to be remarked that the above definition of /r is purely 
,eometncal, and is merely a consequence of the assumed incom- 
tressibihty of the fluid. If we make any assumption whatever as 

the form of this function, the formulas (6) or (8) will give us a 
lossible type of motion; but it by no means follows that it will 
>e a possible type of permanent or steady motion. To ascertain the 
ondition which must be fulfilled in order that this may be the 
ase we must have recourse to dynamics, but before doing this it is 
onvenient to introduce the notions of circulation and vorticity. 

The circulation round any closed line, or circuit, in the fluid is 
tie line-integral of the tangential velocity taken round the curve 

1 a preset ibed sense In symbols it is 

ds, . . .(10) 

/here x is the angle which the direction ot the velocity q makes with 
tiat of the line-element 8s In lectangulai co-oidmates, resolving 
and v in the direction of Bs, we have 



dx , dy 

q cosx = u + v-+, 
ds ds 



o that the circulation is 



Kdx . dy\j [/ j , j \ t \ 

u~ + v~- )ds, or (udx + vdy) (i i) 
A ds] i 

t will appear that the circulation round the contour of an infinitely 

mall area is ultimately proportional to the area. The ratio which 

bears to the area measures (m the present two-dimensional case) 

ic vorticity] we denote it by . Its value, in terms of rectangular 



62 THE MECHANICAL PROPERTIES OF FLUIDS 

co-ordinates, is found by calculating the circulation round an 
elementary rectangle PQRS whose sides aie $x and Sy. The por- 
tions of the line-integral (n) due to PQ and RS 
S * R aie together equal to the difference in the cor- 

icspondmg values of uSx, i.e. to -~SySx. 
\ I The portions due to QR and SP are in like 

manner equal to SxBy. Equating the sum to 
p >- Q x 

Fig. 7 &*%> we have 

c. dv __ du , ^ 

dx 9y' 
or, from (6), 

f\e* i r\n t 

o O^/r O^ift 



The value of at any point P is related to the aveiage lotation 
relative to P of the particles in the immediate neighbourhood To 
examine this, we calculate the circulation round a circle of small 
radius r having P as centre The velocity at any point Q on the circum- 
ference may be regarded as made up of a general velocity equal to 
that of P, and the velocity relative to P. The former of these contri- 
butes nothing to the required circulation The latter gives a tan- 
gential component cur, where cu is the angular velocity ol QP The 
circulation is therefore 



fZtr rZt 

I corrdO = r 2 / 
./ o J o 



where 6 is the angular co-ordinate of Q. Since the same thing is 
expressed by nr 2 , we have 



TTJ 



which is twice the average value of co on the circumference. For this 
reason a type of motion in which is everywhere zero, i.e in which 
the ratio of the circulation in every infinitesimal circuit to the area 
within the circuit vanishes, is called irrotational. 



MATHEMATICAL THEORY OF FLUID MOTION 63 



3. Condition for Steady Motion 

We can now ascertain the dynamical conditions which must 
e satisfied in order that a given state of motion may be steady, 
'or this purpose we consider the forces acting on an element, 
'QQT', of fluid included be- 
tfeen two adjacent stream-lines 
tid two adjacent normals. The 
itter meet at the centre of cur- 
ature C. We write PQ = &, 
P' = Bn, PC = R. The mass 
f the element is therefore p$s8n. 
'he forces acting on it may be 
ssolved in the direction of the 
ngent and normal, respectively, 
1 the stream-line PQ. Tan- 
in tial resolution would merely 
ad us again, after integration, 
Bernoulli's equation (2). 




ormal resolution 
e help of (i), 



gives, 







with 



Fig 8 



SsSn 



3V 
dn 



here dp fin and 3V /dn are the gradients of p and V in the direction 
- Hence a z 



lie cnculation lound the cucuit PQQ'P' will be equal to Ss8n 
calculating this cnculation, we may neglect the sides PP', QQ' 
>ce they aie at right angles to the velocity The contributions of 
e remaining sides are 

q$s and - 



lere $s r = P'Q'. Now from the figure we have 
8/ = CP' ^ R - Sn 
Ss ~ CP ~ " R ' 

that the circulation is 



64 THE MECHANICAL PROPERTIES OF FLUIDS 
omitting teims of higher oidei than those retained Hence 

=,, _^ + l .......... ( I6 ) 

if <-\ I Tpk \ / 

dn R 
The formula (15) may now be written 

-pql ..... (I?) 



Comparing this with (2), we have 

8C = pqt$n, .... . . (18) 



where C is the quantuy which was proved bcloie to be constant 
along any stream-line, but will in general vary fiom one stieam-lme 
to another. If we fix our attention on two consecutive stieam-lmes, 
SC will be a constant, and q$n will also obviously be constant The 
dynamical condition for steady motion is therefoie that the vorticity 
should be constant along any stream-line. When it is fulfilled, the 
distribution of pressure is given by (2) and (17) We may expiess 
the result otherwise by saying that any fluid element letains its 
voiticity unchanged as it moves along. This is a paiticular case oi 
a theoiem in vortex-motion to be proved later 

An obvious example is that of fluid rotating with uniform angulai 
velocity co about a vertical axis, and subject to giavity The law ot 
distribution of pressure may be deduced from (17), or moie simply 
from first principles. If r be the ladms of the ciicular path of a 
small volume Q, the resultant force upon it must be radial, of amount 
pQo>V. Hence, and since there is no veitical acceleiation, we have 

2 ty ^/> / \ 

*"' = <>=--' ...... (-9) 



the positive direction of % being that of the upward veitical. It 
follows that 

p = %pa> z r z gpz -\- constant ......... (20) 



The free surface (p p Q ) is therefore the parabola 



x = - 



if the origin of x is where the free surface meets the axis (r o). 



MATHEMATICAL THEORY OF FLUID MOTION 65 

If we imagine the fluid contained within the cylindiical surface 
= a, rotating in the above manner, to be surrounded by fluid 
loving irrotationally, we have in the latter region dq/dr + q/r = o, 



om (16), or x 

#r a= constant = w^ 

ernoulli's equation then gives 

p = constant gpz \p 

he equation to the free surface is therefore 

ry _L 

~ o ~ 



(22) 
(23) 



g 

here the additive constant has been chosen so as to agree with 
i) when r = a It appears that these equations also give the 




Fig 9 

ne value of dz/dr for r = a Putting r = oo in (24), we find 
it the depth of the dimple formed on the free surface is cuPaP/g. 

4. Irrotational Motion 

We proceed now to consider more particularly the case of irro- 
ional motion. The condition for steady motion is fulfilled auto- 
itically if = o everywhere, provided, of course, the necessary 
undary conditions are satisfied, as they are in the case of the 
w of a liquid past a stationary solid. The geometrical condition 
i) reduces to , 2 , , 2 , 



(D312) 



66 THE MECHANICAL PROPERTIES OF FLUIDS 
or, in polar co-ordinates, 

?! -1- I ^ -t- I ^ = o (26) 

dr z ^ r $r r 2 d6* ............ V ' 

and the pressure- distribution is such that 

P 4- |p# 2 -f pV = constant ....... (27) 

throughout the fluid. The particular value of the constant is for 
most purposes unimportant, since the addition of a unifoim pres- 
sure throughout does not alter the resultant force on any small 
element of fluid, or on an immersed solid. 

Some simple solutions of (25) or (26) aie easily obtained. Thus 

= \] y = Ur smS ...... (28) 

gives a uniform flow with velocity U from left to right. Again, 
take the case of symmetiical radial flow outwards from the ongm. 
The stream-lines are evidently the radii, so that ifi is a function of 
Q only. Since the total flux outwards across any circle r constant 
must be the same, we have froin (8) 

9.4 



- = constant = m, say, 
rod 

, m n t -. 

or = 0. ..... (29) 

27T 

If m be positive we have here the fictitious conception of a line-source 
which emits fluid at a given rate If m be negative we have a sink 
Since (29) would make the velocity infinite at the origin, these 
imaginaiy sources and sinks must be external to the legion occu- 
pied by the fluid. The formula (29), for instance, would be 
realized by the expansion of a circular cylinder whose axis passes 
through O. Again, since the differential equations (25) and (26) 
are linear, they are satisfied by the sum of any number of sepaiate 
solutions. For instance, the combination of a source at A, and an 
equal sink at a point B to the left of A, gives 

= --&-**), ............... (30) 

27T 

wheie 1} a are the angles which the lines drawn from A and B to 
any point P make with the direction BA. Since 1 2 = APB, 
the lines ^ = constant are a system of circles through A, B. This 



MATHEMATICAL THEORY OF FLUID MOTION 67 

ind of motion would involve infinite velocities at A and B, but if 
e combine (28) with (30) we get the flow past an oval cylinder 
hich encloses the imaginary source and sink. If the points A and 
be made to approach one another, whilst m increases so that the 
roduct mAE is constant, we have ultimately t Z APB = 
B sm#/r. We thus get the form 



ombining this with (28), we have 

/ - Ur + - 



(32) 




Fig 10 

ie stream-line j/r = o now consists of the radii 6 = o, 6 = 
d the circle r a, provided C = Ua z . The formula 

a 



d, = - \j 



(r- 
\ 



-\smO 
j 



(33) 



jrefore gives the flow past a circular cylinder. The normal velocity 
the surface is of course zero, whilst the tangential velocity is 



^ = - U/i + *\in0 
dr \ r 1 / 



(34) 



cutting the external forces, if any, represented by V, which have 
rely an effect analogous to buoyancy, the pressure at the cylinder 

p = constant 2/>U 2 sin 2 ............. (35) 



68 



THE MECHANICAL PROPERTIES OF FLUIDS 



Since this is unaltered when is replaced by 0, or by + (TT 6}, 
it is evident that the stream exerts no resultant force on the cylinder. 
Some qualification of this result will be given presently. Meantime 
we note that if we superpose a general velocity U from light to left, 
we get the case of a cylinder moving with uniform velocity (and 
zero resistance) through a fluid which is at rest at infinity. The 
stream-function then has the form 




Fig n 



so that the relative sti cam-lines are portions o the circles r = 
Csinfl, which touch the axis of x at the origin. If we calculate the 
square of the velocity from (36), we find 



(^ l\ ' 
Oijj\' 
to) 



W ' \rW/ V* 

The total kinetic energy of the fluid is therefore 



(37) 



where M' is the mass of fluid displaced by the cylinder. The effect 



MATHEMATICAL THEORY OF FLUID MOTION 6g 



f the presence of the fluid is therefore virtually to increase the 
tertia of the latter by M'. 

Another simple type of motion is where the fluid moves in 
mcentric circles about O. The velocity is then a function of r 
ily If the motion is irrotational we must have, by (22), 



qr 



nr 

UJ. 



constant 





Or 



K 

__ 

, 

27T 



iere K is the constant value of the circulation round 0. Thus 



additive constant being without effect. This corresponds to the 
se of a concentrated hne-vortex at the origin, and would give 
mite velocity there. For this icason (38) can only relate to cases 
iere the origin is external to the space occupied by the fluid 

The combination of (33) and (38) makes 



- U r 



-} sind + -1 logr 
r J 2TT 



(39) 






Fig 12 



! tangential velocity at the cylinder is now 
= - 2U 



of 



2ira 



70 THE MECHANICAL PROPERTIES OF FLUIDS 

whence 

p = constant 2pU 2 sin 2 + /> sm0 ...... (40) 

TtCL 

The last term is the only one which contributes to a resultant force. 
Since it is the same for 9 and -n 0, there is on the whole no force 
parallel to Ox. The force parallel to Oy is 



r 

J 



- p/clL. .(41) 



This resultant effect is due to the fact that (if K be positive) the 
circulation diminishes the velocity above the cylinder and increases 
it below, and that a smaller velocity implies (other things being the 
same) a greater pressure. It may be shown that the icsult is the 
same for a cylinder of any form of section, as might be expected from 
the fact that it does not depend on the radius a. This theoiem is 
the basis of Prandti's theory of the lift of an aeroplane. 

5. Velocity-potential 

We may imagine any area occupied by fluid to be divided by 
a double series of lines ciossing it into infinitesimal elements. The 
circulation round the boundary of the area will be equal to the 
sum of the circulations round the various elements, piovided these 

circulations be estimated in a con- 
sistent sense. For, in this sum, a 





Fig 13 



Fig 14 



side common to two adjacent elements contributes amounts which 
cancel. Hence if the motion be irrotational the circulation round 
the boundary of any area wholly occupied by fluid will be zero. 
We have here assumed the boundary to consist of a single closed 



MATHEMATICAL THEORY OF FLUID MOTION 71 

urve. If it consists of two such cuives, what is proved is that the 
urn of the circulations round these in opposite senses is zero. In 
ther words, in irrotational motion the circulation in the same sense 
s the same for any two circuits which can by continuous modifica- 
ion be made to coincide without passing out of the region occupied 
y the fluid. For example, in the case to which (38) refers, the cir- 
ulation in any circuit which embraces the cylinder is /c, whilst that 
i any other circuit is zero. 

This leads to the introduction of the function called the velocity- 
otentzal, in terms of which problems of irrotational motion are often 
iscussed. This is defined by the integral 



& 

== I (udx -f- vdy) . (42) 

J A 



A 

ken along a line diawn from A to P. The integral has the same 
due for any two such lines, such as ABP, ACP in the figure, pro- 
ded the space between them is fully 
:cupied by fluid. For, reversing the 
rection of one of these lines, the 
iths ABP, PCA together form a 
osed ciicuit, round which the cncu- 
tion is zero It follows that so long 
A is fixed, cf) will be a function of 
e position of P only. If P' be any Fig 15 

>int adjacent to P, the mciement of 

in passing from P to P' is S</> = qfis, where q t is the component 
locity m the direction PP', and PP' = 8s. Hence 




lere d(f>/ds is the gradient of <j> in the direction PP'. For instance, 
rectangular co-ordinates, putting first Ss = Sx, and then = Sy, 
' have 

dA d< / . . v 

U=. __L i} = ~ . . (44) 

dx' dy 

nilarly, the radial and transverse velocities in polar co-ordinates 
; given by 

d<f) d<f> f \ 

u = ~ x, v -^ (45) 

dr red 



72 THE MECHANICAL PROPERTIES OF FLUIDS 
From (7) and (44) we deduce 

8V . 8V 

sl + iy ' ....... ......... 

whilst in polar co-ordinates, from (9) and (45) 
9 / 8fv . i 8V 



, . 

(47) 



It is the similarity between these iclations and those met with in 
the theories of attractions and electrostatics that has suggested the 
name " velocity-potential ". For the same reason the cuives foi 
which <j) is constant are called equipotential lines If in (43) Ss be 
taken along such a line we have q s = o, showing that the equipotential 
lines cut the stream-lines at light angles If on the other hand Sn 
be the perpendicular distance between two adjacent equipotential 
lines, we have 80 == q$n If, therefoie, we imagine a whole 
system of such lines to be drawn for equal small increments S(f>, 
the perpendicular distance between consecutive lines will be evciy- 
where inversely propoitional to the velocity. If, further, we suppose 
the stream-lines to be drawn for intervals Si/f each equal to S</>, we 
have Si/r = #3$', where Ss' is the interval between consecutive 
stream-lines of the system. Hence 8s' = 8n, showing that the 
stream - lines and equipotential lines drawn for equal inciemcnts 
of the functions will divide the region occupied by the fluid into 
infinitesimal squares. 

The functions and fi are connected by the relations 



=, = - 

dx 9y' dy fix 

in which the equations (25) and (46), expressing the incompressi- 
bihty and the absence of vorticity, are implied. If we write 



For this makes 

dzo .- . dw 



= a -f zy, ........... (49) 

are satisfied by any assumpti 
=/(*)... ....... (50) 



where i +/( i), the relations (48) are satisfied by any assumption 
of the form 



MATHEMATICAL THEORY OF FLUID MOTION 73 

vhence, substituting the value of w, and equating sepaiately real and 
maginary paits, we repioduce (48). 
For example, if zo Uz, we have 

I = ~U*, = -Uy, ............ (52) 

xpressing a uniform flow paiallel to Ox. Again, if w ~ CJz 

- i sin0) ..... (53) 



This corresponds to (36), if C = U<z 2 , and shows that in the 
ase referred to 

(54) 



r 

A moie general assumption is 

to = Cs", 
or $ + lift = C(x + y) B = Cr n (cosn6 

The stream-function is now 

= O" 



/hich vanishes both for 6 o and 6 a, provided n rr/a. 
Baking these lines as fixed boundaries we have the flow in an angle, 
r round a salient, according as a > TT. The radial and transverse 
elocities are, by (8), 

nCr n ~ l cosnd and nCr n ~ l smnd, 



espectively If a < TT, n > i , and these expressions vanish at the 
ertex where r = o If, on the other hand O.>TT, n<i, and 
tie velocity theie is infinite Even if the salient be rounded off, 
tie velocity may be very great, with the result that the pressure falls 
luch below the value at a distance It is otherwise obvious that if 
tie fluid is to be guided round a sharp curve there must be a lapid 
icrease of pressuie outwards to balance the centrifugal force If 
bis is not sufficient a vacuum is formed and " cavitation " ensues. 

If w = C log#, where C is real, 

^ -f njj = C log(x + ty) = C logre' 9 = C logr + iC6. (55) 

Tiis represents a line-souice of strength m, if to agree with (29) 
re put C WZ/27T. The corresponding value of ^ is 

(D312) 4 * 



74 THE MECHANICAL PROPERTIES OF FLUIDS 
If on the other hand C is a pure imaginary, zA, say, 

^ -j- ^ = ~ AO + tA logr (57) 

This repiesents the case of the line-vortex to which (38) refers, if we 
put A = K/ZTT, and so make 

(58) 



The function <{> has an important dynamical inteipietation. Any 
state of motion in which there is no circulation in any circuit, and 
in which, theiefore, cj> has a definite value at every point, could be 
generated instantaneously from rest by a proper application of 
impulsive pressures over the boundary. For the icquisite condition 
for this is that the resultant of the impulsive pressures (oi) on the 
surface of any small volume Q should be equivalent to the momen- 
tum acquired by this. Hence if q s is the component velocity in the 
direction of any linear element Ss we must have 



which is satisfied if 

& = P^ .............. (59) 

Hence <f> determines the impulsive pressure requisite to stait the 
actual motion in the above manner. 

As an example, we may take the case of a cylinder moving through 
a laige mass of liquid, without circulation, to which the foimula (54) 
refers. The resultant of the impulsive piessuies on the surface 
of the cylinder is paiallel to Ox, of amount 

= M'U, . .(60) 

if M' = 7r/>a 2 as befoie. The total impulse which must be given to 
the cylinder to start the motion is therefore (M + M')U. This 
confirms the former result that the inertia of the cylinder is viitually 
increased by the amount M'. 



MATHEMATICAL THEORY OF FLUID MOTION 75 

6. Motion with Axial Symmetry. Sources and Sinks 

The second type of motion to which reference was made on 
54 is where the flow takes place in a system of planes passing 
rough an axis, which we take as axis of x, and is the same in each 
ch plane. We denote by x^y the co-oidmates in one of these 
ines, by r distance from the origin, and by 6 the angle which r 
ikes with Ox. The conditions for steady motion are obtained by 
e previous process. Resolving along a stream-line we should be 
i to Bernoulli's equation (2); whilst the normal resolution in an 
ial plane yields equations of the same form as (15) and (17), pro- 
led now denotes the vorticity in that plane The inference 
to the distribution of vorticity is however altered. The space 
tween two consecutive stream-lines now represents a section of 
thin shell, of revolution about Ox, and the flux in this is accord- 
*ly q 27ryS Comparing with (18), w.e see that along any stream- 
e must vary as y. We may conceive the fluid as made up of 
nular filaments having Ox as a common axis. The section of such 
filament, as it moves along, will vary inversely as y, hence the 
oduct of the vorticity into the cross-section must remain constant. 
us is a particular case of a general theorem that the strength of a 
rtex-filament (in this case a vortex-ring) lemams unaltered as it 



If = o, the aigument for the existence of a velocity-potential 
11 hold as befoie One or two simple cases may be noticed If 
: imagine a point-source at O, the flux outwaids across any con- 
Qtnc sphencal surface of radius r must be equal to the output 
') per unit time whence 

d<{> . m i ., , 

-^ ATT?-" = m, or d> = -- ..... (61) 
or 47? r 

We may apply this solution to the collapse of a spherical bubble. 
R be the radius at time t, we have 

rh R2 ^ R /< x 

#- T -^ - - (62) 

ice this makes 90 Jdr dR/dt for r R. The corresponding 
netic energy of the fluid is 



y6 THE MECHANICAL PROPERTIES OF FLUIDS 

If pQ be the pressure at a distance, the rate at which work is being 
done on the fluid enclosed in a spherical surface of laige radius r is 

-p q4<7rr* = - 4 77/> R a ~, (64) 

the pressure inside the bubble being neglected. Equating the rate 
of increase of the energy to the work done, 



dt 

whence 

W-* n / W-J.X \ o A'ft/TTfc Q T^ ON //" /" \ 

R3( \ = fP(R 3 R 3 ), (66) 



where R is the initial radius of the cavity It is not easy to integrate 
this further in a practical form, but the time of collapse happens to 
be ascertainable, it is 

' (6?) 

Thus if pQ be the atmospheric pressure, and R = i cm., r is less 
than the thousandth part of a second. The total eneigy lost, or lathci 
converted into other foims is, from (63) and (66), 

irf> R 8 . . . ..(68) 

3 

In the particular case referred to, this is 4 19 X io 6 eigs, 01 o 312 
of a foot-pound 

The expansion of a spherical cavity owing to the piessuie of an 
included gas can be treated in a similar way This illustrates, at all 
events qualitatively, the early stages of a submanne explosion 
The potential energy of a gas compressed under the adiabatic con- 
dition to volume v and pressure p is pvf(y i), wheie y is the 
ratio of the two specific heats. If p be the internal picssure when 
the radius of the cavity is R, and p its initial value, we have by the 
adiabatic law . /~ v 3V 



The potential energy is therefore 



MATHEMATICAL THEORY OF FLUID MOTION 77 
xpressmg that the total energy is constant, we have 



here 

/* ^^; ../ f *f\ /n ) 1^72 I 

his quantity c , which is of the dimensions of a velocity, is a 
easure of the rapidity with which the changes take place. It is not 
isy to carry the solution further except in the particular case of 

= -. If we write 

3 R/R = i + ,, (73) 

5 have then . 

at R 
tience 

bis gives the time taken by the radius of the cavity to attain 
y assigned value R The following table gives a few examples. 

R/RO C O */R O I 

1 O 

2 2 64 

3 6-27 

4 11-76 

5 1942 

3 a concrete illustiation, suppose the initial diameter of the cavity 

be i m., and the initial pressure p Q to be 1000 atmospheres, so 

at c = 3-16 X io 4 cm /sec. We find that the radius is doubled 

-rrhj- sec., and multiplied five-fold m about -$V sec. It must be 

membered that m this investigation, as in the preceding one, the 

iter has been assumed to be incompressible With an initial 

ternal piessure of the order of 10,000 atmospheres, we obtain 

lues of dR/dt compaiable with the velocity of sound m water. 

tie influence of compressibility then ceases to be negligible. 



78 THE MECHANICAL PROPERTIES OF FLUIDS 

The combination of a somce m at a point A and a correspond- 
ing sink m at B gives 

(76) 



If we imagine the points A and B to approach one another, whilst 
the product mBA is constant (= /A), we have ultimately r z r^ = 

AB cos#, and n 

, u. cose/ / x 

< = ~ -o- .......... 77) 

47? r 2 

We have here the conception of a double-source. If we combine 
this with a uniform flow $ = U# = U> cos# parallel to Ox 
we have 



, 

= - ( 

\ 



477T 2 / 



cos0 



This makes d(/>]dr = o for r = a, provided fi = 
The formula 

(78) 



therefore gives the steady flow past a sphere of radius a. The tan- 
gential velocity at the surface is 



= - 
rod 



and the pressure is accordingly 

p = constant fpU 2 sm a # .......... (79) 



Since this is the same when 6 is replaced by TT 6, the resultant 
effect on the sphere is ml. If we superpose a general velocity U, 
we get the case wheie the sphere is in motion with velocity U in 
the negative direction of x; thus 

............ (80) 



If we imagine this motion to be produced instantaneously liora 
rest, the impulsive pressure of the fluid on the sphere, in the direc- 
tion of ^-negative, is 

/ R r 

I m cosd.ZTTa sindadd I pfi cosd 2ira s'mOadd 
Jo Jo 

(81) 



MATHEMATICAL THEORY OF FLUID MOTION 



79 



or |MTJ, where M' is the mass of fluid displaced. The impulse 
which must be given to the sphere to counteract this is fM'U, 
and the total impulse in the direction of the velocity is (M + $M')U, 
where M is the mass of the sphere itself. It is a proposition in 
Dynamics that the kinetic energy due to a system of impulses is 
got by multiplying each constituent of the impulse by the velocity 
produced in its direction, and taking half the sum of such products. 
In the present case this gives $(M -f |M')U 2 . The case is analogous 
to that of the cylinder, already treated, except that the virtual 
addition to the mass is $M' instead of M'. 

This result, viz. that the effect of a frictionless liquid on a body 
moving through it without rotation consists merely in an addition 
to its inertia, is quite general. Whatever the form of the body, the 
impulsive pressme necessary to start the actual motion of the fluid 
instantaneously from rest will evidently be proportional to the 
velocity (U), and the reaction on the body in the direction of motion 
will therefore be &M'U, where k is some numerical coefficient. 
The impulse necessary to be given to the solid is therefore 
(M + M')U. A similar conclusion would follow from the 
consideration of the energy produced The value of k will, of 
course, depend on the form of the solid and the direction of its 
motion. The following table gives values for the case of a prolate 
ellipsoid, the ratio c\a being that of the longer to the shorter semi- 
diameter The column under " 7^ " relates to motion " end-on ", 
and that under " k z " to motion " bioadside-on " 

c\a &! k z 

i (sphere) | | 

1-5 0305 0621 

20 o 209 o 702 

30 O 122 O 803 

40 o 082 o 860 

50 o 059 o 895 

60 o 045 o 918 

70 0036 0933 

8 o 0-029 "945 

90 o 024 o 954 

10 o 0-021 0-960 

oo (cylmdei) o i j 



Any line AP drawn in a plane through the axis represents an 



8o THE MECHANICAL PROPERTIES OF FLUIDS 



annular portion of a surface of revolution about Ox The flux 
across this portion, say from light to left, will be the same for any 
two lines from A to P, provided the space between them is occupied 
by fluid. If A be fixed, this flux will therefore be a function 
only of the position of P; we denote it by zmft. If PP' be a linear 
element 8s, drawn in any direction, the flux across the surface 
geneiated by its revolution about Ox will be 



,(82) 



where q n is the velocity normal to Ss Hence 

i 9 

** 



y * 

It was to simplify this formula that the factor 2-77 was introduced 
in the definition of $. As paiticular cases of (79), the component 
velocities parallel and perpendicular to Ox are 



= 



V = - 

y 



(83) 



The lines for which 




Fig. 1 6 



is constant are stream-lines, and /r is called 
the stream-function 

To find for the case of a 
point-source, we calculate the 
flux act oss the segment of a 
spherical suiface, with OP as 
radius, cut off by a plane 
* through P perpendiculai to Ox 
The radial velocity across this 
segment is mf^.7rr z , and the aiea 
is 27rr(r A?), wheie r = OP, 
x ON. Hence, omitting an 
additive constant, the flux in the 
desired sense is 



, or tft 



m 
qrr 



cos#. 



(84) 



The combination of an equal source and sink at A and B 
gives 

m 



iff = 



4" 



COS0 2 )>. 



MATHEMATICAL THEORY OF FLUID MOTION 81 

whilst if A and B are made to approach coincidence in such a way 
hat wAB = ft, we have ultimately 

8 (cos0) = sm0 80 = sin0 (AB sin0)/r, 

and therefore $ = (86) 

477 r 

'or a uniform flow parallel to Ox, we have 2$ = Uy 2 , and if 
re supeipose this on (85) or (86) we get stream-line forms, one of 
rhich may be taken as the profile of a stationary solid in the stream, 
'or instance, combining with (86), and putting fj, = - 



= - 4:2 _ f 
2 \ r 



""he line iff o now consists of the circle r = a and of the portions 
f the axis of x external to it. If we now remove the uniform flow 
r e get the lines of motion due to the sphere moving in the direction 
f ^-negative with velocity U. 

The process just indicated admits of great extension By taking 
senes of sources and sinks, not necessarily concentrated in points, 
long the axis of x, subject to the proviso that the aggregate output 
< zero, and superposing a uniform flow, we may obtain a variety of 
urves which may serve as the profile of a moving solid This pro- 
sdure was originated by Rankine from the point of view of naval 
rchitecturc, and has recently been applied to devise profiles which 
mtate those of airships. Since the motion of the fluid is known, 
ic pressure distribution over the suiface can be calculated and 
ompaied with model experiments 

7 Tracing of Stream-lines 

Theie aie vanous methods by which diawmgs of systems of 
tream- lines can be constructed. For example, suppose that the 
trcam-function consists of two parts ^ 15 /r 2 , which are themselves 
eadily tiaced Drawing the curves 

J/T! = ma, ifj z = na, 

fheie. m, n aie integeis, and a is some convenient constant (the 
mailer the better), these will divide the plane of the drawing into 
curvilinear) quadnlaterals. The cuives 

* == na 



82 THE MECHANICAL PROPERTIES OF FLUIDS 

will form the diagonals of these quadrilaterals, and are accordingly 
easily traced if the compartments are small enough. For instance, 
in the case of (33), where we may put U = i, a = i, without 
any effect except on the scale of the diagram, we should trace the 

straight lines . n 

r sine? = ma, 

which aie parallel to the axis of x and equidistant, and the circles 

r = sin# 
no. 

Another method is to write the equation (as above modified) 
in the form 



and to tabulate the function i/(i i/r 2 ) for a series of equidistant 
values of r, beginning with unity. This is easily done with the 
help of Barlow's tables. The values of y where a particulai stream- 
line crosses the corresponding circles are then given by 




Giving i]i in succession such values as 0*1, 0-2, 0-3, ... a system of 
stream-lines is leadily drawn The same numerical work comes in 
useful in the case of (39) A similar process can be applied to tracing 
the stieam-lmes past a sphere, to which (87) refers 

A more difficult example is presented by equation (99) later. 
Nothing is altered except the scale if we write this in the form 

? i 2 

^z 
whence 

(x -f- i ) -|- y .^ 

i \2 i 2 c > 

and therefoie 

r 5 -f- i = 2/ , ,, + J = 2,sc coth^-llr. 



The hyperbolic function on the right-hand has been tabulated, so 
that we can calculate the values of r (the distance from the origin) 



MATHEMATICAL THEORY OF FLUID MOTION s 3 

at the points where any given stream-line curve cuts the lines 
x = constant. 

8. General Equations of Motion 

The general equations of hydrodynamics have so far not been 
required. To obtain them in their full three-dimensional form we 
denote by u, v, w the component velocities parallel to rectangular 
axes at the point (, y t #) at the time t. They are therefore functions 
Df the four independent variables x, y t #, t. If we fix our attention 
m a particular instant t , their values would gives us a picture 
Df the instantaneous state of motion throughout the field. If on 
he other hand we fix our attention on a particular point (a? , y , # ) 
n the field, their values as functions of t would give us the history 
)f what takes place at that chosen point. We introduce a symbol 
D/Dt to denote a differentiation of any property of the fluid con- 
sidered as belonging to a particular particle. Thus D/D* denotes the 
component acceleration of a particle parallel to Ox; this is to be dis- 
inguished from Bu/dt, which is the rate at which u varies at a parti- 
cular place. The dynamical equations are obtained by equating the 
ate of change of momentum of a given small portion of the fluid 
o the forces acting on it. Considering the portion which at time t 
iccupies a rectangular element SacSyS*, we have, resolving parallel 
o Ox 

P S%S* D " - - 

L)t ox 

vhere the first term on the right hand is the effect of the fluid pres- 
ures on the boundary of the element, as determined by (i), whilst 
he second term is due to extianeous forces (such, for example, as 
gravity) which aie supposed to be conservative, V being the potential 
nergy per unit mass. Thus we find, 

DM dp __ 8V > 

P D* " dx P bc' 
Dv dp 9V 

p Dt = -5TV 

Dzy _ __dp_ 9V 
P Dt ~~ dz P 8F; 

To find expressions for Du/Dt, &c., let P, P' be the positions 
iccupied by a particle at two successive instants %, t z . Let %, / 



84 THE MECHANICAL PROPERTIES OF FLUIDS 

be the values of u at the points P, P', icspectively, at time t it and 
u &) # 2 ' the corresponding values at time t z . The average acceleration 
of the particle parallel to Ox in the interval t z ^ is theiefore 

u z ' ~ Wj u z HI . u% u z 



The limiting value of the left-hand side is Dw/Dtf; that of the first 
term on the right is dujdt, the rate of change of u at P. Again, 
u z ~ w a i s tne difference of simultaneous component velocities at 
the points P' and P, so that 

, du TVTJ/ du , . .v 

u 2 ' - U2 = _.PP' == _(/ 2 ~ tj, 

where # is the resultant velocity 

x/(tt a -f v 2 + w 2 ), 
and 9#/3.s is the gradient of u in the direction PP'. Thus 



n a 

Dt at os 

Now if I, m, n be the direction cosines of Ss, 

dti ,du , du . du 

/ - -f- ?W ~ + W ~5 

oi 1 ox ay az 

and/g 1 = u, mq = ^, raw = ;. Hence, finally, 

DM 3^ . du . du . du , x 

D* = -s +I fe + ^; + "5 ........ (90) 

Similar expressions are obtained for DvjDt, Dw/Dt Substituting 
in (88) we get the dynamical equations in their classical foim 

To these must be added a kinematical i elation, which expi esses 
that the total flux outwards across the boundary of the element 
SflSyS^ is zero. The two faces perpendicular to O^; give uBySx, 

on 
and (u -f ;r-8#)SyS# respectively, the sum of which is dujdxBxSySz 

C/JC 

Taking account in like mannei of the flux across the remaining 
faces, and equating the total to zero, we have the equation of con- 
tinuity &< &> 8 



of which (7) is a particular case. 



MATHEMATICAL THEORY OF FLUID MOTION 85 
When the motion is irrotational we have 



dy dydx dx* 

ind similar relations, so that (90) becomes 

92 = -J!. + j?, ............ (92) 

Dt dxdt ' dx ' 

Vhen this is substituted in (88), it is seen that the dynamical equations 
lave the integral , 

P - = f-W-V + FW, ....... (93) 

p ul 

inhere F(f) denotes a function of t only which is to be detei mined 
>y the boundary conditions, but has no effect on the motion. It is 
vident beforehand that a piessure uniform throughout the liquid, 
ven if it varies with the time, is without effect The occurrence 
f F(t) m the present case is merely a consequence of the fact already 
lentioned that m an absolutely incompressible fluid changes of 
ressure are transmitted instantaneously. 

The equation of continuity (91) now takes the form 

?* + ?V + ?X = o 

8^8/^ 8* 2 

i steady motion d^/dt o, and (93) i educes to our foimer result 
'.?) 

Vortex Motion 

i Persistence of Vortices 

Turning now to the consideration of voitcx motion, the funda- 
icntal theorem in the subject is that the circulation in any circuit 
>ovmg with the fluid (i e. one which consists always of the same 
irticles) does not altei with the time. For, consider any element 

>x of the mtegial r 

I (udx + vdy + wdz}, 

hich expresses the circulation. We have 

, N 
(95) 






86 THE MECHANICAL PROPERTIES OF FLUIDS 

Now D(Bx)jDt is the rate at which the projection on the axis of 
#, of the line joining two adjacent pai tides, is increasing, and is 
therefore equal to SM. Hence, 

Dt Dt p dx dx 

and therefore 



= - 8 + V - ftf 2 ) (96) 

When this is integrated round the circuit, the result is zero. Hence 
l(udx + vdy -f- wdsi) = o. . . ... (97) 

It is important to notice the restrictions under which this is 
proved. It is assumed that the density is uniform, that the fluid is 
frictionless, and that the external forces have a potential. The fitst 
of these assumptions is violated, for instance, when convection 
currents are produced by unequal heating of a mass of water, owing 
to the variation of density. The second assumption fails when the 
influence of viscosity becomes sensible. 

Irrotational motion is characterized by the property that the 
circulation is zero in every infinitesimal circuit We now have a 
general proof that if this holds for a particular portion of fluid at 
any one instant, it will (under the conditions stated) continue to hold 
for that particular portion, whether there be rotational motion in 
other parts of the mass or not. Again, in two-dimensional motion 
we have seen that the circulation round any small area is equal to 
the product of the vorticity into the area Since the area occupied 
by any portion of fluid remains constant as it moves along, we infer 
that the vorticity also is constant. This has already been pioved 
otherwise in the case of steady motion. The value of is, of couise, 
constant along a line drawn normal to the planes of motion. Such 
a line is a vortex-line according to a general definition to be given 
presently, and the vortex-lines passing through any small contour 
enclose what is called a vortex- filament, or simply a vortex. The 
strength of a vortex is defined by the product of the voiticity into the 
cross-section, i.e. by the circulation immediately round it. 

Still keeping for the moment to the case of two- dimensions, 
we have seen that the circulation round the boundary of any area 



MATHEMATICAL THEORY OF FLUID MOTION 87 

occupied by the fluid is equal to the sum of the circulations round 
the various elements into which it may be divided, provided these 
be estimated in a consistent sense. In virtue of the above definitions 
an equivalent statement is that the circulation in any circuit is equal 
to the sum of the strengths of all the vortices which it embraces. 



2. Isolated Vortices 

The stream- and velocity-functions due to an isolated rectilinear 
vortex of strength K have already been met with in (38) and (58) 
The velocity distributions due to two or more parallel rectilinear 
cortices may be superposed. 

Suppose, for instance, we o^ j- Oj- 

lave a vortex-pair composed A B 

)f two vortices A, B of equal Flg I7 

ind opposite strengths K 

tach produces in the other a velocity K/27ra, where a is the 

listance apart, at light angles to AB. The pair advances 

herefore with this constant velocity, the distance apart being un- 

iltered. The lines of flow are given by 

^-log- 1 , (98) 

27r r z 

vhere r^, r z are the distances of A, B respectively from the point P 
o which i/r icfeis The lines for which the ratio r^\r z has the same 
alue are co-axial circles having A, B as limiting points If we super- 
)ose a uniform flow K/z-na in the direction of y negative, the case 
s reduced to one of steady motion, and the stream-function is now 



277 



, . 

(99) 



The stream-line i/r = o consists paitly of the axis of y, where r { = r z 
nd x o, and paitly of a closed curve which surrounds always 
he same mass of the fluid. This portion is canied forward by the 
ortex-pair in the original form of the problem. 

If a flat blade, e g a paper-knife, held vertically, be dipped into 
rater, and moved at light angles to its breadth for a short distance, 
nd then rapidly withdiawn, a vortex-pair will be produced by 
liction at the edges, and will be seen to advance in accordance with 
lie preceding theory. The positions of the vortices are maiked by 



88 THE MECHANICAL PROPERTIES OF FLUIDS 

the dimples produced on the watei suiface In this way the action 
of one vortex-pair on another may be studied. 

The detailed study of vortex motion in three dimensions would 
lead us too far, but a brief sketch of the fundamental relations may 
be given. It is necessary in the first place to introduce the notion 
of vorticity as a vector. Through any point P we draw three lines 
PA, PB, PC parallel to the co-oidinate axes, meeting any plane 
drawn infinitely near to P in the points A, B, C. It is evident at 
once from the figure that the circulation round ABC is equal to the 
sum of the circulation round the triangles PBC, PCA, PAB, pro- 
vided the positive diiection of the circulations be light-handed as 
regards the positive directions of the co-ordinate axes. Now, if 




Fig 18 

/, m, n be the diiection-cosines of the normal drawn horn P to the 
plane ABC, and A the area ABC, the areas of the above tuangles 
are M, mA, nA, icspectively. Hence if f, 77, be the vorticities in 
these planes, i.e. the ratios of these circulations to the respective 
areas, the ciiculation round ABC will be 

( -f )7 + )J (100) 

We may regard f, 77, as the components of a vector w, and the 
expression (100) is then equal to co cos# A where 9 is the angle 
which the normal to A makes with the direction of to. In other 
words, the voiticity in any plane is equal to the component of o> 
along the normal to that plane. 

The value of has been given in (12). Writing down the 
corresponding formulae for and 07, we have altogether 

.. dw dv du dzo v dv du , . 

= _ rj = _ =5 r (lOl) 

ay 02 8ar 9 fix dy 



MATHEMATICAL THEORY OF FLUID MOTION 89 

We have, of course, 




A line diawn from point to point always in the direction of the 
vector (, 77, ) is called a vortex-line. The vortex-lines which meet 
any given curve generate a surface such that the circulation in every 
circuit drawn on it is zero. If the curve in question be closed, and 
infinitely small, the fluid enclosed by the 

surface constitutes a vortex-filament, or c 1 (^ ^ B 1 

simply a vortex. Consider a circuit such A ' 

as ABCAA'C'B'A'A in the figure, drawn 
on the wall of the filament. Since the 
circulation in it is zero, and since the por- 
tions due to AA' and A' A cancel, the circu- 
lation round ABC is equal to that round 
A'B'C'. Supposing the planes of these two 
curves to be cross-sections of the filament, 
we learn that the product of the resultant 
vorticity into the cross-section has the same 

value along the vortex. This product is called the strength of the 
vortex. The dynamical theorem above proved shows that under 
the conditions postulated the strength of a vortex does not vary 
with the time. The constancy of the strength of a vortex-ring has 
already been proved in the case of steady motion. 

The argument by which the circulation m a plane circuit was, 
under a certain condition, proved to be equal to the sum of the 
strengths of all the vortices which it embraces, is easily extended 
(under a similai condition) to the geneial case. 

The most familiar instance of isolated vortices is that of smoke 
rings, which are generated in the first instance by viscosity, but retain 
a certain degree of persistence. A voitex-nng at a distance from 
other vortices, or from the boundanes of the fluid, advances along 
its axis with uniform velocity. The mutual influence of vortex- 
rings is closely analogous to that of vortex-pairs. 

Wave Motion 

I. Canal Waves 

Water-waves are by no means the simplest type of wave-motion 
met with in Mechanics, and the general theory is necessarily some- 
what intricate, even when we restrict ourselves to oscillations of 



go THE MECHANICAL PROPERTIES OF FLUIDS 

small amplitude. The only exception is in the case of what are 
variously called long waves, or tidal waves, or canal waves, the charac- 
teristic feature being that the wave-length is long compared with 
the depth, and the velocity of the fluid particles therefoie sensibly 
uniform from top to bottom, 

Taking this case first, we mquiie under what condition a wave 
can travel without change of form, and therefore with a definite 
velocity. Supposing this velocity to be c, from left to right, we may 
superpose a general velocity cm the opposite direction and so 
reduce the problem to one of steady motion. The theory is now the 
same as for the flow through a pipe of gradually varying section, 
except that the upper boundary is now a free suiface, instead of 
a rigid wall. If h be the original depth, the velocity where the 
surface-elevation is 77 will be 

ch f ^ 

*=-*+} .......... (103) 

The pressure along the wave-profile, which is now a stream-line, 
is given by Bernoulli's equation 



P 



/ \ _ t> 
I TI \ 
constant f# 2 g*rj = constant ik 2 ( i -f- ~) g-q 

\ nj 

= constant $c a (i ^J gr), . (104) 
\ fa f 



approximately, if we neglect the square of 77 [h. This pressuie will 
be independent of 77, provided 

c = *J(gh) ...... (105) 



The required wave-velocity is therefore that which would be acquired 
by a particle falling vertically under gravity, from rest, thiough a 
space equal to half the depth. 

If we now restore the original form of the problem, by imposing 
a velocity c in the positive direction, we have 

ch 77 , ,. 

q = c - - = c-L ........... (106) 

h + T] h 

approximately. The velocity of the water itself is therefore forward 
or backward, according as 77 is positive or negative, i.e. it is forward 
where there is an elevation, and backward where there is a depression. 
The potential energy per unit area of the surface is Igprf, and the 
corresponding kinetic energy is \pfh = %pc*v) 2 /h. Since these are 



MATHEMATICAL THEORY OF FLUID MOTION 91 

qual by (106), the energy of a progressive wave is half-potential 
nd half-kinetic. 

The condition for permanence of form has not, of course, been 
xactly fulfilled in the above calculation. A closer approximation 
o fact is evidently obtained if in (105) we replace h by h -j- if)', this 
rill give us the velocity of the wave-form relative to the water in 
he neighbourhood, which is itself moving with the velocity given 
>y (106), if 7]//j is small. The elevation ^ is therefore propagated 
n space with the velocity 



ippioximately. The more elevated portions therefore move the faster, 
with the result that the profile of an elevation tends to become 
steeper in front and more gradual in slope behind. 

2. Deep-water Waves 

Proceeding to the more general case, we will assume that the 
motion takes place m a series of parallel vertical planes, and is the 
same in each of these, so that the ridges and furrows are rectilinear. 
Fixing our attention on one of these planes, we take rectangular 
axes Ox, Oy, the former being horizontal, and the later vertical 
with the positive direction upwards. The problem being reduced to 
one of steady motion as before, the stream-function will be 

j^ = cy -J- j/fj, . .(108) 

wheie ^ is supposed to be small By Bernoulli's equation 

P L ^ i (/ P 9 ( Ai \ 2 , /9<Ai\ 2 l 

= constant - gy - M (c + 'M -|- (-^) \ 
p l\ cy J \dx/ j 

O I 

= constant gy c~, (IOQ) 

ay 

if we neglect small terms of the second order. We assume the motion 
to have been originated somehow by the operation of ordinary forces, 
and therefore to be irrotational, so that 



We further assume, in the first instarj.ee, that the depth is very great 



9 2 THE MECHANICAL PROPERTIES OF FLUIDS 

compared with the other lineal magnitudes with which we are con- 
cerned. The simplest solution of (no) which is periodic with respect 
to x, and vanishes for y oo , is 

^ = Ce^sinkx (m) 

If we take the origin O at the mean level of the suiface, the con- 
dition that the wave-profile may be a stream-line is, by (108), 

C C 

. = _ e ky s'mkx = smkx, (112) 

c c 

if we neglect an eiroi of the second order in C. We have still to 
secuie that this is a line of constant pressure. Substituting in (109), 
the result will be independent of x, provided 

g- - kcC = o, or c 2 = f , (113; 

C R 

to our order of approximation. The wave-length, i.e. the distance 
between successive crests or hollows is A = 27rjk, so that 



//#A\ , N 

Vw ^ II4 ^ 



This gives the wave-velocity relative to still water 

The original form of the problem is restored if we omit the first 
term in (108), and replace x by x ct. Thus, if we denote the sur- 
face amplitude C/c by a we have 



tjs = ace ky smk(ct 

To find the motion of the individual particles, we may with consistent 
approximation write 

___ w _______ o sm (ct #) , 

Dv dib , kv ,, . I ^ 

_^: = v = _L = &0c g'y. cosk(ct X Q ), 
Dt ox 

where (^ , ^ ) is the mean position of the particle referred to. Inte- 
grating with respect to t, and recalling (113), we have 

x = X Q a e ky " cosk(ct x), 



MATHEMATICAL THEORY OF FLUID MOTION 93 

ie particles therefore describe circles whose radius a e ky diminishes 
m the surface downwards. At a depth of a wave-length, y = A, 
= e~ zv = 0-00187. The preceding investigation is therefore 
ictically valid for depths of the order of A, or even less. 
For smaller depths, provided they are uniform, the solution (in) 
to be replaced by 

^ = C smh/e(jy + h) sinkx, .......... (118) 

ice this makes v = o for y = h. We should now find 



c* = - tanhM = tanh ....... (119) 

k 27T A 

r small values of A/A this gives c = \/(gh}> and so verifies the 
-mer theory of long waves As /z/A increases, tanh/z tends to 
ity as a limit, and we reproduce the result (114). The paths of the 
dividual particles are ellipses whose semi-axes 

coshk(y + h) smhk(y ~r h) 
smhM ' smhkh 

e horizontal and vertical, icspectively 

The eneigy, per unit aiea of the surface, of deep-water waves is 
und as follows The potential energy is 

2 sm z k(ct x), . (120) 



e mean value of which is {gpa z The kinetic energy is 

..(121) 



r (115) Since c 2 = gjk the energy is, on the whole, half-potential 
id half-kinetic The total energy per wave-length (zTr/k) is TrpaV. 
his is equal to the work which would be required to raise a stratum 
the fluid, of thickness , through a height |. 
The theory of waves on the common boundary of two supei- 
)sed liquids, both of great depth, is treated in a similar manner. 
he formulas (108), (109), (in) may be retained as applicable to 
ie lower fluid. For the upper fluid (of density p') we write 

'A = cy + </V> ............ (122) 

id 

0/ = C'e~ ky smkx ........... (123) 



94 THE MECHANICAL PROPERTIES OF FLUIDS 
since i/r/ must vanish when y is very great. This makes 



C' 

= smkx, 
c 



We have also 



- = constant c~- ( I2 >5) 



The two values of p will be equal provided 

C C' 

gp- kcpC gp (- hcp'C' (126) 

c c 

By comparison of (112) and (124) we have C = C', and therefoie 



If (p p')/(p + />') is small, as in the case of oil over water, the 
oscillations are comparatively slow, owing to the relative smallness 
of the potential energy involved in a given deformation of the common 
surface. A icmaikable case in point is where there is a stiatum of 
fresh water over salt, as in some of the Noiwegian fiords, where 
an exceptional wave-resistance due to this cause is sometimes 
experienced 

The preceding theory of surface- waves is restricted to the case 
of a simple-harmonic profile It is true that any othei form can be 
resolved into simple-harmonic constituents of different wave-lengths, 
and that it is legitimate, so far as our approximation extends, to 
superpose the results. But the formula (114) shows that each con- 
stituent will travel with its own velocity, so that the form of 
the profile continually changes as it advances. The only exception 
is when the wave-lengths which are present with sensible amplitude 
are all large compared with the depth, in which case there is a 
common wave-velocity \/(gh) as found above. 

3. Group Velocity 

One consequence of the dependence of wave-velocity on wave- 
length is that a gioup of waves of approximately simple-harmonic 
type often appears to advance with a velocity less than that of the 
individual waves. The simplest illustration is furnished by the 



MATHEMATICAL THEORY OF FLUID MOTION 95 

combination of two simple-haimonic trains of equal amplitude but 
slightly different wave-lengths, thus 

T) = a cosk(x ct) + a cosk'(x c'l) 
'k - k' kc - k'c'\ 

X t}( 



2 2 / 

If k and k' are nearly equal, the fiist trigonometrical factor oscillates 
very slowly between -f- i and i as x is varied, whilst the second 
factor represents waves travelling with velocity (kc + k'c')l(k -|- /'). 
The surface has therefore the appearance of a series of groups of 
waves separated by bands of nearly smooth water. It is evident 
then that the motion of each group will be practically independent 
of the rest. The centre of one of the groups is determined by 

k - k' kc - k'c' 

x t o; 

2 2 

the group as a whole is therefore propagated with the velocity 
U = = , (120) 

7 if jj * v y / 

k k dk 

m the limit This is called the group-velocity If c is constant, as 
when the wave-length is laige compaied with the depth, we have 
U = c. On the other hand, lor waves on deep water, c 2 gjk, 
by (113), so that 

2 dc _ i 

c dk k' 

whence U = \c, . . . ..(130) 

or the group- velocity is only one-half the wave-velocity The geneial 
foimula, obtained fiom (119), is 

U , . kh 



This expression diminishes from i to | as kh increases from 
o to oo. 

The group-velocity U determines the rate of propagation of 
energy across a vertical plane. To take the case of deep-water waves 
as simplest, the rate at which work is done on the fluid to the right 



96 THE MECHANICAL PROPERTIES OF FLUIDS 

of a plane through the oiigin perpendiculai to the axis of x is 

/o 

I pudy .......... (132) 

J _co 

The value of p is given by Bernoulli's equation provided we put 
q* = ( c -|- uf -f- v 2 c 2 zcu, to our order of approxima- 
tion. The only term in the resulting value of p which varies with 
the time is pcu. Now 



f u z dy = pW sm z kct t e* ky dy = %pka z c* sitfkcl. . . (133) 

J 00 * 00 



pc 



The work done in a complete period (zTrjkc) is therefore 
which is half the energy of the waves which pass the above plane 
in the same time. The apparent paradox disappears if we lemember 
that the conception of an infinitely extended tram is an aitificial 
one. In the case of a finite tram, generated by some peiiodic action 
at the origin which has only been in operation for a finite time, the 
profile will cease to be approximately umfoim in chaiactei and 
sinusoidal near the front, there will be a gradual diminution of 
amplitude, and increase of wave-length, by which the tiansition to 
smooth water is effected. We infer from the piecedmg aigument 
that the approximately simple-harmonic portion of the tiain is 
lengthened only by half a wave-length in each period ol the 
originating force. 

The principle that U lather than c detei mines the rate of pro- 
pagation of energy holds also, not only in the case of waves on water 
of finite depth, but in all cases of wave-motion m Physics. 

Some further results of theory must be merely stated in geneial 
terms. A localized disturbance travelling over still watei with velocity 
c leaves behind it a train of waves whose length (2ir/k) is related to 
c by the formula (113) or (119), as the case may be. In the same 
way a stationary disturbance in a stream pioduccs a liam of waves 
on the down-stream side. In the former case the encigy spent in 
producing the train measures the wave-resistance expencnced by 
the disturbing agency. If E be the mean energy per unit length of 
the wave-train, the space in front of the disturbance gains in unit 
time the energy cE, whilst the energy transmitted is UE, where 
U is the group-velocity. The wave-resistance R is therefore given 
by 

Rc= (c-U)E ................ (134) 



MATHEMATICAL THEORY OF FLUID MOTION 97 

"lie value of E has been found to be $gpa 2 , but unfortunately the 
alue of a can be predicted only in a few rather artificial cases. 

A curious point arises in the case of finite depths. It appeal's 
-om (119) that the wave-velocity cannot exceed \/(gh). The above 
tatements do not apply, therefore, if the speed of the travelling 
isturbance exceeds this limit The effect is then purely local, and 
L = o. A considerable diminution in resistance was in fact observed 
y Scott Russell when the speed of a canal boat was increased m 
lis way; and an analogous phenomenon has been noticed m the 
ise of torpedo boats moving m shallow water. 

Viscosity 
I. General Equations 

The subject of viscosity is treated in Chapter III, which deals 
lainly with cases of steady motion wheie this influence is pie- 
aimnant The general equations of motion of a viscous fluid 
ive the forms 

Dti dp 3V . -_ , , 

'Di"- -'to* ' iV " l/ ' ..... (I35) 

ith two similar equations m (v, y) and (w, z), where 

v 2 = a 2 /a*: 2 + a 2 /^ 2 + a 2 /as 2 . 

he formal proof must be passed over, but an mteipietation of the 
]uations, which differ only from (88) by the terms at the ends, 
n be given as follows. Considering any function of the position 
a point, let F be its value at P, whose co-oidmates are (x, y, #). 
s value at an adjacent point (x -[- a, y + ft, z + y} will exceed its 
lue at P by the amount 

r J_ 9F L 9F 

-a + ft + y 
dy 3# 



iproximately. If we mtegiate this over the volume of a sphere 
small radius r having P as centre, the first three terms give a 

ro result owing to the cancelling of positive and negative values 
a, 0, y. The terms containing j5y, ya, a, also disappear for a 

nilar reason. The mean value of a a or p* O r y 2 on the other hand 

(D812) 



5 



98 THE MECHANICAL PROPERTIES OF FLUIDS 

is -i-r 2 , by the theory of moments of ineitia. The mean value over 
the sphere of the aforesaid excess is therefore T y- 2 V 2 F. The 
reason why this should vary with the radius of the sphere is 
obvious. It is also clear that the expression V 2 F gives a measure 
of the degree to which the value of the function F in the immediate 
neighbourhood of P deviates from its value at P. In particular 
V 2 M measures the extent to which the a: - component of the 
velocity in the neighbourhood of P exceeds the component at P. 
The first of the equations (135) accordingly asset ts that in addition 
to the forces previously investigated there is a force propoitional to 
this measure. An excess of velocity about P contributes a force 
tending to drag the matter at P in the direction of this excess. 

The coefficient //. in (135) is called the coefficient of viscosity. 
In cases of varying motion we are often concerned not so much by 
the viscosity itself as by the ratio which it bears to the inertia of 
the fluid. It is then convenient to introduce a symbol (v) for the 
ratio pip This is called the kinematic viscosity. 

An important conclusion bearing on the comparison of model- 
and full-scale experiments can be drawn from the mere form of these 
equations Omitting the term repiesenting extraneous force, the 
first equation is in full 

du . du . du . du I dp , , ^ 

_]_ u j_ v + w = __ _ Jl 4. v ^ u . . .(136) 

ot ox oy o% p ox 

Now consider another state of motion which is exactly similar except 
for the altered scales of space and time. Distinguishing this by 
accented letters, a comparison of corresponding terms in the respec- 
tive equations shows that we must have 

u' . u __ u' 2 . w 2 __ p' , p __ v'u' vu 

' 



The equality of the first two ratios requires that 

f t/V VV 

u :u = -7 : -, 

as was evident beforehand. The equality of the second and fourth 
ratios requires , 

^r,'- <'3) 

use ux 

A necessary condition for the similarity of the two motions is there- 
fore that VJ?/i> should have the same value in both, where V is any 



MATHEMATICAL THEORY OF FLUID MOTION 99 

racteristic velocity, and / any linear dimension involved. The 
o of corresponding stresses is then 

JL/ ',.,'2 

p p U 



p pu z 

It is to be noted that the viscous terms disappear from the 
ations (135) if the motion is irrotational, since we then have 
= o, and therefore V 2 u = o, V 2 # = o, V 3 z# o. But it is 
,eneral impossible to reconcile the existence of irrotational motion 
i. the condition of no slipping at the boundary, which is well 
blished experimentally. The above remark suggests, however, 
, when the motion is staited, vorticity originates at the boundary- 
is only gradually diffused into the interior of the fluid. ^~ 

2. Two-dimensional Cases 

The diffusion of voiticity is most easily followed in the two- 
ensional case The equations may be written, in virtue of 
, in the forms 




where X = - + l(u* + ^ 2 ) -f- V, ........ (141) 



icntiating the second of equation (140) with respect to x, and 
irst with lespect to y, and subtracting and making use of the 
tion of continuity (7), we have, finally 

a-*'* ................ (I43) 

is exactly the equation of conduction of heat, with the vorticity 
place of the temperature, and the kinematic viscosity v( /n//>) 
ace of the thermometric conductivity. Consequently, various 
r n results in the theory of conduction can be at once utilized 
e present connection. 



ioo THE MECHANICAL PROPERTIES OF FLUIDS 

For instance, the known solution for the diffusion of heat from 
an initially heated straight wire into a surrounding medium can be 
applied to trace the gradual decay of a line vortex initially concen- 
trated in the axis of #. Since there is symmetry about Os the 
equation (143) takes the form 



dt 

as may be seen by a comparison of the left-hand membeis of (25) 
and (26) It is easily verified by differentiation that this equation 
is satisfied by 



which vanishes for t = o except at the origin. Moreover, this gives 
for the circulation in a circle of radius r 

2-jrrdr = K(I e~ t ~'^" t ) ( 1 4 ( ->) 

As t increases from o to o , this sinks from K to o. The value of , 
on the other hand, at any given distance r increases from zeio to a 
maximum and then falls asymptotically to zero. 

A comparatively simple application of the equations of motion 
is to the case of " laminar " flow in parallel planes, or of smooth 
rectilinear flow m pipes, but the results have only a restricted 
application to actual phenomena To take an example due to 
Helmholtz, consider the flow of a hypothetical atmosphere of 
uniform density, and height H, over a horizontal plane If it is 
subject to meitia and viscosity alone, the equation of motion is 

"du d 2 u 

= v } . (147) 

with the conditions that u = o for y = o and du/dy o for 
y = H. These are all satisfied by 

u = Ac-"* 1 ' sinky (148) 

provided 

cos/di = o, or k = (zn -f 1)^5, (149) 

where n is an integer. By addition of such solutions with different 
values of n and suitable values of the coefficients A we can represent 



MATHEMATICAL THEORY OF FLUID MOTION ror 

the effect of any initial state, e.g. one of uniform velocity. The 
most persistent constituent in the result is that for which n = o. 
This will have fallen to one-half its original value when 



v&t - loga, or t = ........ ( IS o) 



Putting v = 0-134 (air), H = 8026 metres, this makes t = 305,000 
yeais! The fact is that in such a case the laminar motion would be 
unstable, turbulent motion would ensue, by which fresh masses of 
fluid moving with considerable velocity are continually brought 
into contact with the boundary, so that the influence of viscosity is 
enormously increased. 



CHAPTER III 
Viscosity and Lubrication 

A. VISCOSITY 

All motions of actual fluids, as distinguished from the " peifect 
fluid " of the mathematician, are accompanied by internal forces 
which resist the relative movements and are theiefore analogous to 
frictional forces between solid bodies The origin of the frictional 
resistances is in all cases referred to the property of viscosity, common 
in varying degree to all fluids, which has already been defined in 
general terms in Chapter I. The present chapter is devoted to a 
fuller explanation of the theory of this property and to discussions 
of some of its direct applications, one of the chief of these being to 
the theory of lubrication 

There aie other direct applications of the theoiy of viscosity 
which are of importance to engineers, most though not all of which 
relate to the motion of fluids in narrow channels or in thin layers 
between solid surfaces, and these applications are met with in all 
branches of engineering. The fluid frictions, however, which chiefly 
concern hydraulic and other engineers, who deal with fluids such as 
water or air in large volumes, though physically referable in origin 
to viscosity, cannot be directly calculated by means of its theory 
The appropriate methods applicable to such cases are discussed in 
Chapters IV and V. In the meantime it may be said of the direct 
applications of the theory, in Rayleigh's words (30, p 159),* that m 
these cases "we may anticipate that our calculations will correspond 
pretty closely to what actually happens more than can be said of 
some branches of hydrodynamics ". 

* Arabic numerals in brackets after names of authors refer to the short biblio- 
graphy at the end of thia chapter. 

102 



VISCOSITY AND LUBRICATION ro3 

Laminar Motion 

The law of viscous resistance is most clearly conceived in the 
se of laminar motion, which may be defined as a state of motion of 
body of fluid m which the direction of the motion of the particles 
the same at all points and the velocity is the same throughout 
ch of a series of planes parallel to one another and to the direction 
motion A volume of fluid in laminar motion can thus be roughly 
garded as a series of very thin 
pers of solid material, sliding one ^ 

)on another in a common direc- / / / / 

m Quantitatively, if the face k JP u 

p , By of the rectangular element 
, 8y, S# (fig i) is parallel to the &!/ X* 

iiinse, and if the laminar motion 

in the direction of X, the velo- 
y of flow, u, at any point P, 
11 depend only on the distance, 

of the point P from the plane 
Y If the element is sufficiently 

lall, u may be taken as varying uniformly with z over the small 
stance Sz, so that if w , % are respectively the velocities of the 
rimas which form the lower and upper faces ol the element 

= z/ -f- Bz, in which can be regarded as constant over 

dz dz 

e small distance $z 

In a viscous fluid there will then be exerted a shearing force, or 
iction, parallel to X, between the portions of the element of fluid 
ove and below any section of the element parallel to the face 
By, tending to retard the portion which is moving with the higher 
locity, and the magnitude of this force will be 

S = pBxBy, (i) 

oz 

ou 
being a quantity, independent of x, y, z, u, and , known as the 

Efficient of viscosity. 




104 THE MECHANICAL PROPERTIES OF FLUIDS 

Coefficient of Viscosity 

The value of the quantity ^ varies greatly from one fluid to 
another, and in any one fluid it changes with the temperature, and to 
a smaller extent with the pressure, of the fluid. Its value is in 
general much higher for liquids than for gases Liquids m which 
the value of p is low are said to be " limpid ", " thin ", or " light ", 
while those in which it is comparatively great aie said to be " vis- 
cous ", " thick " or " heavy ". There is however no necessary, or 
general, correspondence between the density of a liquid and its 
viscosity. Thus mercury, the heaviest of known liquids at atmos- 
pheric temperatures, is one of the least viscous. 

The fact that the coefficient of viscosity, for a given liquid at 
constant temperature, is independent of the rate of shear was fiist 
experimentally proved with great accuracy by Poiseuille (i), not, 
howevei, by duect measmement of plane laminar flow, but by in- 
vestigation of the flow of water in small cylindrical tubes 

The flow of fluids in such tubes, as well as the motion of viscous 
fluids in many other cases which are of practical inteiesl, is closely 
analogous to plane laminar flow. 

The importance of the coefficient of viscosity /A, however, anscs 
from the fact that it is the sole physical constant connecting the 
internal frictional resistances of fluids with their iclative motions, 
not only in the case of such simple types of motion, but of all kinds 
of fluid motion however complicated, provided that they aie not 
discontinuous or unstable 

The explanation of this unique prop city of the coefficient of 
viscosity icqmres some analysis of the types of deformation ot which 
a fluid element is susceptible This analysis is given briefly in the 
following paragraphs, from which it will be seen that the iclations 
between the internal motions and stresses in a fluid aie similar to, 
but essentially simpler than, those between the deformations and 
stresses in an elastic body The failuie, already icf cried to, of the 
law of viscosity when fluid motions become discontinuous or unstable 
may be regarded as analogous to the failure of the laws of elasticity 
in solids when fracture takes place or the " yield-point " is exceeded. 
In such cases the conditions which result are no longer amenable to 
theoretical calculation. 

We proceed to show that, when no such discontinuities exist, 
there is in fluids only one kind of internal resistance and only one 
coefficient of viscosity. 



VISCOSITY AND LUBRICATION 
Relative Velocities 



105 



If M, v, w (see fig 2) are the components of the velocity parallel 
to the rectangular axes X, Y, Z of n par- , 

tide of the fluid at the point x, y, #, the z Ar - 

correspondmg components for the neigh- 
bouring point x -f- Bx, y + By, 2 + Bz are 

, . du <j . du j, . du <,, 

n = M -(- ojc -f- dy -f- o#, 

C/JC OV O*^ 



v = 




and the components of the velocity of the second point relatively 
to the first are u' u,v' v, w' w, or 



\JtAi j"^ 



(3) 



dx ' 3j 



3^; 



Of the derivatives in these expressions it is clear from inspection 

c r i ;. 3 dv , dw , . , 

of iig 2 that , , and - repiesent rates of stretching or elongation 
ox ay ds 

of the element in the directions of X, Y, and Z icspectively, while 
by the pairs of sums of denvatives* 



dzv . dv 
dy 3V 



du dw 

dz dx' 



dv . du 
dx dy 1 



are icpiesented respectively rates of change of the angles between 
the edges By and Bz, Bs and S#, and Bx and By of the element 

Thus by means of these six expressions any deformation of the 
element can be expressed. 

As a hypothesis which is suggested as probable by the experi- 
mental law proved by Poiseuille, but which depends for its real justifi 
cation on the consistent correspondence of the results of theory 
with experience, it is assumed that the frictional forces arise from 



(D312) 



5* 



o6 THE MECHANICAL PROPERTIES OF FLUIDS 

he rates of deformation of the elements of the fluid, and are linear 
unctions of these lates. ~ 

As to the three rates of elongation , , , it is a well-known 

ox oy oss 

heorem that they can be resolved into a rate of dilatation or com- 
>ression of the elementary volume, uniform in all three directions, 
ombined with three lates of shearing deformation respectively in 
he directions of the diagonals of the faces of the element supposed 
ubical.* 

As there is no experimental evidence of any internal lesistances, 
ither in liquids or gases, depending on rates of change of volume 
>y dilatations or compressions equal in all directions, resistance to 

o 

n elongation, such as -- Sx, can only arise from its shearing com- 

ox 

>onents. Such resistances are theiefore of the same kind as those 
vhich depend on the purely shearing deformations whose rates are 

w . dv 

_L ATP 

I n ) tx ' 1 - / 

y as: 

In the applications which follow, the axes X, Y, Z will be so 

hosen that the rates of elongation, such as --, and consequently also 

ox 

heir component rates of shear, are everywhere small compared to 

tie rates of shear lepresented by + , &c. 

oy dz 

Now in a homogeneous liquid or gas there is no physical difference 
i the properties depending on the direction of the co-ordinates, 
onsequently (the fnctional forces being linear functions of the rates 
f shear) the only foices that will arise may be expressed as: 

o , dv^ 



dw 



; Bu\ 

-.dx dyj 'j 

ivolving the single coefficient ft By comparison with fig. i, and 

*Cf the similar theorem for stresses (Morley, Strength of 'Materials, 2nded ,p 12). 

f In this notation the first subscript indicates the direction of the noinial to the 
ane on which the foice acts, the second the direction in which the force acts 
us Syz is the sheaung stress on a plane a normal of which is parallel to Oy and 
<g acts in the O# direction 



VISCOSITY AND LUBRICATION 107 

equation (i) with the second of the above equations (4), in 

r\ 

ich is taken as zero, it is seen that this constant is the same 

dx 

the coefficient p introduced in the special case of laminar 
ition. 



Conditions at the Bounding Surfaces of Fluids 

Before the laws of fluid friction can be applied to fluids as we 
ually have to deal with them, account must be taken of the be- 
dour of the fluid where it is in contact with the solid bodies which 
itam it. In the case of liquids the condition of a free upper 
face, usually a surface of contact with air at atmospheric pressure, 

also to be considered. 

It is clear in the first place that the presence of a boundaiy in- 
ves that on the bounding surface the 
itive velocity of the fluid normal to 
t suiface is zero. The noimal velo- 
r will furtheimoie be very small at 

points near the bounding surface. 

, let W, fig 3, be the fixed bound- 

suiface, and W a suiface in the 

d parallel, and very near, to W. Fig 3 

simplicity W and W may be 

sidered plane. Let the average velocity towards W over a 

le of radius R in the plane W be v, the normal distance between 

and W being Sn. Then a volume of fluid rrR 2 v flows through 
circular area in unit time In the same time a volume 27rRw.cr 

re outward between the surfaces past the circumference of the 

le, o- being the mean outward ladial velocity parallel to the sur- 

s. Thus 

Sn t \ 

v = 2o-~, (5) 

he noimal velocity is veiy small compared to the velocity parallel 
lie surface. 

[n the case of a solid boundary it will be seen from the next 
igraph that the velocity a is itself very small close to the surface, 
hat in this case the normal velocity v is a small quantity of the 
nd order. 




10 8 THE MECHANICAL PROPERTIES OF FLUIDS 

Motion Parallel to Bounding Surfaces 

With regard to the motion of fluids parallel to solid walls w 
which they are m contact, there is strong evidence that in the a 
of liquids at least the relative tangential velocity a at the wall is ze 
Some of the evidence will be referred to later in connection with t 
flow of liquid through tubes under great pressure, and in the d 
cussion of the theory of lubrication. 

Even when the mutual molecular attraction of a liquid and so 
appears to be compaiatively small, so that the liquid does not tend 
spread over, or " wet " the suiface of the solid, as is the case w 
mercury and glass, there is no observable sliding or slipping of 1 
fluid over the solid at theii common surface. 

If the tangential tractional force between liquid and solid, a 
consequently the late of shear in the liquid neai the suiface, ; 
finite, the relative tangential velocity, being zero at the suiface, mi 
be still small at all points of the liquid near the surface, as was , 
serted in the last paragraph 

In gases, the same rule as to the relative velocity being zero 
a soHd surface is found to apply under ordinary circumstances, 
least as a very close approximation When, however, a gas is at su 
low pressure that its molecules aie at distances apart comparable w 
the dimensions of the volume of gas which is being dealt wi 
phenomena are observed which can be regarded as ansing from 
appreciable velocity of slipping of the gas over the solid surfa 
According to Maxwell,* the motion of the gas is very nearly the sa 
as if a stratum, of depth equal to twice the mean fiee path of the j 
molecules, had been removed from the solid and filled with the g 
there being no slipping between the gas and the new solid suiface. 

At free surfaces, which, of course, can only exist in liquids, 1 
normal velocity relative to the surface is again obviously zeio. 1 
liquid surface may, however, have a tangential velocity, and it 
usual to assume that the law of viscous shear holds up to the suif; 
and that either the tangential traction there becomes zeio, or, if 1 
liquid surface is exposed to a stream of air, that the traction is c 
only to the rate of shear in the air near the common surface. Expc 
ments by Rayleighf and others have shown that, at least in the ca 
of water with an uncontammated surface and of oils and other Hqu 
which are capable of dissolving solid grease films, theie are 
fnctional resistances peculiar to the surface film 

* Collected Papers, Vol II, p 708 f Collected Papers, Vol. Ill, p 363. 



VISCOSITY AND LUBRICATION 109 



Viscous Flow in Tubes 

On the principles which have been explained, we can proceed 
calculate the flow of viscous fluids in various cases which are 
piactical inteiest Take first the case of a uniform tube of 
cular section of winch the diameter is small compared to the 
gth of the tube. A fluid flows through the tube as the icsult 
a constant difference of pressuie between its two ends. The 
ition, except veiy neai the ends, will be sensibly parallel to the 
s of the tube, and the piessuie (and consequently the density) 
I be sensibly umfoim over every noimal section. By symmetry, 
any one section the velocity must be the 
ae at all points at any given radius r from 
axis. 

If w be the velocity (upwards in fig. 4) 
this radius, p the density, and p the 
ssure at any section, the radius of the 

e of the tube being a, the mass dis- 

1 1-1 i 1 

rged per unit time, which must be the 

le for all sections, is 



f fl 

I pzo 2rrrdr = m, constant . . (6) 

J o 



Fig 4 



2 axis of the tube is taken as the axis Z, and is supposed to be so 
rly straight that effects due to its curvature can be neglected, and 
the fiist instance the motion will be supposed so slow that the 
2tic energy of the fluid is inappreciable. The fluid may be either 
quid or gas. The effect of gravity is disregarded, or, if included, 
- gpz is to be written instead of p. 
From equations (4), p. 106, since the velocity w varies radially 

r 

he rate -, but not circumferentially, there is a ti action in the 

Of 

Action of Z on each unit of area of the cylindrical body of 
i inside radius r of amount 

8U=/ ........... (7) 



isidering a section of this cylinder of length 8^, the total traction 
its cylindrical surface, whose area is 2ur8#, must be equal to the 



io THE MECHANICAL PROPERTIES OF FLUIDS 
ifference of the total pressures on its upper and lower ends, so that 

~z = 7rr*-~-Sz t 
dr dz 



dw r dp 

or -=-, 

dr 2u dz 



and therefore w 



= -- -- -- 



) . 

/ 



Since w = o when r = a, C = , and 

2 



i which the negative sign expresses the obvious fact that the direc- 
on of flow is opposite to the direction of increase of pressure. 
Now fiom (6) and (8) 



= / pw.2Trrdr = / 
J o J o4/A 



-- 
dz 



In the case of a liquid, p and [Jt, may usually be taken as constant, 

dp 
that is constant along the length of the tube, being equal to 



_ 

2 Fl where p lt p z are the pressures at the lower and uppei ends 
f the tube whose length is /. 

m, TTCflp pi p 9 , N 

Then m = -^ ^ 1 ra .......... (io) 

OjU. / 

"he limits of application of this formula will be more fully explained 
i a later chapter For the present it may be stated to be applicable 
) the flow of all liquids through " capillary " tubes (that is to say, 
ibes whose diameter is only a fraction of a millimetre), unless the 

* In all numerical applications of this and other formulse thioughout this 
lapter all quantities must be expressed m the C.G.S. or other absolute system of 

uts. 



VISCOSITY AND LUBRICATION in 

lifference of pressures, p p z , is greater than is ordinarily met with 
i engineering practice, provided that proper correction is made for 
he disturbing effects of the ends of the tube. 

In the case of viscous lubricating oils, the formula is applicable, 
vith certain restrictions, to their flow through ordinary lines of 
)ipmg, but it must be regarded as subject to correction, or even 
wholly inapplicable, to the flow of the less viscous oils especially 
inder considerable pressures.* 

In the case of a gas, p = -^- , T being the absolute temperature 

R 1 

md R a constant Thus from (9) 



If T and ju, can be regai ded as constant throughout the length of 
he tube, integrating (n) we have 



is the equation connecting the flow and the fall of piessure. 

In the preceding discussion the kinetic eneigy of the fluid has 
Deen assumed to be negligible All the foimulas given, however, 
-emain coirect for the case of a liquid even when the kinetic eneigy 
s appicciable, provided that they aie applied only to the middle 
portion of the tube and not to its end poitions where the flow is 
iffected by the acceleiation and letardation of the fluid which occur 
near the inlet and outlet It is well known that the kinetic energy 
which a fluid acquires in enteung an orifice is not wholly lestored 
is pressuie eneigy at its discharge There is theiefore a resistance 
to the flow aiising fiom the acceleration and retardation at the inlet 
and outlet of a tube, additional to the factional losses within the tube 
itself. In the case of a square-ended tube opening into large vessels 
at each end, the loss of pressure is approximately 1-12 X u z J2g, 
where u is the mean velocity at the outlet. \ 

There aie further sources of resistance not taken into account 
in our calculations, arising from viscous friction between the streams 
at the ends, where the lines of flow aie not parallel to the axis 
of the tube. Fig 5! shows the course of the particles of fluid 
at the inlet and outlet of a square-ended tube when the kinetic energy 
is appreciable and both ends of the tube are immersed in the fluid, 

* See e.g. (13), p 159 t See Hoskmg, Phil Mag , April, 1909, Schiller, 

Zeits Math, u. Meek , Bond, Proc. Phys, Soc , 34, IV. j Fiom (10), p. 158. 



12 THE MECHANICAL PROPERTIES OF FLUIDS 

Fig. 6 illustrates the condition which occurs when the outlet of such 
a tube is not immersed but discharges the fluid in a series of drops. 
In this case there is another resistance to the 
flow, due to the excess of internal pressuie 
which is necessary to extend the surfaces of the 
drops during their formation. 





OuHel 

vessel 



[ J 





Fig 5 Fig 6 

The calculation of the resistances due to these disturbing effects 
is rathei uncertain, and on this account an accurate correspondence 
between the results of calculation and those of expenment can only 
be expected when the tubes are very long compared to then diameteis 

Use of Capillary Tubes as Viscometers 

The experimental determination of coefficients of viscosity is 
earned out by instmments of vaiious kinds, known as " viscometeis " 
or " glischiorneteis " These are divided into two classes, namely 
" absolute " viscometers, by the use of which the coefficient of 
viscosity can be determined in absolute measure dnectly iiom the 
dimensions of the instrument itself (combined with measurement of 
a time interval), and " secondary " or " commercial " viscometers, 
which require to be calibrated by comparison of then results with 
those of an " absolute " viscometer. 

The best absolute viscometers, for liquids at least, depend on the 
measurement of flow through capillaiy tubes, //, being determined 
from the equation (10) given on p. no, after instrumental measure- 
ment of the other quantities involved. The appaiatus by which 
Poiseuille made the first accurate determinations of the viscosity of 
water was of this. class. The tubes which he used varied in diameter 
from o-ooi to 0-014 cm., their lengths being a few centimetres, and 
the pressure was applied by a column of mercury up to 77 cm. in 
height. Such instruments are capable of very considerable accuracy 



VISCOSITY AND LUBRICATION 




FiS 7 Stone's Absolute Viscometer 

hen used with proper precautions, and when the necessary cor- 
'Ctions are applied for the various disturbing factors. The viscosity 
"water, for instance, at atmospheric temperatures is probably known 
ithin Tilth of i pei cent of its true value.* 

See(u),p. 158. 



ii4 THE MECHANICAL PROPERTIES OF FLUIDS 




The consistency to this order of accuracy of determinations made 
with different instruments and under different conditions is con- 
clusive evidence of the correctness of the basic assumption of the 
linear connection of traction with shear, and of the absence of slipping 
of the fluid over the walls of tubes The puncipal 
precaution which has to be taken in the use of the 
capillary viscometer, in addition to the elimination of 
(01, in so far as that is not possible, the correction 
foi) the end-disturbances which have been pointed 
out, is the accurate determination of the tempera- 
ture of the fluid undei test. The latter requiiernent 
is usually met by sin rounding the capillary tube 
with a water-jacket, means being provided for 
warming or cooling the water, and measuring its 
temperature. 

The most convenient form of " absolute " capil- 
lary viscometer for liquids is that descubed by W. 
Stone (18, p. 159). In this instrument the pressure 
is applied by a column of mercuiy of which the 
height is automatically maintained constant, and other 
devices aie provided which further simplify the 
manipulation of the instrument and the calculation 
of the results from the observations The Stone 
viscometer is illustrated in fig. 7, the capillaiy tube 
and its attachments being shown separately in fig 8. 
The following is an abbreviation of the designer's 
descuption cited above. 

The instrument consists of thiee essential ele- 
ments, viz the viscometer burette, the adjustable 
constant-head apparatus, and the piessuie-gaugc. 
The viscometer burette consists of two glass vessels 
A and B (fig 8), of equal internal diameters and suitable lengths, 
connected at their lower ends by means of a wide-boie tube C, and 
of a capillary tube D of suitable dimensions for the desired purpose. 
The three portions of the burette are held together by the brass 
clips and tension-rods R. Several interchangeable tubes D may 
be provided for fluids of different viscosities. 

The measuring vessel A is provided with two platinum wires 
sealed into its wall, and so bent that the inner end of each wire lies 
on the axis of the tube. The capacity of the vessel between the two 
platinum points can be thus accurately measured. A glass tap T 




R 



Fig 8 



VISCOSITY AND LUBRICATION 113 

is provided on the inlet to the burette to control the staitmg of a 
test. The whole of the burette is immersed in water contained in 
a glass tube (see fig. 7) having a brass bottom. A brass cover is also 
fitted having a slot for the insertion of a stilling rod and a ther- 
mometer A Bunsen burner selves to heat the water. 

The adjustable constant-head apparatus consists of two glass 
vessels, the lower one F being furnished with a tap V at the top and 
the upper one G suspended by a spring from a hook attached to a 
sliding clip H which can be clamped to the standard S at any desired 
height. Through the outer end of the clip a glass siphon pipe passes 
to the bottom of the vessel G when the latter is at its highest point, 
i.e against the clip H The siphon is connected to the lower vessel 
F by means of a rubber tube. The strength of the spring is so 
adjusted that as the mercury flows from G to F, the former, 
being thereby lightened, will rise so as to maintain the surface 
of the mercuiy m it at constant height above that of the 
mercury in F. 

The pressuie-gauge K is of the oidmary U pattern, with meicury 
as the woiking fluid A thiee-bianch pipe P connects the burette, 
pressure-gauge, and constant-head appaiatus. 

The instrument must be set up veitical. As the liquid to be 
tested is fed into the burette A (fig 8), the vessel F is removed from 
the socket J and raised to a sufficient height above G to reduce the 
lir-pressure in B (fig 8) and thus draw the liquid under test into it, 
lowering the suiface in A below the lowei platinum gauge-point. 
The glass tap T is then closed and the pressure apparatus adjusted 
to the deshed piessure. Then the tap is opened, and the time 
slapsing between the moments of contact of the liquid surface with 
the gauge-points in A is taken by means of a stop-watch or suitable 
:hionogiaph. 

By the use of this instrument the viscosity of a sample of oil 
2an be determined at ten or twelve different temperatures within an 
houi The pressure can be varied fiom about 5 to 50 cm of mercuiy 
in order to give (without changing the tube D) convenient intervals 
D time for measurement according to variations in the viscosity of 
the oil 

Vanous other forms of apparatus have been used for the absolute 
deteimination of viscosities, their action depending, for instance, 
Dn the torsional oscillations of a disc or cylinder (a method which 
is convenient for measurement of the viscosity of gases, on account 
of the accuracy with which the very small forces involved may be 



ii6 THE MECHANICAL PROPERTIES OF FLUIDS 

measuied by this means), the continuous rotation of a cylinder or 
disc or sphere, or the fiee fall of a sphere in a body of fluid 
Foi geneial purposes, however, no other method is so convenient 
01 accurate for absolute measurements of viscosity as that of 
Poiseuille. 



Secondary or Commercial Viscometers 

Tube viscometeis are also commonly employed foi making 
practical or commercial measurements of viscosity In ordei to 
reduce the time occupied by the measurements, and to simplify 
the apparatus and to reduce its delicacy, much shoiter tubes 
are used in these instiuments than are admissible for absolute 
instruments. 

In the Redwood viscometer, for instance, the tube is appioxi- 
mately 1-7 mm. in diameter, and 12 mm m length, being a hole 
drilled through an agate plug fixed in the bottom of a vessel 
which is ananged to contain a measured quantity of the liquid to be 
tested. The liquid flows out of the hole under the force of gravity, 
the time of efflux of the measured quantity being taken by a stop- 
watch. Means are provided for warming or cooling the liquid to 
any temperature at which it is desired to make the test, but the 
determination of the actual tempeiature of the fluid as it is passing 
through the hole is one of the chief difficulties in the use of this and 
similar instruments. 

In some of these the " tube " Is so much i educed in length as 
to become a meie orifice. It will be readily undei stood that the 
conections for the end effects of the tube, which have been 
pointed out as necessary in connection with all cases ol viscous 
flow in tubes, become i datively much rnoie consideiable in the 
case of such shoit-tube instiuments. In these, except for the 
moie viscous liquids, the times of efflux are no longer propoi- 
tional to the viscosity of the fluid. It is therefoie necessary, in 
order to obtain reasonably accurate results, that such instruments 
should be calibrated over the range of their intended application 
by comparison with an absolute viscometer. Such a system of 
cahbiations not having been generally adopted, an unfortunate 
practice has become common of expressing viscosities, not in 
terms of physical or engineering units (by which alone the value 
of the unit can be applied in calculations), but by the number of 
seconds or minutes required for the efflux of a certain volume through 



VISCOSITY AND LUBRICATION 117 

ic or cither of the best-known forms of commercial viscometers, 
heie aie thus in use as many arbitrary, irreconcilable, and dynami- 
illy meaningless units of viscosity as there are manufactureis of 
>mmercial viscometeis. 

A different type of secondary viscometer recently intioduced 
the cup-and-ball viscometei The action of this instiument 
epends on the viscous flow of the fluid, not in a tube, but between 
vo nearly parallel and closely adjacent surfaces. The instiument 
id its mode of operation will be moie fully described below, after 
iscussion of the theoiy of that type of viscous motion. 



Coefficients of Viscosity of Various Fluids 

In Table I (p. 118) aie given values of the viscosity constant 
of a few of the fluids which aie of chief interest to engmeeis, 
specially in connection with lubrication The table contains also 
^proximate numeiical data, for the same fluids, of certain other 
hysical properties, the significance of which, as affecting the utility 
f the fluids as lubricants, will be made more appaient by the later 
ortions of this chapter. The constants are expiessed in all cases 
i C G S units. The value of ju for instance is the ratio of a stress 
leasmed in dynes pci sqiuic centimetie to a late of sheai measuied 
i centimeties pci second pei centimetre 

The values oi /x aie given lor various tempeiaturcs between o 
ad 1 00 C The other constants, which for the most pait do not 
01 y lapidly with temperatuic, aie stated for atmospheric tempeia- 
ues m the neighbouihood ot 15 C. The mle which is apparent 
om the table as to the values ot ^ for liquids, namely that the value 
31 each liquid diminishes as the temperature uses, is tme geneially. 
t will be noticed that the late of vaiiation is much less lapid for 
icicury and carbon bisulphide than foi the other liquids In all 
ases, as in au, on the other hand the viscosity mci eases with the 
cmperatuie. 

Variation of Viscosity with Pressure 

The viscosity of both liquids and gases varies veiy little with 
raiiations of piessure over a range from many times less, to many 
lines greater, than atmospheric piessure. At pressures, however, 
)f the order oi intensity of hundieds of atmospheres most liquids 
ippear to have greatly increased coefficients of viscosity. 



pqpqpq 






COCO 



PH 





X XX X 



u S 

a K 



CO O VO COOO 
O co xfJ> 
O O N in 

M O O O O O 



^ , 

ooo 




a 







X w 

ON o 



w 

PQ 



n 

oo 

ON 



o o 
o o 



O vO 

o o 



O O O I> CO N 

o o o oo o r-. 

ON O Th 
J" 1 * {** *T" 

M H O >O CO 

O O O <* H I 
000 M 



--o o 
oo M 

X i 



o 




-a 

D4 



O 



Kt W^- ^T5 
tu JJ S O 

J3- <us p 

i>iis 



118 



<1 



VISCOSITY AND LUBRICATION 119 

The following table (Table II) from Hyde (21, p. 159) shows 
how the viscosities of a few lubricating oils vaiy with pressures of 
this order Although such pressures do not usually exist m ordinary 
bearings, there are cases in the application of the theory of viscosity, 
as will be seen later in this chapter, in which the changes of the viscous 
constant by mciease of pressure cannot be neglected. 

Within very wide limits, the viscosity of gases is independent 
of pressure, the viscosity of air for instance being practically invariable 
from a piessure of a few millimetres of mercury up to pressures of 
many atmospheres This law, originally predicted by Maxwell from 
the kinetic theory of gases, has been confirmed by numerous experi- 
ments 

TABLE II 

VISCOSITIES OF VARIOUS LUBRICATING OILS AT VARYING 
PRESSURES. TEMPERATURE 40 C 

Absti acted from a table by J H Hyde Proc Roy Sot, , A 97 

Pressme,Kilo- Mineral Oil Troltei Oil Rape Oil ~ 1Tn n , 
^"^ <" Bayonne ) (Animal) (Vegetable) S P eim Ol1 



Coefficient of Viscosity, M, C G S 

o o 47 o 344 o 375 o 154 

157-5 062 o 413 0422 0190 

3150 092 0550 0539 0236 

472 5 i 32 o 686 o 703 o 299 

630 o i 86 o 824 o 880 o 368 

787 5 2 51 i 089 

945 3 6 5 I 21 7 l 3 10 

1 102 5 5 32 i 578 o 619 

1260 755 1731 



Viscous Flow between Parallel Planes 

As one of the typical conditions of flow met with m problems of 
lubrication and other practical applications of the theoiy of viscosity, 
it is convenient to consider m detail the flow of viscous liquid between 
two paiallel and closely adjacent plane walls, supposed fixed 

In rectangular co-oidinates, let z = o, and z = h be the 
parallel planes, h being small compared to their dimensions m the 
X and Y directions, as indicated in fig 9 

For the reasons already explained the components of velocity 



I2O 



THE MECHANICAL PROPERTIES OF FLUIDS 



normal to the planes must be eveiywhei e negligible In other words , 
the rates of shear and the momentum in the Z diicction are 
very small; consequently the fluid pressuie p does not vary in that 
direction but is a function of x and y only. Also the rates of change 
from the values of the finite velocity components, u, v, in the fluid 
to their values, known to be zero, on the walls are lapid compared 
to their rates of change m the X and Y diiections. Thus, considei- 
mg a rectangular element, as in fig 10, anywheie between the 





Fig 9 



Fig 10 



planes % = o and g = h, the viscous ti actions on its lower face 
in the directions in which x and y increase, aie. 



-- 

dz 



and 



03 



The corresponding tractions on the uppei face aie 

* \ o ' ' 

\dz 

//liri 

and 



The sums of these pairs of tractions added to the differences of the 
fluid pressures on the faces parallel to the YZ and ZX planes are 
respectively equal to the rates of increase of the momentum of the 
element m the X and Y directions, thus 



-T.* 

dt 



VISCOSITY AND LUBRICATION 121 

d z u dp . du 

or = 




and similarly /* ? = 

o being the density of the liquid 

The rates of increase of velocity -^, _? are of the order of the 

du dv dt dt 

products UTT, and v~~, and are thus, if, as we assume, u and v are 

ox ay 

small, of the order of squares of small quantities. These momentum 
terms will therefore be neglected in this and the following discus- 
sions. With this stipulation the equations (13) reduce to 

d z u dp 

dz z dx' 

, d 2 v dp 

and LL = --. 

as 2 dy 

These can be directly integrated, since p is independent of #, and 
hus 

du i dp, , ^ . 

_(V _J_ ( i 

o o V 6 i ^IJ) 

G% jU. OX 

and u = --( + C,# + C 2 ), 

M QX\ 2 / 

and similarly v = ( -f- D,^ + D, ) 
udv\2 / 



Now since ?^ and v arc zero on the plane z = o, the integration 
constants C 2 and D 2 are each zero, and since u and v are also zero 
Dn the plane z = A- 

A 2 A 2 

2. + dA = - + D^ = o, 

2 2 

so that Cj. = D x = --. 

2 



i dp z(z h] 
" 
j 

and ' 



rp. 

Thus a 



122 THE MECHANICAL PROPERTIES OF FLUIDS 
and the resultant velocity of the fluid at any point is 



being in the diiection of, and proportional to, the most rapid fall oi 
pressure, and varying accoiding to a paiabolic law along each normal 
from one plane to the other, having its maximum value midway be- 
tween them. 

The total flow across a width dy (see fig 9) from plane z = o to 
plane % = h (m the direction of x increasing) is 

* TT Q { h j ^y^P[ h f 2 r\j 

S-yU = $y f ud% ~ I (^ %h)d% 

J o 2ft OXJ o 

_ ... A 8 3? 



or U = --*!.* (I7 ) 



Similarly the flow pei unit width in the y direction 

^ 7 h z "dp , ON 

is V = -- ~. . . ...... (18) 

I2[JiOy 

Thus the total flow in any diiection across a unit width perpendiculai 
to that diiection is equal to the rate of decrease of piessure in that 

h 3 
direction multiplied by the constant - 

I2/A 

The same relation evidently holds for the flow of a viscous liquid 
in the space between two concentric, fixed cylinders, in either the 
axial or the circumfeiential direction, provided that the ladn of the 
cylinders are so nearly equal that their difference can be neglected 
compared with either of them. 

In both of these cases, as well as in all othei cases of flow between 
parallel surfaces plane or curved, it is evident, considering any small 
rectangular element S#, Sy, which extends in the z direction from 
one surface to the other, that since the same amount of fluid must 
flow out of, as flows into, the element in unit time, 



= * 

d s t> 2 
or from (17) and (18) 



VISCOSITY AND LUBRICATION 123 

it being lemembeied that the surfaces # = o and % = h, are 
assumed to be fixed. 



Flow between Parallel Planes having 
Relative Motion 

If the plane % = h is moving parallel to the plane % = o, with 
components of velocity % and z^ in the X and Y directions, uniform 

rates of shear ~ and -~ in these two directions will be superimposed 
h h 

on the fluid velocities u and v of (15) and (i$d). The components 
of velocity at # will become 

, i dp z(z h) . % 

___ '_ ~L n 

2 h' 



. . i dp z(z A) i % 
and v' = -- --- - + iy 
\Loy 2 h 

but neither the pressures nor the relation 

^ + ?* = o 

d x z dy z 

will be affected 

If, on the othei hand, the plane % = h is caused to move nor- 

mally away from the plane % o, with velocity -, so that the 

at 

distance between the planes continually mci eases at this rate, it is 
evident that an excess of inflow over outflow must take place thiough 
the sides (at light angles to the planes) of the elemental y volume 
hSxSy to supply the additional volume which is continually being 

added to the element at the late - SxSy 

dt 

Expressing this equality in symbols, 



, au, 

8y 8; 

J dx 


c -)- S# By = 
3y 


- ,;8*8y, 
dt 




au av __ 


dh 


01 


3^v 8y 


7t' 



and consequently from (17) and (18), 



= 
dx* d z dt 



i2 4 THE MECHANICAL PROPERTIES OF FLUIDS 



If we take the planes as being circular, of radius a, and suppose that 
the fluid between them at this radius is in direct communication 
with a large volume of the same fluid at constant pressure IT, it is 
evident from symmetry that the flow will 
be everywhere radially inwaids and that the 
pressure will diminish from II at radius a to 
a minimum at the centre. Taking, instead 
of a rectangular element, a cylindrical ele- 
ment extending from one plane to the 
other and contained between radii r and 
r + Sr as well as between two radial planes 
at a small angle Sa apart, its rate of in- 
crease of volume, see fig. n, will be 
This must be equal to the rate of increase of the 




Fig ii 



dh 



dt 



Sr.rSa. 



inward ladial flow as r increases by Sr, so that from (17), p. 123, 



or 



Integrating, 



s <j 
or roa 
dt 



dt 

dh , 



a / k 5 dp 



dr \i2ju, 9r 



Sr. 
J ' 



~ 
dr\ drJ' 



But from (17), since the radial velocity is zero at the centre, 

dp 

~ = o when r = o, 

or 

r A dp 6/x. dh 

^ = o and = -- r, 

dr h* dt ' 



so that integrating again 



3/z dh 

P = f-^ 
h 3 at 



When r = <z, = II ^ 2 + C, 

/t 3 dt 

so that ^> = II ^ ~( 2 r 2 ) ( 2 i) 

The force, P, necessary to move the plane at % h, against the 



VISCOSITY AND LUBRICATION 125 

viscous resistance is equal and opposite to the difference of pressure 
p II integrated over the whole circle, or 



. 
h* df 



/a* _ a* 
~ 4" 



} 

(22) 




Gup-and-ball Viscometer 

The type of viscous motion which has just been discussed is 
that on which is based the action of the cup-and-ball viscometer 
already mentioned on p. 19, and illustrated in figs 
12 and 13. In the actual instrument, however, 
as illustrated m fig. 12, the two parallel surfaces 
which are drawn apart aie not planes but segments 
of two spheres, one concave and the other convex. 
The fixed surface is the concave lower surface of 
a metal cup, to which is attached a hollow handle by 
which the instrument is suspended In the cup 
fits a steel ball, but its surface is prevented from 
making actual contact with the sphencal surface of 
the cup by three very small projections (j, fig. 12) 
horn the cup's spherical surface The two spheri- 
cal sui faces are thus maintained parallel and about 
o 01 mm apart, when the ball rests on the pro- 
jections The narrow interspace is filled with the 
liquid to be tested, and in addition a groove G formed 
aiound the edge of the cup, and having a capacity 
of a few cubic millimetres, is also filled with the 
fluid, which is held in both the groove and the 
interspace by capillary tension The groove forms 
the reservoir at constant pressure II from which the 
interspace is fed with fluid when the two surfaces 
are drawn apart, as in the preceding calculations. 

The force P employed to draw the surfaces apart is the weight 




Fig 12 



126 THE MECHANICAL PROPERTIES OF FLUIDS 

of the ball, which is usually of steel and I inch in diameter The 
method of making a test is merely, after placing sufficient liquid m 



f- f 1 -'.*--* v_ -.,, TTJBMW. \*"ff- r . 

? ^'.iTT^CaSNi;?/ 




Fig 13 Cup-and-ball Viscometer 



the cup to fill the groove and interspace, and pressing the ball home, 
to suspend the whole instrument and note the time by stop-watch 
which the ball takes to detach itself. The temperature of the 
instrument, which, on account of the good conductivity of the 



VISCOSITY AND LUBRICATION 127 

netal and the very small mass of liquid, is also veiy nearly the 
ernperature of the latter, is observed at the same time by means of 
L thermometer inserted in the hollow handle. 

The time of fall, the dimensions of the instrument being given, 
an be calculated approximately from formula (22), the spherical 
egments concerned, which m the actual instrument are compara- 
ively flat, being treated as circular planes of the same area. 

dt __ 

dh ~ 



J h 



'''^TTjLtfl* dh ___ 37TjllflY I __ I 

2 hJ 



n which t is the time of fall of the ball, of weight P, from its initial 
listance h^ to a final distance h z from the surface of the cup. 

This fall is to be consideied to be complete when the volume of 
luid drawn into the interspace is equal to the volume initially con- 
amed in the groove, i e 



/here S is the sectional area of the groove, 

thus ho = 
nd the time of the complete fall is 



+17 7 L 2S 

thus h z = //!]_ -| 8 



S(ah l -j- S) 



P 

nd if S is laige compared to ah 1} as it should be, 

77TU<2 4 , APh, 2 t , . 

t = 3 r and u= 3 i- ........ (23) 

v ^ 



It will be seen from the formula (22) that the velocity of fall 
- varies as the cube of the distance fallen through. It is thus very 

b 

mall at first, but increases very rapidly in the later stages, and there 
; no difficulty in practice in deciding the moment when the fall is 
irtually complete. 



128 THE MECHANICAL PROPERTIES OF FLUIDS 

Although the action of the cup-and-bali viscometer can be cal- 
culated with sufficient accuracy when its dimensions, including the 
initial thickness of the fluid film, are known, the determination of 
this thickness, that is to say the height of the three projections in the 
cup, with sufficient accuracy would be so difficult that in practice 
the mstiument is employed as a secondary viscometer only. Each 
mstiument requires, however, only a single calibration test, which 
suffices to determine a single constant for the instrument, applicable 
over its whole range. The corrections for the momentum of the 
fluid and for capillarity are negligible, the foimei because the velocity 
of the fluid is exceedingly low and the latter because the radius of 
curvature of the meniscus of the liquid in the groove is very laige 
compared to the thickness of the liquid film sublet to viscous 
traction. 

B. LUBRICATION 
The Connection between Lubrication and Viscosity 

Although viscous liquids and plastic solids have been used from 
the earliest times to dimmish friction between solid bodies moving 
in contact with one another, and although the practice of thus " lubri- 
cating " the bearings of machines has doubtless been universal since 
machines were first constructed, no rational explanation of the 
action of the lubricant was known until Osboine Reynolds (5), in 
1886, gave a clear interpietation of the phenomena in teirns of the 
theory of viscosity. Reynolds' explanation was only complete in a 
quantitative sense in the case of journal beanngs furnished with 
special, and at that date unusual, means for supplying ample quan- 
tities of lubricant. He showed that in such cases the solid surfaces 
aie completely separated from one another by fluid films of ap- 
preciable thickness, and that such films are maintained and enabled 
to support the pressure imposed on them quite automatically by the 
relative motion of the parts. The theory has since been extended to 
bearings of other kinds than journal bearings, and by its application 
new types of bearings have been devised for various pui poses which 
have proved far more efficient than the foims which they were 
designed to replace. 

While this " viscosity theory " of bearing lubrication is not 
quantitatively complete in all cases, and while there are probably 
other modes of lubrication in which viscosity does not play an 
essential part, it is at present true that all the most efficient known 



VISCOSITY AND LUBRICATION 129 

ypes of bearings which operate with sliding, as distinguished from 
oiling, contact utilize the principle of lubrication which was dis- 
:overed by Reynolds. The expenmental and theoretical work by 
vhich the principle has been developed may be followed in the 
>apers quoted in the bibliography attached to the end of this chapter, 
t is only possible in the present chapter to give an outline of the 
heory and a few of the leading results which have been established, 
vith examples of the practical forms of bearings in which the theory 
las been utilized 

The feature common to all the bearings to which Reynolds' 
heory can be applied is that the surfaces of the relatively moving 
tarts are not exactly parallel but slightly inclined to one another. 
? or instance, in ordei that a journal bearing of the usual type may 
>e lubricated according to Reynolds' principle, it is necessary that 
he journal shall be slightly eccentric in the bearing, so that the 
ilm of lubricant shall be of a thickness varying around the 
ournal 

Similarly, for the proper lubrication of a slipper moving rela- 
ively to a plane surface, it is necessary that the suiface of the 
lipper, if plane, shall be slightly inclined to the plane surface 
>ver which it moves. 

Essential Condition of Viscous Lubrication 

The explanation of this essential condition is readily given as 
n extension of the calculations contained in the first part of this 
hapter. 

Usually in the bearings to which the theory is applicable one of 
le surfaces can be considered as 2 

ontinuous or unlimited in dimen- 
lon in the direction of the relative 
lotion (as for instance the surface 
f a journal, or a thrust collar, or 
n engine cylinder), while the sur- 
ice of the other member is essen- 
lally limited or discontinuous in 
tie same direction (as the surfaces ' FIB 14 

f the corresponding bearing-brass, 

tirust-bearing shoe, or engine piston). In fig. 14, let XY be axes 
f co-ordinates (straight or curved) in directions at light angles to 
ach other along the surface of the continuous element, and Z the 

(D812) 6 




i3o THE MECHANICAL PROPERTIES OF FLUIDS 

co-oi dinate axis normal to this surface, i e. in the direction of the 
thickness of the film, and as before let u, v, w be the components of 
the velocity of the fluid at any point in these three directions The 
surfaces of the continuous and discontinuous elements are assumed 
to be nearly parallel, and the distance between them h to be small 
compared to their radii of curvature. The discontinuous surface 
is supposed to move with components of velocity u lt % in the X and 
Y directions, parallel to the continuous surface, at scy. The problem 
of finding the motions and pressures of a viscous film between the 
surfaces is the same as that discussed on p 123, except that the sur- 
faces are not now parallel. Considering, as before, the rate of change 
of volume and the flow of fluid into and out of an element extending 
from one surface to the other and standing on the base 8x, Sy, it 
is seen from equations (17), (18), that the rate of increase of 
volume of fluid m the element due to the rates of change of piessuie 
and of film-thickness, in the X and Y directions is 

8 / h* 8p\ s a . 3 / W 9p\ a s , v 

/ - ~ Y>xBy -f ( ; Sy8tf , ...... (24) 

ox\ 1 2ju ox/ dy\ 



while the rate at which fluid passes out of the element in consequence 
of the shearing deformation due to the movement of the upper 
surface over the lower is 



is $ , is % , ^ 

_1 _8#8y -f - 1 - Sj/Stf ........... (25) 

2 ox 2 oy 

The volume of the element is however diminishing, in conse- 
quence of the movement of the upper plane, at the rate 



8/L ^ 9/z~ -, 

u^oxoy -j- Vj_- oybx, 
QOC oy 



consequently 

3 / W dp\ , 9 / h 3 dp\ (u v dh , v, dh\ / dh , dh\ 

_ ( __ _ ) _L_ _ ( _ * J | _ f _ JL. _ i - j I 11 _ _j_ <y _, j 

dx\i2pdx/ dy \i2ju- dy/ \2 dx z dy/ \ dx dy/' 

9/ 7 o9*\ , 9/ 78 9p\ . , / dh . dh\ , ,. 

or A 3 -i- ) H- 7r + 6w u^~- + > 17r - = o (26) 

dx\ dx/ 2y\ dy/ ^\ l dx l dy/ v ; 

This is the general differential equation determining the value of p 
at every point, being solved by integration for each particular case 
when h is given as a function of x and y (thus defining the forms of 
the surfaces), and when the velocities %, ^ are assigned. The 
complete solution is often not practicable, but exact or approximate 



VISCOSITY AND LUBRICATION 131 

lutions can be obtained in a number of the simpler cases which 
n be regarded as sufficiently close approximations to the actual 
nditions of various types of bearings. 



Inclined Planes Unlimited in one Direction 

Take the case of two plane surfaces, the lower, # = o, being 
ilimited in the directions of X and Y, while the upper, also un- 
nited in the direction of Y,* extends only from x = a^ to x = a z , 
d intersects the plane ss = o on the line x = o. Thus the dis- 
ace between the planes, 
erywhere small, is proper- Z 

tnal to x, so that h = ex, 
icie c is the tangent of the 
lall angle between the planes 
;e fig. 15). Let us assume 
at the upper plane moves 
er the lower with velocity v/ 
, in the direction of X, v 
mg zero, and that the 
lole is immersed in fluid, so that the piessure both in front 

and behind the moving plane is H and is constant. Ob- 
Dusly none of the conditions vaiy in the direction of Y, so that 

and ~ are both zero. Thus equation (26) becomes 
oy 




'X 



Fig 15 



_ h , 6 u = 
dx\ dx/ 1 dx 






and therefore 



dp 

z ~ 

ox 



-J- 



= o, 



.(2?) 



> 
being the value of h where = o, that is to say at a point, 

ox 

? x 1} where p has a maximum or minimum value 

rpi dp f 

Thus - OW 

dx ^ 



ice -- is positive when h<h 1) and negative when h>h lt it is seen 

*The dimension of a beanng in the direction of the motion will in all cases be 
sried to as its length, and the transverse dimension as its width, regardless of 
ich of these is the greater. 



I 3 2 THE MECHANICAL PROPERTIES OF FLUIDS 

that p has a maximum value (at x = x ) : between x == a lt and 

Integrating, p = -^(- -~ - C), (28) 

but sincej> = II, both when x = %, and when x = a z , 

H = ^ 
from which 






2(2 1 <3:o / \ 

or ^ = - ; " ..... . ....... (29) 

&i + a* 

(the point of maximum pressure thus being nearer to a^ than to <7 2 ), 



and - 
n C 2 C 2 [a a. 2V 2 

= n 



Thus by substitution for ^ 15 and C in (28), 

* = n 

c 

6^% /% + a ^1^2 T \ /, n \ 

= 1J - "" , , v ( ~~ 5- ~~ I ) ---- (3) 

c 2 ^ + a 2 ) \ x X* ) 

this equation deteimining the pressures at all points between the 
two planes. 

The total upwaid pressure on the upper plane per unit width in 
the direction Y is 



2^ 

3 (<2 1 H- #2 



VISCOSITY AND LUBRICATION 133 

icing dependent only on the ratio of #., to a v for a given value of c, 
nd the mean pressure is 

P 6uMi i a* 2 ] f N 

log,- 2 - f ...... (32) 



Jso the total frictional resistance to the motion of the upper plane 
er unit width is 



F/ **1 J I f*' M '1. J f \ 
- I -t-sjtM . I I */7'V* I 1 ^ 1 
i r(, ^ _ i . w^ *, . , I -S { J 

117 f yin-i/i ^*^ *'* 

= ^ log,, (33^) 

c ^i 

ependent, like P, only on the ratio a 2 : a and c; and the ratio 
f traction to load, or " coefficient of friction " is 

F lo g fi " 

f = _ = f ^ f-7^ 

J p /- U't/ 

log,,-^ 2 X 2 

Iso the position of the centre of the upward piessuie on the upper 
ane is given by 

~ i [ a ~ 6ww r rtj ( ci ci \ 

* = pj - ] l ) xdx = p 6 2^_L Q J |( fl l + 2 ) - -^ ~ # j<& 



^ o 1 "2 

<2o" <2i 2^7i (2 2 1O&. 
*s i i j o^ 

- 1 --- (35) 

2 / \ i / i 



ing independent of . 

Applications to Actual Bearings 

The solution of the problem in viscous motion illustrated in 
;. 15 has been worked out in some detail because it affords m a 
igle case a general view of the nature of Reynolds' theory of lubri- 
tion. 

If we imagine the lower plane % o replaced by the surface of 
:ylinder whose axis is parallel to the Y axis of co-ordinates, and 




I 3 4 T HE MECHANICAL PROPERTIES OF FLUIDS 

the upper plane, extending from x = %, to x = a z , leplaced by 
a cuived surface which, at every point of co-ordinates x, y, measured 
respectively circumferentially from a generating line of the cylinder 
corresponding to x = o, and axially from a circumferential cncle 
of the cylinder corresponding to y = o, is at the same normal 
distance h from the cylinder as are the two planes from one another, 
the results which have been obtained will still apply This ideal 
form of a cylindrical journal beanng is illustrated in fig. 16 The 
cylinder can be regarded as the journal of an axle, and the upper 
surface as the bearing surface of the beai ing-brass of the axle. 

The lesults as to the fluid pressure which have been calculated 
above evidently remain true if, instead of 
the bearing-brass moving in the direction x 
with linear velocity w ls the journal revolves 
in the opposite direction with the same sur- 
face velocity. 

Actual bearings are, of course, not of 
unlimited width, but for the middle por- 
tions of a beanng whose dimension in the 
Fis l6 direction transverse to the relative motion 

is not less than two or thice times that 

in the direction of motion, the calculated results apply with fair 
accuracy. In such middle portions of the bearing the oil will flow 
in lines approximately at right angles to the geneiatmg lines of the 
cylinder. In the lateral portions of the bearing, on the othei hand, 
the oil being under pressure will tend to flow towaids the nearest 
side, and the theoietical conditions will on this account be depaited 
from If, however, the sides of the beanng be closed by some ar- 
rangement, such as a stuffing-box, preventing the escape of oil, the 
flow of oil will be everywhere, except within distances fiom the 
closed sides compaiable with h lt circumferential, and the conditions 
assumed for unlimited surfaces will be precisely icalized, piovided 
always, of course, that the bearing-brass is of such a form that 
h = ex, which is tiuc only to a first approximation for the form 
which is usually given to such brasses. 

The calculations apply more accurately to the case of a conical 
sleeve moving longitudinally on a cylindrical rod as illustiated in 
fig. 17. In this figure the axis of X is a generating line of the cylin- 
drical surface of the rod, the axis of Y is a circumferential circle, and 
that of Z as befoie is normal to the surface. As before, we assume 
that the normal distance between the surfaces is given by h = cx t 



VISCOSITY AND LUBRICATION 



135 



that the conical and cylindrical surfaces, which are coaxial, inter- 
ct at x = o. The sleeve extends from sc = <% to x = a%, and 
supposed to move parallel to the axis of X with velocity u^. 

From symmetry the motion of the fluid must be everywhere 
rallel to the axis of X, and as the cone and film of fluid have no 
undaries in the direction of Y, the solution given above will hold 
curately provided that the thickness of the film is very small 
mpared to the radius r^ and length, a z a^ of the cone. Thus, for 
ample, the resistance to the motion of the cone, from (33^), p. 133, is 



le curve /> x in fig. 18 shows the mode in which the fluid pressure 



r 



\j 




Fig 17 



Fig 1 8 



ween the surfaces of figs 15, 16, and 17 vanes m the direction 
x for the particular case in which > = za . It will be seen that 

' maximum pressure occurs at x 1 = -a 1 , or at one-third of the 

3 
gth of the sleeve or bearing-brass fiom its rear end, and, as may 

seen by wilting 2% for a 2 in (3^), the resultant piessure 
urs at a = i'43i 1 , or 0-431 of the length from the same end. 

Table III, p. 130, shows the actual numerical results in 
j.S units for a moving surface carrying a resultant pressure of 
Cgm. with a lubricating fluid of viscosity i C G.S The surface 
issumed to be i cm long in the direction of motion (i e. a 2 % 
i cm), and the results are expressed for i cm. of width in the 
tisverse direction The quantities tabulated are' 

h lt the thickness of film at x = a lt unit io~ 3 cm.; 
h z , the thickness of film at x = a 2 , unit io~ 3 cm.; 

- flp the distance of the centre of pressure from the trailing end. 



136 THE MECHANICAL PROPERTIES OF FLUIDS 

Unit, i cm", f, the effective coefficient of friction, = F x the tractive 
force in kilograms. 

The independent variable in the fiist column of the table is the 



a* 
ratio - 2 = 



TABLE III 

h z a i- / 



I 


o 


o 


o 5000 


00 


1-2 


02775 


0333 


04818 


3-285 X 10-3 


1-4 


03465 


o 4851 


o 4664 


2-428 


1-6 


03793 


o 6079 


04532 


2 065 


1-8 


03955 


o 7119 


04416 


1858 


2 


0-4026 


o 8051 


04313 


I 722 


2 2 


0-4043 


08895 


04221 


1-025 


24 


o 4027 


o 9665 


04137 


1-553 


2-6 


03991 


10375 


o 4061 


1-497 


28 


o 3942 


1 1037 


03991 


1451 


3-0 


03884 


I 1652 


o 3926 


1414 


4-0 


03559 


14237 


0-3662 


i 298 


50 


03247 


I 6237 


03465 


i 239 


60 


0-2982 


I 7892 


03310 


I 202 


II 


02115 


2 3269 


o 2832 


I I 34 



The corresponding results for any other dimensions and con- 
ditions of loading may be derived from the following dimensional 
formulas, viz 

If the length of the surface, velocity, resultant load, and viscosity, 
instead of being each unity in the units employed, are respectively 

Length, L centimetres, 
Velocity, V centimetres per second, 
Load, P kilogiams per unit width, 
Viscosity, M C.G.S. units, 

then, for any given value of , h t and h z aie to be multiplied by 

LV*M* a a-, , , l 1 _ . , ,.,.,, ^TITM-H-TI 

.p, is unchanged, and F is to be multiplied by V*P*M*, 

* a z a i m ]y[iyi 

while c and/ are to be multiplied by -^ . 

It will be seen from Table III, combined with these dimensional 
formulae, that the thicknesses of the films of viscous fluid concerned 



VISCOSITY AND LUBRICATION 137 

i lubrication are small, and comparable to the smallest linear mea- 
irernents which the mechanical engineer is accustomed to make. 
t is therelbie necessary in order to effect lubrication in the manner 
itended, and to secure the low frictional resistances which the 
icory indicates as attainable, that the workmanship of the bearings 
lall be of a relatively high order of accuracy. 

The fact, otherwise inexplicable, that the conditions and laws 
F viscous lubrication were not discovered until the end of the 
ineteenth century, is doubtless due to the circumstance that it was 
tily at about that epoch that mechanical workmanship became 
merally of such a quality that the necessary conditions were often 
)mphed with. With rougher workmanship the necessary con- 
nuous films cannot be formed, but the two members of the bearing 
)me into actual or viitual contact, at least at some points, and thus 
ting about mixed conditions of solid and viscous friction incapable 
* being referred to any simple or consistent laws 

Even with woikmanship which may be legarded as perfect the 
Itimate stage of failure initiated by any cause is contact of the solid 
irfaces, either directly or through the small particles of solid im- 
imties which aie always to some extent present in the lubricant 
'here is thus suggested as a cnterion of the safety of any bearing 
om such failure, the thickness of the lubricating film, at its thinnest 
nt under the working conditions which reduce this thickness to 
minimum 

It will be seen from Table III, p 136, that for a bearing surface of 
iven length, with given velocity, load, and lubricant, the thickness of 
ie lubricant at the point of closest approach to the other surface, is a 

laximum when -2 = 2-2 . . . It is usual to adopt this ratio as that 

. 1 

be preferred in designing beatings The table shows that with 

ds ratio the coefficient of friction, though higher than is attainable 

ith greater values of the ratio , is nevertheless already so small 

i 
lat its further reduction may usually be considered of little moment. 

must, however, be remembered that the optimum ratio, = 2-2 

j 

iften taken as z o as a sufficiently close approximation) has, strictly 
)eakmg, been derived only from the special case of a bearing surface 
" infinite length and for the condition h = ex. 

It is hardly necessary to remark that if the velocity HI in the above 
ilculations be reversed, the equations for the pressures will be still 

(D312) 6* 



138 THE MECHANICAL PROPERTIES OF FLUIDS 

valid, with merely a change of sign for both u and p It must, 
howevei, be lemembered that whereas in the case of positive values 
of p the intensity of pressure has no necessary limit, negative values 
of p, that is to say tensions, are not in general sustainable in fluids 
such as ordinary oils, and indeed in most forms of bearings positive 
values of/> less than H, the atmospheric pressure, aie usually incon- 
sistent with the assumptions made in the calculations, since, under 
those conditions, air will be drawn into the spaces assumed to be 
occupied by oil. 

The volume of fluid flowing between the surfaces per unit time 
may be calculated as follows: 

From (30), p. 132, the rate of change of the piessuie with x at 
the tear end of the bearing, i e. at x = a^ is 

dp _ _6/x 

dx c z 



Theiefore from (17), p. 122, and (25), p 130, the volume rate 
at which the fluid passes through unit width of the noimal plane 
at the reai end of the moving surface is 



- -^ .. (36) 



The same result would be obtained by calculating the inflow at the 
front edge, and it may also be seen at once from the considciation 
that at the point of maximum piessure x = x lt theic being no flow 
due to late of change of pressure, the volume rate at which fluid 

11 

passes the normal plane is entiiely due to the mean velocity 1 , 

2- 

acting ovei the film thickness, which is 



Under the same assumptions as in Table III, p 136,1110 value 

of Q for the condition = 2-2 is 2-78 X io~ 4 c. c. per second 

<h. 
per centimetre of the transverse dimension of the bearing. 



VISCOSITY AND LUBRICATION 139 

Self-adjustment of the Positions of 
Bearing Surfaces 

The question naturally arises how it is possible to secure in actual 
:arings the exact locations of the bearing parts shown to be necessary 
r the preceding calculations, and as illustrated in figs. 15-17, and 
>w it is that so delicate an adjustment is not liable to be destroyed 
r inevitable wearing of the parts The explanation is that in suc- 
ssful types of beaiings the paits are self-adjusting, their correct 
utual location being automatically brought about by their relative 
otion and continually corrected for any slight wear which may 
ke place. 

Take for instance the case of the infinite plane slipper illustrated 

fig. 15, of which fig. 19 is a sec- 
m on any plane parallel to XZ. 

It has been seen from Table 
I that if the latio of 2 to a { is 
z the resultant piessuie of the 
lid acts at the point x = a, 
'iere a a = 0-4221 X ( 2 ~ a i)> T 

id that with the value of h t Flg I0 

ven in the table and unit values 

ju. and n the total resultant piessurc is i Kgm per unit 
uisverse width. Conveisely, if a load of I Kgm per unit 
dth be applied to the slippei at the point x a as indi- 
ted by the anow in fig 19, and the slippei be moved with 
lit velocity and supplied with fluid of viscosity I C.G S , it will 
ke up the same position. Expeuence, moi cover, shows that such 
[iiilibnum is stable for the displacements which are liable to occur 

the operation of the bearings. In the case of plane slippers the 
id must in practice be applied as shown in fig 19, that is to say, 
lough an actual or viitual pivot ol some kind with which the slippei 
provided at the correct point. Actual examples \vill be illustrated 

the descriptions of thmst bearings given in the later parts of this 
apter. 

In the case of cylindrical journal bearings, however, there is 
other mode of self-adjustment possible, which, though not so 
icient as the pivot method, is even simpler, and which undesignedly 
ok place in bearings of this class long befoie Reynolds' principle 
is discovered, and rendered them superior in efficiency to all other 
isses of bearings known at that time. 




MO THE MECHANICAL PROPERTIES OF FLUIDS 

Self-adjustment in Journal Bearings 

This action, is illustrated for the ordinary foim of fixed journal 
bearings in figs. 2oa, b, c. We will assume that the bearing is one 
of a pair of journal bearings, as for the shaft of an electuc motor, 
consisting of a cylindrical brass, or pair of semi-cylmdiical half- 
brasses, of which only the lower half cylinder is noimally effective. 
The radius of the bearing is, necessarily, gi eater than that of the 
journal. The load W is assumed to be the weight of the shaft and 
parts attached to it, acting vertically downwards 

When the journal is at rest its position m the bearing is that 




shown in fig zoa. The journal and bearing are then in contact along 
the lowest generating lines of their cylindrical surfaces When, 
however, the shaft begins to rotate, foi example, in the clockwise 
direction as indicated m the figures, the oil at the light-hand side ot 
the journal is subjected to a traction directed fiom the wider to 
the narrower part of the interspace between the journal and the 
bearing. On the principles which have been explained, the oil in 
this space will consequently exert a fluid pressure. On the opposite, 
or left-hand side of the journal, on the contraiy, the mtci space 
increases m thickness in the direction of motion, and consequently, 
as explained on p. 137, the pressure m the oil film will fall, becoming 
negative unless, as is sometimes the case, air is fice to enter, when 
atmospheric pressure will tend to be established. The journal will 
consequently tend to move towards this left-hand side, the point of 
contact between journal and bearing shifting from the lowest gener- 
ating lines to some higher line towards the left hand. Oil tinder 
piessure will thus be admitted between the paits of journal and 
bearing, and this action will be progressive until the resultant up- 
waid piessure becomes equal to the load W on the bearing. At 




VISCOSITY AND LUBRICATION 141 

onslant speed a stable condition will be reached as shown in fig. 2ob. 
'he point of closest approach, C, will be somewhere on the left-hand 
de of the vertical, with a portion of the interspace above and to 
ic left of C still diverging in the direction of motion The oil in 
us latter space will in consequence exert a negative pressure on 
ic journal, as indicated by the arrow P 2 The resultant of this force 
ad the positive resultant pressure P ls exerted by the oil in the 
ght-hand converging portion of the interspace, will be equal and 
pposite to W, the load on the journal. 

If the speed of the journal is increased, the amount of con- 
ergencc and divergence of the respective parts 
f the journal, for a given load W, will auto- 
mtically dimmish, the limiting condition with 
ifimte speed (or zero load) being that illus- 
ated in fig 2oc, the journal becoming then 
Dncentric with the bearing 

It may be noticed that in all cases the 
ivergmg portion of the film, and the neaily Flg 3I 

arallel portions in the immediate neighbourhood 
f C, though of respectively negative and zero value for the support 
f load, are subject to sheai of equal or greater intensity than the 
ffective pressure-producing film on the right hand and lower 
11 faces of the journal For this icason such journal bearings, with 
ic biasses embracing a semicircle or other relatively large arc, are 
ecidedly inefficient compared to a pivoted bearing of small arc such 
> that illustrated in fig 16, in which self-adjustment takes place 
i the same mode as that described in connection with fig 19 

It is also readily seen that, in all cases, the interspace between 
ic journal and a segmcntal cylindrical bearing surface can only be 
mvergent throughout its length if the arc of the bearing surface 
less than 90 It is indeed desirable, in order to secure a fairly 
ipid rate of convergence thioughout, that the arc should be limited 
i 45 at most 

It will be seen that in such a case as that illustrated in fig 21, 

is possible, without pivoting the brass, for the icsultant fluid 

ressme to be vertical and thus in equilibrium with the load W, 

ithout the formation of any diverging interspace, and this even 

hen the radius of cuivature of the brass is the same as that 

the journal The latter is a convenient condition, as it 
Imits of the simplest and most accurate method of accurately 
rmmg the bearing surface, namely by scraping or lapping it. 



142 THE MECHANICAL PROPERTIES OF FLUIDS 

It is to be lemarked, however, that automatic self-adjustment in 
journal bearings with rigid (i.e. non-pivoted) brasses, as m fig. 2ob 
or 21, is only possible when there are not more than two bearings on 
a shaft, or if the shaft is flexible, as otheiwise, since a thud beaimg 
will be invariably out of alignment to an extent comparable with or 
greater than the thickness of effective oil-films, it is not possible for 
each of the journals to adjust itself to its correct position. With any 
number of pivoted bearings, however, if the pivots are appioxi- 
mately in the vertical plane through the axis of the shaft, each bearing 
will exert a vertical resultant piessure, and by providing adjustments 
for the pivots in the vertical direction only it is possible to divide 
the total load carried by the shaft equally between the beanngs. 



Exact Calculation of Cylindrical Journal 
and Bearing 

The mathematical solution of the viscous motion foi the case 
illustrated in figs, 2oa, 21 was given by Reynolds (5, p 158) The 
solution was simplified by Sommerfeld (8, p. 158), of whose process 

a brief lesume will now be given. In 
fig. 22, O and O' are the centies and r 
and r -f- 8 the radii of the cylindrical 
journal and semi-cylindrical bearing, 
both of infinite extension m the dnec- 

tion of their axes. 

<> 

Let OO' = e, and - = a, a having 

therefore different values in difleient 
cases, varying fiom i, when the journal 
Fig 33 and bearing are in contact, to oo when 

they ate concentric. 

Let i/r be the angle between OO' and the vertical and (j> an angulai 
co-ordinate measured from the direction OO', the co-ordinates for the 

ends of the bearing-biass being 




~, and iA -f - as indicated in 
2 2 



the figure. Thus the linear co-ordinate, x, in the direction of motion 
of the brass relatively to the journal is now constant rfi. Then 
from (27), p. 131, 



rd$ 



h* 



(37) 



VISCOSITY AND LUBRICATION 143 



md since p II when < = i/i - , and when $ = ^ + - , 



r 

J 



2 a 

, if p is the fluid pressure and q the circumferential traction per 
unit width at <f t and P the total load on the bearing per unit width, 

/> + - 
P cosi/r * (p II) cos^rJ^ 

J z J ~z 

md 

/> + E A|, + I 

P sin^ w (p IT) sm<l>rd(l> + I 2 # coafod(f> = o. 



But since 



. . jr 

fa r 

(p H) COS(bd<f) = \ (p 
, 7T L 



ind 



r5 r + r"/> 

(/> - H) smfW = - (p - H) cos./, + -77 

^ -E "- -",-1 / -E ^ 






md since p = II, both when $ = ift -\- - and when cf> = ijj -, 

2 2 

30 that the terms not under integral signs vanish, 

r-H 



. 



P 

r COS! ^' 

(38) 



/-V "1* 

md 

2 

Now from (33), p. 133, and (37), p. 142, 



in which the second term on the right can be neglected on account 
of the smallness of h compared to r. 



144 THE MECHANICAL PROPERTIES OF FLUIDS 
Thus the equations can be written 

/l/f +7T/2 
{(h hj)lh?}sm<f*ty = (P/r) cosi/, 

*::", 

/\li + tr[Z 
{(h hj)/h 9 }cos(f>d(f> = (P/r) smift, 

V-rr/2 

or, since h = e(a + cos</>), 7/j = e(a -j- 
these equations become 



J (a 
J (a 



(1 JL\2 I ( 

(X "1 COSC&I G * \ Cff - i - vwoyy i \jp*t\j(t i ifc-i . v 

/ /* COS(P o w i S] 

+ -// , . -,^^ = 



(a + cos<ji) 2 (a H- cos0) 3 6r X 



7T 



the integrals as before being from -- to tft 

2 2 

These integrations can be effected by usual methods,* and from 
the results Sommerfeld calculated the following numerical table, 

Table IV, in which r] -, and the " coefficient of fnction ",/= - -, 

r Pt 

where M is the moment of the frictional tractions about O. 

TABLE IV 



a a 


A 


co^ (1 


z 


/ 


I 


90 


I'D 





f\ X i oo 


I '02 


1 20 


o 998 


PYI 2 

f J?o. x o 012 


X o 94 


I-I3 


129 


- 098 


X 0-04 


X 091 


1'5 


135 


- 093 


X 008 


X o 92 


2-4 


133 


-088 


X 0-14 


X i oo 


63 


128 


o 72 


X 0-29 


X 134 


33'9 


120 


o 50 


X 0-62 


X 2 17 


00 


90 


o 


X oo 


X oo 



If the coefficient of friction / be plotted with z^ as the variable, 

*T cos^ . f(a + cos<) a f d<f> 

I .am I drp = ff> a / - . - 7 

J a -f cobf/> ' J a + cosf/p J a H- cosip 

= ^ _ a ? =tan -x / tan 
Va a -i \ 



Differentiate this with icspcct to to get intcgials in second line of (40). In- 
tegials in first line come at once, since d(a + cos^) = suvfxty. 



VISCOSITY AND LUBRICATION 145 

s 

P being constant, then for various values of v) Q = - we have a series 

of cuives, m which as % increases from zero, the coefficient of 
friction falls at first to a minimum value about 8 per cent lower 
than its initial value, then with further increase of % the coefficient 
gradually rises and finally increases to an asymptotic approximation 

to the straight line/ = i. 

The value of % for which the coefficient of friction is a minimum 
is approximately 

= P82 
12-5 X ftf* 2 " 

In actual bearings the initial value of the coefficient of friction 
will be much higher than that calculated, since with very low velo- 
cities, and values of a only slightly greater than i, the journal and 
bearing will be, owing to minute roughnesses of their surfaces, in 
metallic contact instead of being separated by a very thin continuous 
film, as assumed in the theory 

It is to be observed that in these calculations of Sommerfeld's 
the portion of the film between the point of closest approach, <f> = < , 

and <}> = i/f -f- is subject to a negative pressure. The possibility 

2 

ot such a condition may icasonably be postulated in very wide 
beanngs, but can hatdly be assumed in bearings of usual proportions 
unless special means are employed for preventing the entry of air at 
the sides 

Approximate Calculation of Cylindrical Bearings 

The method and results of Sommerfeld's investigation given 
above apply to the case of a cylindrical bearing whose angulai length 
in the circumfeiential dnection is 180. A similar process may be 
applied to beanngs of smaller angulai length, as in fig 21, such 
beanngs, as explained on p 141, being pieferable in practice In 
these cases, howevei, there is little value in the assumption that the 
suiface of the brass is a circular cylinder, and, especially in the 
pivoted type, it is usually sufficient to assume that the thickness of 
the mtei space is a Imeai function of </>, that is to say to apply in the 
case of a veiy wide bearing the method and results of pp. 131-133. 

If a closer approximation is desired, the form of the bearing may 



146 THE MECHANICAL PROPERTIES OF FLUIDS 

be approximately represented by the equation h cx m = C(n/>)"' 
with an appropriate value of m differing from unity The solution 
of this case has been given by Rayleigh (20, p. 159), who, however, 
found that in the numerical applications which he made of it, the 
results did not differ very materially from those derived from the 
simpler formula, h = ex. 

Plane Bearings of Finite Width 

A more important modification of the Reynolds' theory of beanngs 

with uniformly varying interspaces, is that which it requires for its 

application to bearings which are of limited width, and in which, 

consequently, there is a transverse flow of the fluid under pressure 

i to the sides, i e m the 

direction of Y, as well as 
flow m the direction of the 
relative motion X. 

The solution, as given 
A i a by Michell (9, p 158), m- 
*/"' volves rather lengthy cal- 

f a s culations, and we can give 

* 2 only an indication of the 

method and a few working 
formulas and constants 

In fig 23 (which corresponds to fig 15 for the case of infinite 
width), ABDC is a rectangular plate m the plane z = ex (the length 
of the plate being # 2 a^ and its width b) sliding m the dnection of 
X with velocity u^ 

The pressure is assumed to be uniform eveiy where except in 
the inteispace between the plate ABDC and the infinite fixed plate 
in the plane % = o, i e the boundary conditions of the plate are 
p = IT, when x = 15 or x = fl 2 , for all values of y, and also when 
y = QJ or y = b, for all values of x Between the two plates p 
must satisfy the differential equation (26), p 130, i.e 

9 /;<M , 9 fi$P\ ^ f ( dh . dh\ 

/r*-f -f -( h + M is- + v i^~ } J 

dx\ dx dy\ dyJ ^\ *dx l dy 




7 j 

or, since h = ex, and = o, 

dy 



a<t , j 

xdx d z cV 



VISCOSITY AND LUBRICATION 147 

This equation may be written in the form 

ox x ox oy c x TT 

{ . TTV . I . i^ry . . i miry . \ 

( sin- + - sini- +...+- sin -^ + . . . ) = o, 

\ b 3 b m b / 

.ince the sum of the series in brackets, for all the values of y with 

vhich we are concerned, viz. y = o to y = b, is -. 

4 

To solve this differential equation so as to give p as a function 
>f x and y, it is assumed that there is a solution of the form 

p = n + pj_ + p 3 -f . . . + p m + . . . ad inf. (42) 

.. . mrry 

f oirt ' 

S m *^ J "** * 

in which p m == > 

ITITTX 

m being a function of x only. 

The integer m can have only odd values, because p H must be 

ymmetrical on both sides of y = - 

. ^ miry 
Thus p n = S 00 '" b , 



/here ? is odd. 

If for brevity we write 24^^ = A, and = , 

oc 2 6 



. miry 
sin =-, 
o 



Thus the coefficient of sin ^ in equation (41), p. 146, is 



4 - i -i- 

2 



r 4 8 THE MECHANICAL PROPERTIES OF FLUIDS 

Every such coefficient must vanish, and consequently the factoi 
within the brackets may be equated to zero, of which equation the 
particular integrals are the BessePs Functions, I x (^) and K^), and 
the complete integral may be written in either of the foims 

&, = AA() + BJ^O - k(i + 2 -f -^ + ...),... .(44) 

or = A' w I 1 a)H-B' M K 1 (0-^( 

The second form, useful when is veiy large, being " asymptotic " 
The coefficients A^, A' OT , B OT , W m are to be deteirmned so as 
to make g m vanish for x == a 1} and x = 2 , and hence p m vanish 
for all values of y on these two lines 

These coefficients can only be determined aiithmetically, numeii- 
cal values being given to the quantities <z 1} a Z) and b. The steps of 
the calculation, with tables, aie given in the paper (9, p 158) 

The coefficients A m , &c , having been calculated, the values of 
p for as many points x, y, as may be desired aie also calculated aiith- 
metically, and when p is known the total fluid pressure supporting 
the block is determined by aiithmetical or graphical summation, 
from the relation 

P = f a *f b pdxdy ...... (46) 

J a^J 

The fnctional traction by (33), p 133, is F = log e , pei unit 
width, and a i 



M = Hog? = Alog .(47- 



for the whole of the square shoe, of area A = b(a z a). 

The point of action of the resultant pressuie is found by an 
arithmetical summation of moments By way of examples, a lew 
numerical formulae will be given. 

The total pressuie on a square bearing in which 

flg flj - - d^ o is Jr . 

c 

being, by comparison with formula (31), p. 132, only 0-421 of the 
total piessure on a portion of equal area, equal length and inclina- 
tion of a plane of infinite width, thus showing the effect of the 
escape of oil fiom the sides of the bearing. 



VISCOSITY AND LUBRICATION 149 

The position of the centre of pressure for the finite square block 
is at a distance 0-42% from the rear edge, as compared with 
o 431%, in the infinite beaiing. (See Table III, p 136.) 

The coefficient of friction is 10-3^. A further calculation 

serves to show that of the total quantity of oil which enters the inter- 
space at the leading edge of the square shoe approximately one-sixth 
passes out at each of the sides and the remaining two-thirds at the 
rear edge. 

Similarly, in the case of a bearing whose width transverse to the 
motion is only one-thud of its length, so that 

a z ~" a i a \ ~ 3^ 
the total pressure is 



ind the centre of piessure is 0-39%, from the rear end. 

These results, as already explained, are equally applicable to 
ournal bearings as to plane slide bearings provided that the foim of 
he beaiing surface and the position of the pivot are such that h = ex, 
ind <2 a a a^ 

Arithmetical evaluations of the pressures and frictional co- 
'ffiuents given by the above theory have been calculated for an 
>xtensive series of beaiing blocks of varying proportions by Torao 
Cobayashi (30) 

Cylindrical Bearings of Finite Width 

Mathematical treatment of cylindncal bearings of finite width, 
'onespoiidmg to the theory given above for plane bearings, does 
tot yet exist This is unfortunate, since the limitation of width 
ias an even gi eater cflect in a cylmdiical than in a plane bearing 
n reducing the pressures generated, and particularly so if the arc 
ubtended by the cylindrical bearing approaches a semicircle as it 
isually does in the conventional type of journal bearing. 

By way of illustration of this statement, we may take a journal 
'earing in which the bearing-shell subtends an arc of 120, 
nd in which the thickness of the film at the inlet is double its 
hickness at the outlet (as in the plane bearing previously discussed). 

Such a bearing is illustrated in fig 24,* in which r is the radius 
* Fiom (28), p 159. 



150 THE MECHANICAL PROPERTIES OF FLUIDS 

of the journal, r ~\- 8 that of the bearing-shell, the distance between 
their centres being e, while 26 is the width of the bearing. 




It is readily seen that in such a bearing the areas through which 
oil may escape at the sides of the bearing are much greater relatively 
to the areas at the front and rear (at which the oil would enter and 
leave in a two-dimensional bearing) than is the case in a plane 
bearing of similar length and width. 

In the particular case in which the width of the bearing is equal 
to the radius of the journal, and in which the radii of journal and 
bearing are equal, so that 

8==o 
2,b = r, 

the respective areas of the leading, trailing and side openings are in 
the proportions 

2:1: 10-3. 

In other words, the oil which enters the interspace at the leading 
edge has more than 10 times greater area by which to leave at the 
sides than at the rear Pressures are consequently determined almost 
entirely by the conditions at the sides, and a two-dimensional solution 
would convey a very false idea of the actual conditions 

A more useful approximation in such a case can be obtained by 
treating the bearing as infinitely long in comparison with its width. 
On this assumption the pressures over the portion of the bearing 
near its trailing end (which is the only portion m which effective 
pressures will be generated) are given by 

P 



VISCOSITY AND LUBRICATION 151 

being the axial co-ordinate measured from the middle circum- 
irence, and h the varying thickness, determined by 

h = 8 -j- cos#. 



Experimental Results 

The curves given in the right-hand half of fig 24 are derived 
rom an extensive series of tests of a pivoted journal bearing of which 
he circumferential length was 6 98 cm. and the width was 6-35 
m , the block being thus not quite square From examination of 
his diagram it will be found that for a given load the coefficient of 
action varies approximately as V/^'u while for a given value of fiu lt 
varies nearly inversely as the square root of the load, both these 
esults being m accordance \\ith the formulas above The facts 
tated are brought out more explicitly in the following table, which 
hows that the values of F, P and pu^ as read off the right-hand 
art of fig 24, make Fy'(P//xM 1 ) approximately constant. The 
ift-hand part of fig 24^ will be explained on p. 157. 

TABLE V 

F P/ju/! V(P//ii) io 3 FV(P//<i) 



o 0008 o 12 o 35 o 28 

OOI2 067 26 31 

0016 037 19 30 

OO20 023 15 30 

0024 017 -13 31 



Types of Pivoted Bearings 

The chief practical field of application of plane pivoted bearings 
, to thrust bearings These usually take the form of an annular 
incs of " shoes " or " blocks " pivoted upon fixed points in the 
ationary casing and presenting their plane working surfaces to a 
lane-surfaced annular collar fixed on the rotating shaft. Such a 




N 



E3 

01 

am 

u 

S 

"s 



152 





I'K, _>() I IHU SI SllOl S 




I I(r 28 FlIKl'Sl Hi V.KIM, [OK IIOKI/OMVI SlIUlS, 



I'ating piigf i ^, 



VISCOSITY AND LUBRICATION 153 

rust bearing arranged for a vertical shaft is shown in fig. 25. In 
is bearing the thrust shoes, t, are fixed in the lower part of the 
sing of the bearing which also serves as the casing of a journal 
taring for the thrust shaft. (The journal bearing is of the flexibly- 
voted type described on p. 155.) 
In order that the casing may form a reservoir for oil to 






Fig 23 

ibiicate both bearings, the journal surface is formed not diiectly 
pon the shaft, but on the outer surface of a collar T, attached to 
ic sleeve S, and forming also the thrust collar which revolves upon 
le annulus of thrust shoes t. This annulus is shown separately in 
o; 26 (The flexible journal ring is shown in fig. 31, facing p. 154). 
In figs 27 and 28 is shown another type of thiust bearing, con- 
snient for application to horizontal shafts. In this form, which 
. adapted to take thrusts in either axial direction, two pivoted thrust- 
loes R only are employed for each direction of thrust, each pair 
eing mounted in a common housing H, which is itself pivoted on the 





\n\\soi 



Hi \KINC, Snoi 




I' ic, 31 LMU.L JOURNAL BTAKING 



Facing page 134 



>/ 

Q s? 
(\A ^V 



* X 



\ v 



- ? 




VISCOSITY AND LUBRICATION 155 

yvver part of the fixed casing on an edge e at light angles to the 
ivoting nbs, rr, of the individual shoes. 

Fig 28 is a photograph of the parts of the bearing which is shown 
y longitudinal and cross sections in fig. 27. 

In fig. 29 aie shown two views of a pivoted journal bearing shoe, 
icing one of an annular series of four arranged for the journal bearing 
f a vertical shaft The pivoting edge e is clearly seen on the back 
f the shoe 

Flexible Bearings 

The comparatively small clearances and slight relative inclinations 
ictween coacting bearing parts, requisite to produce effective lubri- 
ating films, allow of a modified type of con- 
traction for achieving the same purposes as 
re attained by pivoted bearings It is evident 
hat in these a spring, or other continuous but 
leformable connection, may be substituted for 
he rolling or rotating contact of a pivot. Such 

spring may be either a separate pait attached Fl , 30 

joth to the shoe and its supporting member, or 
nay be an integial part of one or both of these provided such part 
s made with the necessaiy degiee of flexibility to allow of the shoe 
leflectmg under the load. Alternatively, as in a type of construction 
)roposed by Ferranti,* a pair of springs may be used to connect the 
hoe and its suppoit, viz a comparatively stiff spiing at the rear and 
i lighter spring at the front of the shoe This construction, which is 
llustiated in fig 30, will evidently have the effect of applying 
he resultant load at a point P, behind the middle point of the 
,hoe, much as if it were applied to a ngid pivot at that point. 

The chief advantage of a flexible construction is that it enables 
.mail or relatively unimpoitant beanngs to be simplified, by con- 
itmctmg a number of bearing shoes integral with, but flexibly con- 
iccted to, a common suppoitmg member. A serious disadvantage 
s that the flexibility involves more or less lisk of fracture of the 
lexible part, a danger which is to some extent oveicome by giving 
lexibility to a portion of the shoe itself. The large journal bearing, 
ilready mentioned on p. 153, is constructed in thk way, and is 
llustrated in fig. 31, and in fig. 25, p. 153. 

In the former the individual shoes, S, may be seen attached to the 

* British patent No. 5035/1910. 



156 THE MECHANICAL PROPERTIES OF FLUIDS 

supporting ring R, by flexible necks N, and having also their leading 
portions, L, reduced in thickness for some distance from the leading 
edges. 



Limitations of the Theory 

As the shoes of such thrust bearings as are illustrated in figs 
24*2: to 28 are usually of small radial width compared to their mean 
radii, the formute given for rectangular bearing slippers may usually 
be applied to them with sufficient accuracy for practical purposes 
in spite of their sectonal form. A more exact calculation can be 
made when required by a process which refers the co-ordinates of 
the sectonal shoe to those of the rectangular shoe.* 

Of greater practical importance are the departures from the results 
of the calculations which in some cases arise from the insufficiency 
of the physical assumptions which have been made, especially as to 
the constancy of the coefficient of viscosity 

An experimental method of solution, imagined and applied by 
Kingsbury (29), is free from most, if not all, of these limitations 
This method utilizes the identity which exists between the equations 
connecting pressure and volume-flow in viscous liquids, and potential 
and current in an electrical conductor The conductor used is a 
conducting liquid contained within solid, non-conducting boundaries 
shaped to represent in correct proportion (though on an exaggerated 
scale as legards thickness of the conductor) the lubricating film to 
be investigated The results obtained by Professor Kingsbuiy agiee 
closely with those of the mathematical investigations, e g those of 
the plane bearing of finite width given on pp. 146 of the piesent 
chapter. The method has been applied to both plane and cylindrical 
bearings of various ratios of length to width 

It was shown in Table I, p. 118, that the viscosity of lubricating 
oils diminishes rapidly as the temperature rises. In a well-loaded 
pivoted bearing, carrying for instance a mean pressure of 70 Kgrn 
per square centimetre, and with the product \JM-^ amounting to 2000 
C G S., and with usual dimensions, it can easily be deduced from 
calculations of the energy expended in overcoming the viscous 
friction, and of the heat capacity of the quantity of oil flowing through 
the lubricating film, that apart from conduction of heat through the 
metal, the oil would rise in temperature some 50 C. in passing 

* See (16, p. 158) Correspondence. 



VISCOSITY AND LUBRICATION 157 

uough the bearing. Conduction will diminish this rise of tempera- 
ire, but in most cases of heavily loaded bearings it is still sufficient 
i make the viscosity of the oil in the rear portion of the film much 
wer than in the leading portion. Thus, other conditions remaining 
lalteied, the outflow of oil at the rear will take place with a less 
pid fall of pressure m that direction, and the point of maximum 
essure will be shifted towards the front of the bearing. In fig. 18 
LC dotted curve pi, p. 135, Revue B.B.C. (19, p. 159), is figured 
i the assumption that the rise of temperature of the oil is such 
at its viscosity at exit is reduced to one-half of its value at entry, 
ie conditions being otherwise the same as those for the full-line 
irve as already explained on p. 135. 

The lower values of the fluid pressure throughout the film and 
e shift of the point of maximum pressure towards the leading edge 
e clearly seen. The point of action of the resultant pressure is 
so moved forward relatively to its position with constant oil tem- 
TAture, and it may even happen that the centre of pressure is at, 
in front of, the middle point of the bearing block If, for example, 
c direction of motion of a pivoted bearing is reversed, so that the 
vot is before instead of behind the centre of the bearing, it is 
11 possible in many cases for a lubricating film to be formed and 
essures generated m it in equilibrium with the load Such an 
"ect is shown in the left-hand half of fig 24*2, which shows the 
suits of revusing the bearing In such a case the oil film is neces- 
rily thinner, and the coefficient of friction higher than for the 
rrect direction of motion, but neveitheless the capability of being 
versed in this manner, and of even then working with coefficients of 
ction lowci than those of non-pivoted bearings, is a valuable 
opcrty of the pivoted type When, however, pivoted bearings are 
iploycd in this manner, it has always to be remembered that their 
ccess when running reversed depends upon the lubricant having a 
nsideiable rate of diminution of viscosity with rising temperature 
>r example, an experimental thiust bearing which ran very suc- 
ssfully in both dnections with water and with a mineral oil of low 
icosity as lubucants, 01 with carbon bisulphide when running in 
2 noimal direction, completely failed to run in the reversed direc- 
>n with the last-named fluid, doubtless on account of the peculiarity 

its viscosity-temperature relation, which has already been men- 
ned on p 117 

Effects of the same nature, which arise in the use of air as a 
meant in pivoted thiust bearings, have been pointed out and 



158 THE MECHANICAL PROPERTIES OF FLUIDS 

experimentally investigated by Stone * (24, p. 159). With air, owing 
to the viscosity of gases increasing with rising temperatures instead 
of diminishing as in liquids, pivoted bearings tend to be much less 
stable as to the inclination of the pivoted shoe than with liquid 
lubricants. 

On the other hand, as the same author has also remarked, the 
increase of the viscosity of the air film with temperature tends to 
increase the thickness of the film when a rise of temperature takes 
place owing to excessive load or undue resistance. The risk of 
direct contact of the bearing elements thus tends to become less as 
the bearing heats up, instead of greater as with liquid lubricants. 

Calculation and experiment agree in showing that the successful 
use of air as a lubricant demands the highest refinements of work- 
manship, with modeiate loads and relatively high speeds. 

BIBLIOGRAPHY OF ORIGINAL WORKS ON VISCOSITY OF 
FLUIDS AND VISCOUS THEORY OF LUBRICATION 

1 POISEUILLE. " Recheiches expeiimentales sur le mouvement des 
liquides " Memoir es de VAcademie des Sciences, 9, 1846 

2. STOKES, G G "Theories of the Internal Friction of Fluids in 

Motion, etc " Collected Papers, Vol. I, p. 75 

3. HIRN, A " Eludes sur les prmcipaux phenomenes que presentent 

les frottements medials, etc " Bulletin de la Soadti Industnelle de 
Mulhouse, 1855 

4 BEAUCHAMP TOWER Proc. Inst Mech Eng , 1883 and 1884. 

5 OSBORNE REYNOLDS " On the Theory of Lubrication " Phil 

Trans Roy Soc London, 1886, p. 157, also Collected Papers, Vol II, 
p 228 

6 GOODMAN Manchester Association of Engineers, 1890 

7. LASCHE " Die Reibimgsverhaltmsse in Lagern " Zeit thrift 

deutschet Ingcmeure, 1902 

8. SOMMERFELD. " Hydi odynamische Theone der Schmieimittel- 

reibung." Zeitschrift fur Math u Phys., 1901, 50, p 97. 
9 MICIIELL, A. G M. " The Lubncation of Plane Surfaces " Zeit- 
schnft fwr Math u. Phys , 1905, 52, p 123 

10 BRILLOUIN La Viscositd (Gautliiei-Villars, Pans, 1907) 

11 HOSKING " Viscosity of Water " Phil Mag , April, 1909, p 502 

* These expeurnents were made by means of a thrust beating consisting of 
quaitz-crystal thtust shoes and a glass thiust collar, the beanng suifaces being 
worked to true planes by optical methods Monochromatic diffraction bands 
produced by the closely adjacent pair of beaung surfaces at a slight mutual in- 
clination gave an immediate and very accurate measure of the thickness of the 
lubricating film 



VISCOSITY AND LUBRICATION 159 

2 ARCIIBUTT and DEELEY. Lubrication and Lubricants, 2nd ed. (London, 
1912) 

3. CAROTHERS, S P. " Portland Experiments on the Flow of Oil in 

Tubes." Proc. Roy Soc , A, 87, No. A. 594, Aug , 1912 

4. FAUST, O. " Internal Friction of Liquids under High Pressure." 

Gottingen Institute of Phys. Chem., 21st June, 1913. (Quoted in 
Report of British Lubricants and Lubrication Inquiry Committee, 
1920) 

5 GUMBEL " Das Problem der Lagerreibung " Berliner Bezirks- 

verem deutscher Ingemeure, 1st April, 1914 

6 NEWBIGIN, H T. " The Problem of the Thrust Bearing " Mm. 

Proc Imt. C E , 1914 

7 MARTIN, H M " Theoiy of Lubrication " Engineering, July, 1915, 

p 101. 

8 STONE, W " Viscometer " Engineering, 26th Nov , 1915 

9. " De F " " Pahers de butee modernes " Revue BBC, Jan -April, 
1917 

RAYLEIGH, LORD " On the Theory of Lubrication " Collected Papers , 

Vol VI, p 523 

1 HYDE, J H. " Viscosities of Liquids (Oils) at High Pressuies ' 

Proc Roy Soc A , 97, No A 684, May, 1920. 

2 MARIIN, H M "The Theoiy of the Michell Thrust Bearing" 

Engineering, 20th Feb , 1920 

3 LANCIIESTER, F W " Spin Geai Erosion " Engincenng, 17th June, 

1921, p 733 
1 Si ON I , W "A Proposed Method foi Solving Problems in Lubn- 

cdtion " The Commonwealth Engmect (Melbouine), Nov , 1921 
3 STONILY, G " Journal Beatings " Engineering, 3id Maich, 1922 
3 HERSEY, M. D and SIIORF, II "Viscosity of Lubucants under 

Pressure " Amer Soc of Meek Engineers, Dec , 1927 
7 BOSWALL, R O "The Theoiy of Film Lubrication", pp xi, 280 

(Longmans, London & New York, 1928) 
3. MICIIELL, A G M " Piogiess of Fluid-film Lubrication " Trans 

of Amer Soc of Mech Engineers, M S P 51/21, Sep -Dec , 1929 
) KINGSBURY, A " On Pioblems in the Theory of Fluid-film Lubn- 

cation, with an expenmental method of solution " Amer Soc 

of Mech Engineers, Dec , 1930 

) KOBAYASIII, TORAO " A Development of Michell's Theoiy of Lubri- 
cation " Report of the Aeronautical Research Institute, Tokyo 

University, No 107, June, 1934 

A comprehensive bibliography of the whole subject is contained in 
Notes on the Histoiy of Lubncation ", Parts I and II, by M D Hersey, 
wrnal Amer. Soc of Naval Engineers, Nov , 1933 and Aug., 1934. 



CHAPTER IV 
Stream-line and Turbulent Flow 

Stream -line Motion 

The motion of a fluid may be conveniently studied by con- 
sidering the distiibution and history of the stream lines, i.e the 
actual paths of the particles. If these paths or stream lines 
preserve their configuiation unchanged, the motion is called 
steady or stream-line motion. (See Chapter II, p. 57.) 

If the stream line be imagined 
to foim the axis of a tube of finite 
sectional aiea having imagmaiy 

V V \y J J I boundaries, and such that its aiea 

\J\v\ /^s at different points in its length 

V- >A f ^ ^ i s inversely proportional to the 

velocities at these points, this is 
termed a " stream tube ".* 

Such stieam lines must always 
have a continuous cuivature, 
since, to cause a sudden change 

in direction, an infinite foice acting at light angles to the dnection 
of flow would be necessary. It follows that in steady motion 
a fluid will always move in a curve around any sharp coiner, 
and that the stieam lines will always be tangential to such 
boundaiies, as indicated in fig. i, which shows the geneial foirn 
of the stream lines of flow from a sharp-edged orifice. With a 
very viscous fluid, the effect of cohesion may introduce compara- 
tively large forces, and the radius or curvature may then become 
very small. 

* See an alternative way of putting this idea, Chaptei II, p. 57. 




Fig i 



100 



STREAM-LINE AND TURBULENT FLOW 161 

Stability of Stream -line Motion 

Several conditions combine to determine whether, in any parti- 
alar case of flow, the motion of a fluid shall be stream-line or tur- 
ulent. Osborne Reynolds, who first investigated the two manners 
[ motion by the method of colour bands,* came to the conclusion 
lat the conditions tending to the maintenance of stream-line motion 
e 

(1) an increase in the viscosity of the fluid; 

(2) converging solid boundaries, 

(3) free (exposed to ail) surfaces; 

(4) curvature of the path with the greatest velocity at the outside 
the curve; 

(5) a reduced density of the fluid. 

he reverse of these conditions tends to give rise to tuibulence, 
does a state of affairs in which a stream of fluid is projected into 
body of fluid at rest. 

The eflect of solid boundaiies in producing turbulence would 
>pear to be due rather to their tangential than to then lateral 
flness One remaikable instance of this effect of a boundary 
Assessing tangential stiffness is shown by the effect of a film of 

on the suiface of water exposed to the wind The oil film 
eits a very small but appicciablc tangential constiamt, with the 
5ult that the motion of the water below the film tends to become 
stable This results m the formation of eddies below the surface, 
d the energy, which is otherwise imparted by the action of the wind 
form and maintain stable wave motion, is now absoibed m the 
stitution of eddy motion, with the well-known eflect as to the 
lling of the waves. 

Where two streams of fluid are moving with different velocities 
5 common suiface of sepaiation is in a very unstable condition 
ynolds showed this by allowing the two liquids, carbon bisul- 
ide and water, to foim a horizontal surface of separation in a 
ig horizontal tube. The tube was then slightly tilted so as to produce 

elative axial motion of the fluids, when it was found that the 
don was unstable for extremely small values of the relative 
ocity. 

This also explains why diverging boundaries are such a cause 
turbulence. Experiment shows that in such a case as shown in 

*Phil Trans, Roy. Soc., 1883. 
(D812) 7 



i6a THE MECHANICAL PROPERTIES OF FLUIDS 

fig. 2 the high- velocity fluid leaving the pipe of small section is 
projected as a core into the surrounding mass of dead water, thereby 
giving rise to the conditions necessary for eddy formation. 

More recent experiments* tend to show that the foregoing 
conclusions as to the effect of the curvature of the paths in affecting 
the manner of motion, are only true where the outer boundary of 




Fig z 



the fluid is formed by a solid surface, and that in some cases as 
shown at the impact of a steady jet on a plane surface, at the efflux 
of a jet from a sharp-edged orifice, and in motion in a fiec vortex 
curved motion, with the velocity gieatest at the inside and not at 
the outside of the curve, tends to stream-line motion. Generally 
speaking, wherever the velocity of flow is increasing and the pressuie 
diminishing, as where lines of flow are converging, there is an ovei- 
whelmmg tendency to stability of flow In a tube with conveiging 
boundaries it is this which leads to stability, and it is because this 
effect is sufficiently great to oveicome the tendency to turbulent 
motion to which all solid boundaries, of whatever form, give rise, 
that the motion in such tubes is stable for very high velocities. 



Hele Shaw's Experiments 

The fact that stream-line motion is possible at fairly high velo- 
cities between parallel boundaries if the fluid is viscous, and if the 
distance between the boundaries is small, has been taken advantage 
of by Dr. Hele Shaw,f who produced stream-line motion in the 
flow of glycerine between two parallel glass plates, and showed 
the form of the stream lines by introducing coloured dye solution 
at a number of points. By inserting obstacles between the 
glass plates the form of the stream lines corresponding to flow 

* Memoirs, Manchester Lit and Phil Soc., 55, 1911, No. 13. 
f Trans. lust. Naval Architects, 1898, p. 37. 



STREAM-LINE AND TURBULENT FLOW 



163 



ough a passage or around a body of any required shape can 
is be obtained (figs. 3 and 4). 




Fig 3 



The form of the lines to be expected in the case of two-dimensional 
v of a perfect non-viscous fluid around bodies of simple and 




Fifi 4 



164 THE MECHANICAL PROPERTIES OF FLUIDS 



symmetrical 
stream lines 
are identical 
spite of the 



T 



shape, may be calculated,* and an examination of the 
obtained in the Hele Shaw apparatus shows that they 
in form with those thus obtained by calculation, in 
fact that in one case the forces operating are entirely 
due to ineitia, and in the othei 
to viscosity. It has been shown 
by Sir Geoige Stokesf that this 
is to be expected, for if PQ and 
P'Q'(fig. 5) be two boundaries 
of a stream tube, and if PP' and 
QQ' be normals to one of the 
boundaiies, ultimately these will 
become elements of two con- 
secutive equipotential lines, and 
if produced will meet on the 
centre of ciuvatuie of the tube, 
so that if v and v -|- Sv be the 
velocities at P' and P, and it r be 
the curvatme and t the thickness, 

PQ _ r + t. 




P'Q' 



Sv 



Again consideiing the equilibrium of the clement P'Q'QP, now 
imagined as part of a perfect non- viscous fluid, the centrifugal 
force will be balanced by the difference of noimal piessmcs (Sp) 
on the inner and outer faces, and by the resolved part of the diflei- 
ence of pressure due to the difference of level (S^) between the two 
faces If directions towards the centre of curvature be called positive, 
on resolving normally, 



1) t 

On substituting for from (i) this becomes 
r 



-f- 



or 



-j- p 



* = o 
constant, 



* Hydrodynamics, Lamb, p 61, also Trans. Inst N. A., 1898. 
t British Association Reports, 1898, pp. 143-4. 



STREAM-LINE ANt> TURBULENT FLOW 165 

hich is Bernoulli's equation of energy for a perfect non-viscous 
nid. It follows that the velocity relationship indicated in (i), p. 164, 
hich obtains when viscosity is the dominating factor, is also 
msistent with the stream-line flow of a non-viscous fluid. 



Critical Velocity 

The nature of the two modes of fluid motion was first demon- 
rated by Osborne Reynolds* in a series of experiments on parallel 
ass tubes of various diameters up to 2 in. These were fitted with 
ell-mouthed entrances and were immersed horizontally in a tank 




Fig 6 

[ water having glass sides (fig. 6) In these expenments the water 
i the tank was allowed to stand until motionless. The outlet valve 

was then opened, allowing water to flow slowly through the tube, 
little water coloured with aniline dye was introduced at the 
itrance to the tube through a fine tube supplied from the vessel B 
At low velocities this fluid is diawn out into a single colour 
and extending through the length of the tube. This appears to 
e motionless unless a slight movement of oscillation is given to 
le water in the supply tank, when the colour band sways fiom side 
> side, but without losing its definition. As the velocity of flow is 
tadually increased, by opening the outlet valve, the colour band 
ecomes more attenuated, still, however, retaining its definition, 
ntil at a certain velocity eddies begin to be formed, at first inter- 
dttcntly, near the outlet end of the tube (fig. 7). As the velocity 

still further increased the point of eddy initiation approaches the 

*PhiL Trans. Roy, Soc , 1883. 



i66 THE MECHANICAL PROPERTIES OF FLUIDS 

mouthpiece, and finally the motion becomes sinuous throughout. 
The appaient lesser tendency to eddy foimation near the inlet end 
of the tube is due to the stabilizing influence of the convergent 
mouthpiece. 

The velocity at which eddy formation is first noted in a long 



tube in such experiments is termed the " higher critical velocity ". 
There is also a *' lower critical velocity ", at which the eddies in 
originally turbulent flow die out, and this is, strictly speaking, the 
true critical velocity. It has a much more definite value than the 
higher critical velocity, which is extremely sensitive to any distui- 
bance, either of the fluid before entering the tube, or at the entrance. 
Over the range of velocities between the two ciitical values, the 



HEAD 



VELOCITY 
Fis.8 



fluid, if moving with stream-line flow, is in an essentially unstable 
state, and the slightest disturbance may cause it to break down 
into turbulent motion. 

The determination of the lower critical velocity is not possible 
by the colour-band method, and Reynolds took advantage of the 



STREAM-LINE AND TURBULENT FLOW 167 

act that the law of resistance changes at the critical velocity, to 
letermine the values by measuring the loss of head accompanying 
lifferent velocities of flow in pipes of different diameters. On plot- 
ing a curve showing velocities and losses of head (fig. 8) it is found 
hat up to a certain velocity, A, for any pipe, the points lie on a 
.traight line passing through the origin of co-ordinates. From A 
o B there is a range of velocities over which the plotted points are 
/ery irregular, indicating general instability, while for greater velo- 
nties the points lie on a smooth curve, indicating that the loss of 
lead is possibly proportional to v n . 

To test this, and if so to determine the value of , the logarithms 



LOG H 




)f the loss of head h and of the velocity were plotted (fig. 9) 

Then if , 7 

h = k v ", 

log h = log k -{- n log v, 

he equation to a straight line inclined at an angle tan" 1 n to the 
ixis of log v, and cutting off an intercept log k on the axis of log h. 
On doing this it is found that if the velocity is initially 
turbulent the plotted points lie on a straight line up to a certain 
point A, the value of n for this portion of the range being unity. 
A.t A, which marks the lower critical velocity, the law suddenly 
changes and h increases rapidly. There is, however, no definite 
relationship between h and v until the point B is reached. Above 



i68 THE MECHANICAL PROPERTIES OF FLUIDS 

this point the relationship again becomes definite, and within th( 
limits of experimental eiroi, over a moderate range of velocities 
the plotted points he on a straight line whose inclination varies witt 
the roughness of the pipe walls. The values of n determined in this 
way by Reynolds are* 

Mateual of Pipe. n 

Lead . . . i 79 

Varnished . . . . i 82 

Glass ... . i 79 

New Cast lion . . i 88 

Old Cast lion . . 2-0 



those for cast iron being deduced from experiments by Darcy 

When tested over a wide range of velocities, it is found that 
the value of n in the case of a smooth-walled pipe is not constant 
but increases somewhat as the velocity is increased 

Between A and B the value of n is greater than between B and C, 
and the inci eased resistance accompanying a given change in velocity 
is gieatei even than when the motion is entnely tuibulcnt This 
is due to the fact that within this range of velocities eddies aic being 
initiated in the tube, and the loss of head is due not only to the 
maintenance of a moie or less uniform eddy regime, but also to 
the initiation of eddy motion. 

Messis Barnes and Coker* have determined the cutical velocity 
in pipe flow by allowing water to flow through the given pipe which 
was jacketed with water at a higher temperatuie The tcmpeiatuie 
of the watei dischaiging from the pipe was measured by a delicate 
thermometer. So long as the motion is non-sinuous, tiansmission 
of heat through the water is entiiely due to conduction and is 
extremely slow, so that the theimorneter gives a steady reacting 
sensibly the same as that in the supply tank. Immediately the critical 
velocity is attained, the rate of heat tiansmission is increased due 
to convection, and the change from stream-line to tuibulent motion 
is marked by a sudden increase in the temperature of the discharge, 

* Proc. Roy, Soc. A, 74, 



STREAM-LINE AND TURBULENT FLOW 169 

The law governing the relationship between the critical velocity 
I the factors involved was deduced by Reynolds from a considera- 
i of the equations of motion: for if the state of motion be supposed 
depend on the mean velocity in the tube and on the diameter, 
acceleration may be expressed as the difference of two terms, 
: of which is of the nature pvfd, and the other of the nature pv z . 
was then inferred that since the relative value of these terms 
bably determines the critical velocity, the lattei will depend on 
le paiticular value of the ratio p/pvd To test the accuracy of 
s conclusion experiments were made on pipes of diffeient dia- 
ters, and with different values of //- obtained by varying the 
iperature of the water between 5 C. and 22 C. 
The lesults of the expeiiments fully justified the foregoing 
elusions, and showed that the cutical velocity in a straight 
allel pipe is given by the formula 

P 



iie b is a numerical constant, and where P oc /j,fp. If the unit of 
*th is the foot, b equals 25-8 for the lower critical velocity, and 
) for the higher velocity, while if t tempeiaturc in degrees 

itigrade, T 

P = 

I -f- O 033681? + O OOO22lt Z 

Moie iccent expeiiments by Coker, Clement, and Barnes* and 
*rs earned out by Ekmanf on the ongmal apparatus of Reynolds, 
iv that by taking the greatest care to eliminate all disturbance 
ntiy to the tube, values of the higher cutical velocity considerably 
itei than (up to 3 66 times as great as) those given by the above 
nula may be obtained. The probability is, in fact, that there is 
definite higher ciitical velocity, but that this always inci eases 
i decreasing disturbances. 

A general expression for the lower critical velocity in a parallel 
;, applicable to any fluid and any system of units, is 



* dp 

_ 2300^ 
_. 

* Trans Roy Soc , 1903, Proc. Roy. Soc A, 74. 

f "Arluv for Matematic " Ast. Och, Fys , 1910, 6, No 13. 



(D312) 



i 7 o THE MECHANICAL PROPERTIES OF FLUIDS 

Thus foi water at o C., p,/p v~i92Xio~ 5 m foot-pou 
second units, so that 

>v k = -_ zL ft.-sec , wheie d is in feet. 

While for air at o C., 

v = 14-15 X io~ 5 ; 

/. v k 3 ft.-sec. where < is in feet. 

In this connection fig. 10* is of interest, as showing the icsul 
of experiments on a number of pipes of diffeient diameteis, wi 
air and watei flow, in which values of Rfpv z aie plotted as oidmat 
against the corresponding values of vd/v or of log(vd/v) Ilcie R 
the suiface friction per unit area of the pipe wall. The tuive consis 
of two parts connected by a narrow veitical band correspondii 
to a value of vd/v of approximately 2300, over which the points ft 
the various pipes are somewhat irregulaily disposed This ban 
indicates the range of instability between stream line and tiue tin 
bulent flow. The left-hand curve, coi responding to speeds belo 
the critical value, is calculated from the foimula 



, 
d 

theoietically conesponding to stieam-hnc flow f It will be seen tin 
the points for both air and water flow he closely on this cuive, an 
that the break-down of the stream-line motion takes place in a 
cases at appioximately the same value of vd/v 

As may be shown by an application of the pimciplc of dynamic* 
siimlaiity,J formula (3) is a paiticular case of the geneial foimul 

kv 

71, _ 

* I ' 

which is applicable to all cases of fluid motion. Heie / is the lengtl 
of some one definite dimension of the body The value of th 
constant k now depends only on the ioim of the surfaces over whicl 
flow is taking place. Thus in flow past similar plates immeised n 
water and in air, Eden has shown by visual observation that th< 

* Stanton and Pannell, Phil Trans Rov Soc A, 21*4- 
j Chap V, p 200 1 Chap V, p 193 

Advisory Committee for Aeronautics, T.R , 1910-11, p 48. 



AM 



M-LIf 



VD 

V 



4-6 



STREAM-LINE AND TURBULENT FLOW 171 

>e of flow, especially in the rear of the plate, is identical for identical 
ues of vl/v, where / is the length of any particular side of the 
ite. 

Critical Velocity in Converging Tubes 

In a converging tube the angle of convergence of the sides has 
arge effect on the critical velocity. At all ordinary velocities the 
tion in tubes or nozzles having moie than a few degrees of con- 
gence may be consideied as non-sinuous. Experiments on the 
v of water through circular pipes having sides converging uni- 
mly at an angle 6 gave the following approximate values for the 
r er critical velocity, at 14 C.* 

5 Deg 7 5 Deg 10 Deg. 15 Deg 



'At large section (3 in 1 

diametei) .. ..} '* r 94 * 44 3*5 

\ ltlc f l At throat (i| in dia-^ f f 

locit ^ meter) . 6o 77* 977 12 9 

-sec ' 

^dSSSET 1 ' 21 ".} 27 34S +34 " 3 



; lower critical velocity in a i-^-in. parallel pipe at this temperature 
20 ft. per second. Should the ratio of higher to lower critical 
cities have the same value in a conical pipe as in a paiallel pipe, 
would mean that in the case oi a i|-m jet discharging from a 
merging nozzle with steady flow m the supply pipe, the critical 
city would have the following values 

5 Degrees. 7 5 Degrees. 10 Degrees 15 Degrees 

tical velocity, ft -sec 39 50 63 84 

n flow through a pipe bend, the velocity at which the resistance 
es to obey the laws of laminar flow is less than in a straight pipe, 
re is now a considerable range of velocity over which the resis- 
e is proportional to a power of the velocity higher than unity, 
n which turbulence is not developed. This is due to the develop- 
* Gibson, Proc, Roy. Soc. A, 83, 1910, p. 376. 



172 THE MECHANICAL PROPERTIES OF FLUIDS 

ment of a cross circulatory current superposed on the laminar flow. 
The critical velocity is not well defined but experiments * indicate 
that full turbulence is developed at a somewhat higher velocity than 
in straight pipes. 

The Measurement of the Velocity of Flow 
in Fluids 

Several methods are available for measming the flow of fluids 
in pipes. Of these, the use of the Ventun meter or of the Pitot 
tube are the most common Recent investigations into the possi- 
bilities of the hot-wire anemometer have shown that this is capable 
of giving excellent results, and that it is likely to be especially 
valuable for the measurement of pulsating flow. 



The Venturi Meter 

The Ventun meter, invented by Clemens Heischel in iSSi , affords 
perhaps the simplest means of measuring the flow of a liquid When 
fitted to a pipe line of diameter greater than about 2 in its indications 
are, under normal conditions, thoroughly reliable so long as the 



- 2% d 




i\- 



Fit,', ii Venturi Meter 

velocity in the pipe line exceeds i ft. per second, and the discharge 
may then be predicted, even without calibration, to within i or 2 
per cent. 

The meter is usually constructed of approximately the propor- 
tions shown in fig. u, and consists essentially of an upstream cone 
usually having an angle of convergence of about 20, connected to 
* C. M. White, Proc, Roy. Soc. A, 123, 1929, p. 645. 



STREAM-LIKE ANt> TURBULENT PLOW 



downstream cone whose angle of divergence is about 5 30', by 
asy curves. One annular chamber surrounds the entrance to the 
tieter, and a second surrounds the throat, the mean pressures at 
hese points being transmitted to these chambers through a seiies 
>f small holes in the wall of the pipe. The two chambers are con- 
tected to the two limbs of a differential pressure 'gauge which records 
heir difference ot pressure h in feet of water. For this purpose a 
J-tube containing mercury may 
ie used as in fig. n. In this 
ase if the connecting pipes are 
all of water it may readily be 
hown that the difference of 
iressure in feet of water is equal 
3 12-59 times the difference of 
;vel of the tops of the mercury counter 
olumns By using an inverted 
J-tube with compressed air 
applied to the highest poition 

t the tube, the difference of 

icssure may be directly re- 

orded in feet of water When 

n automatic recoid is desned, 

ie type of mechanism shown in 

g 12 may be used. 

If P, A, V and p, a, v re- 
resent the pressures in pounds 

er square feet, the areas in 

q[uaie feet, and the mean velo- Flg 

tties in feet per second respec- 

vely at the entrance and throat of a meter whose axis is hori- 

ontal, neglecting any loss of energy between entrance and throat, 

ieinoulli's equation of energy becomes 




Driving^ 
Mechanism 



u 

I 

4o 
C 
UJ 






-Recording Mechanism for Venlun Metei 



X 2 



W 



W 



or 



= .4. ( 

W 



Y!/(_ 

2g\.\a 



A\ 



ft.-sec. 



(4) 



174 THE MECHANICAL PROPERTIES OF FLUIDS 

Actually owing mainly to fiictional losses the velocity is slightly 
less than is indicated by formula (4), and is given by 

/ 2gh 

v==c \// A \ 2 ; ft -- sec -> (s) 

\~a) 

wheie C varies from about 0-96 to 0-995,* usually increasing slightly 
with the size of meter. When used to measure pulsating flow, the 
value of C is reduced. The effect is, however, small for any such 
percentage fluctuations of velocity as are usual in practice, even with 
the discharge from a icciprocating pump. For accurate results the 
meter should be installed m a stiaight length of pipe lemoved from 
the i fluence of bends. Such bends set up whirling flow in the 
pipe, and this tends to increase the effective value of C. 

The Venturi meter may also be used to measuie the flow of 
gases.f In this case, for air, the discharge is given by 



Q = CAj3P,W lt . . ..(6) 

where P x is the pressure at entrance in pounds pel squat e foot, 
Wj is the weight per cubic foot at P and temperature T; 
and where, if p 1 is the pressure at entrance in pounds per square inch, 
j> 2 is the pressme at thioat in pounds per squaie inch, 
m is the ratio of areas at entiance and thioat, 
n is the index of expansion (1-408 foi dry air 
expanding adiabatically), 




If TJ be the absolute tempeiature at entrance on the Fahrenheit 

scale, on wiiting W x = ' ^ 1 (the value for dry air) (6) i educes to 

T i 

Q = i -ioC%/M~L Ib. per second, (7) 

VT X 

wheie #! is the aiea at entrance in square inches. 

*For a discussion of the vanability of the coefficient C, see "Abnormal Co- 
efficients of the Ventuu Meter ", Proc Inst C E,, 199, 1914-5, Pait I 

\ "Measurement of Aii Flow by Venturi Meter", Proc. Inst. Mech E., 1919, 
P 593, " Commeicial Metering of Air, Gas, and Steam ", Proc. Inst. (7. E., 1916-7, 
Part II, 204, p. 108. 



STREAM-LINE AND TURBULENT FLOW 



175 



Experiments * indicate that the value of the coefficient C is not 
nstant, but that it diminishes as the ratio p 2 /Pi is increased approxi- 
ately as indicated in the following table 



06 



098 



07 

0-97 



0-8 
0-96 



0-9 
0-94 



I'O 



0-91 



Measurement of Flow by Diaphragm in 
Pipe Line 

The coefficients of discharge of standard sharp-edged orifices 
scharging freely are known with a fairly high degree of accuracy, 
d where such an onfice can be used for measuring the steady flow 
her of a liquid or of air, the results may be relied upon as being 





Fig 13 



Curate within i or 2 per cent, if suitable precautions are taken- 
ymg to the convenience of the method and the simplicity of the 
paratus, much attention has recently been paid to the use of 
fices through diaphragms in a pipe line for measuring the flow. 
If D be the diameter of the pipe and d that of the orifice (fig. 13), 
idgsonf states that the coefficient of discharge C for sharp-edged 

*Proc. /Twit Mech E , Oct 1919, p. 593. 
tProe. IKS/. C.E., 1916-17, Part II, p. 108. 



176 THE MECHANICAL PROPERTIES OF FLUIDS 

orifice in a plate of thickness o-ozd, and for ratios of d/D less than 
o 7 is 0-608 for water or air when p z /pi is greater than o 985 and is 
equal to 0-914 0-306 p z /Pi for air, steam, or gas when j& 2 /Pi is less 
than o 98, pressures being measured at the wall of the pipe immedi- 
ately on each side of the diaphragm. 

A rounded nozzle, if well designed, has a coefficient which vaiies 
fiom about 0-94 in small nozzles to 0-99 for large nozzles, either for 
water or air, if p^/Pi is greater than o 6. Fig. 14* shows a form of 
nozzle in which the coefficient lies between o 99 and o 997. 

The Pitot Tube 

For measurements of the flow in pipes or in unconfined streams 
where the velocity is fairly high, the Pitot tube is capable of giving 
excellent results This usually consists of a bent tube teimmating 
in a small orifice pointing upstream, which is surrounded by a second 
tube whose direction is paiallel to that of flow. A series of small 
holes in the wall of the outer tube admit water, at the mean pressuie 



Tubes 03 Jftnn Won Two holes, one on each side 
thick r80D ^ OD through outer wall diam 02" 

Fig is 

in their vicinity, to its mtcnor, which is connected to one leg of a 
manometer. The other leg is connected to the cential tube carrying 
the impact onfice. If v is the velocity of flow immediately upstieam 
from this onfice, the pressuie inside the onfice, wheie the velocity 
is zero, is equal to the sum of the statical picssuie at the point, 
plus kv z /2g ft of water, where k is a constant whose value approxi- 
mates closely to unity in a well-designed tube. It follows that the 
difference of level of the two legs of the manometer equals ktP/2g, 
Figs. 15, 16, and 17 show modern types of this instiument. 
Fig. 15 shows the type used for measuring the air speed of aeroplanes 

* Engineering, ist Dec , 1933, p. 690, 



STREAM-LINE AND TURBULENT FLOW 



177 



and for wind tunnel investigations. A tube of this type having 
the dimensions shown gave K = i-oo within i per cent.* 

The tube illustrated in fig. 16 gave a value of C = 0-926 when cali- 
brated by towing through still water, and 0-895 when calibrated in 
a 2-in. pipe. The low value of C in still water is probably due to 
the fact that the pressure orifices are too near the shoulder of the 
pressure pipe. If this - . 

weie lengthened, with 
the orifices faither back, 
the coefficient would 
probably be higher. The 
difference between the 
calibration in still water 
md in the small pipe is 

o be expected, since 

he velocity at the sec- 
ion of the pipe contam- 

ng the pressure orifices 

s of necessity increased 

>y the presence of the 

ube, and the pi assure 

s lecorded by the static 

>ressure column will 

onsequently be less than 

ti the plane of the im- 

>act orifice. This effect 

/ill increase with the 

atio of the diameter of 

ube to that of the pipe, 

nd unless this latio is 

mall, the tube should be calibrated in a pipe of approximately the 

ime dimensions as that in which it is to be used. It is prefer- 

ble, foi pipe woik, to use a simple Pitot tube having only an 

npact onfice, and to obtain the static pressure from an orifice in 

ic pipe walls in the plane of the impact orifice. Using the tube 

i this way, the coefficient C may usually be taken as o 99, within 
per cent. 

In the type of Pitot tube shown in fig 17, and known as the 
Pitometer ", the pressure at the downstream orifice is less than the 

# " The Theory and Development of the Pitot Tube ", Gibson, The Engineer, 
7th July, 1914, p. 59- 



v///// 




JOIA 

Fig 1 6 Details of Pitot Tube 



178 THE MECHANICAL PROPERTIES OF FLUIDS 

statical pressure in the pipe and the coefficient is less than in the 
normal type. For the tube shown, calibrations in flowing water in 
pipes give a mean value of C 0-916. Owing to eddy formation 
at the downstieam orifice the coefficient of such a tube fluctuates 
within fairly wide limits. 

The Pitot tube may be calibrated either in still water or in a 
current. In the lattei case the mean velocity is computed from 
readings taken at a large number of points in a cross section, and 

TO DIFFERENTIAL 
PRESSURE GAUGE 




the coefficient of the mstiument is adjusted so as to make this mean 
velocity agree with that obtained fiom weir flow, current meter, -or 
gravimeter measurements on the same stieam. 

Without exception, obseivers have found that a still- water rating 
gives a somewhat higher value of C. The explanation would appear 
to be twofold. In the first place, the velocity of flow in a moving 
current is never quite steady, but suffers a series of periodic fluc- 
tuations, and since the Pitot tube is an instrument which essentially 
measures the mean momentum, or the mean (z> 2 ) of the flow and not 
its mean velocity, any such fluctuation superposed on a given mean 
velocity will give an increased head reading. In the second place, 
when metering a flowing current the average tube cannot be used at 



STREAM-LINE AND TURBULENT FLOW 179 

Doints very near the boundary where the velocity is least, and for this 
eason also the mean recorded velocity tends to be too high. 

It follows that although a still-water or still-air rating represents 
he true coefficient of the instrument, this requires to be reduced 
omewhat for use in a current, the effect increasing with the un- 
teadiness of the current. Where a high degree of accuracy is 
equired, the rating should be carried out under conditions as nearly 
s possible resembling those under which measurements have after- 
wards to be made. 

For measurements of the flow in pipes, the instrument should 

e used if possible at a section remote from any bend or source 

f disturbance. For approximate work the velocity of the central 

lament may be measured. This when multiplied by a coefficient 

fcich varies from o 79 in small pipes to 0-86 in large pipes gives 

ic mean velocity Alternatively the velocity may be measured at 

le radius of mean velocity, which vanes from o ja in small pipes 

> 075^ in large pipes, where a is the radius of the pipe. These 

dues, however, only apply to a straight stretch of the pipe, and 

it is neccssaiy to make measuiements near a bend, and in any case 

>r accurate results, the pipe should be traversed along two dia- 

eters at right angles, and the velocities measured at a series of 

dii If <$r is the width of an elemental y annulus containing one 

ncs of such measuiements whose mean value is v, the discharge 

then given by 

( a 
Q = I 2-nrvdr 

Jo 

Scvcuil methods are available for determining the mean velocity 
flow of a liquid in an open channel. This may be obtained: 

(a) by the use of current meteis giving the velocity at a seiies 
points over a cross section of the channel; 

(b) by the use of a standard weir; 

(c) by the use of floats; 

(d) by chemical methods. This method is best adapted to rapid 
d megular sti earns, although it may be applied to the measure- 
atit of pipe flow. It consists in adding a solution of known strength 

some chemical for which sensitive reagents are available, at a 
iform and measured rate into a stieam,* and by collecting and 

* For a description of the method of introducing the solution uniformly, refer - 
<e may be made to Mechanical Engineering, 44, April, 1922, p. 253, or to 
dro-electric Engineering > Gibson, Vol. I, p. 29. 



i8o THE MECHANICAL PROPERTIES OF FLUIDS 

analysing a sample taken fiom the stream at some lower point wheic 
admixture is complete. The solution should be added and the 
sample taken at a number of points distiibuted over the cross section. 
Various chemicals may be used Unless the water is distinctly 
biackish, common salt is suitable. If blackish, sulphuric acid or 
caustic soda may be used. With a solution consisting of 16 Ib. of 
salt per cubic foot of water, a dilution of i in 700,000 will give, on 
titration with silver nitrate, a precipitate weighing i mgm. pei litre 
of the sample, and the gravimetiic analysis of such a sample will 
enable an accuracy of i per cent to be attained. 

If Q = discharge of stieam to be measuied in cubic feet per second, 

q = quantity of solution intioduced in cubic feet per second, 
Cj = concentration of salt in the natural stream watei in pounds 

pei cubic foot, 
C 2 = concentration of salt in the sample taken downstieam in 

pounds per cubic foot, 

C = concentration of salt in the dosing solution in pounds per 
cubic foot, 

then Q = " 



and if V it V 2 , and V are the volumes oi silver nitiate solution lespec- 
tively necessary to titrate unit volume of noimal stream water, of 
the downstieam sample, and of the dosing solution, 

V-V 2 \ 

- v ) q ' 

Vj/ 

(e] By injecting colour and by noting the time icquiied for this 
to cover a measured distance. 

A close approximation to the tiue discharge may be obtained 
either by the use of a weir, of current meters, or by chemical methods 
if suitable precautions are taken.* The colour method would not, 
m general, appear to be so reliable, and float measurements cannot 
be relied upon for any close degree of accuracy. 

The Effect of Fluid Motion on Heat Transmission 

Apart from the effect of radiation, the heat transmission between 
a solid surface and a fluid in motion over it will, for a given differ- 
ence in temperature, be proportional to the rate at which the fluid 

*See Hydraulics, Gibson, 1912 (Constable & Co ),p 346. 



STREAM-LINE AND TURBULENT FLOW 181 

articles are carried to and from the surface, and therefore to the 
iffusion of the fluid m the vicinity of the surface. Such diffusion 
epends on the natural internal diffusion of the fluid at rest, and 
n the eddies produced in turbulent motion which continually bring 
-esh particles of fluid up to the surface In stream-line flow the 
scond source of diffusion is absent; the heat transmission can only 
ike place in virtue of the thermal conductivity of the fluid; and 
le rate of heat transmission is very small. Assuming that H, the 
eat transmitted per unit time per unit of surface, is proportional 
) 6, the difference of temperature between surface and fluid, the 
Dmbmed effect of the two causes may be written 

H = A0 + BM ........... (8) 



here p is the density of the fluid, A and B are constants depending 
a its nature, and v is its mean velocity 

As pointed out by Reynolds,* the resistance to the flow of a 
uid through a tube may be expressed as 

R = Afv + B'/*; 2 , ......... (9) 

id vanous consideiations lead to the supposition that A and B 
i (8) are proportional to A' and B' in (9) For assuming, as is now 
2neially accepted, that even in turbulent flow theie is a thin layer 
f fluid at the suifacc which is in stream-line motion, the heat trans- 
ussion thiough this layer will be by conduction, and fiom the 
oundaiy of this layer to the mam body of fluid by eddy convec- 
on In stieam-lme flow the tiansfei of momentum which gives 
se to the phenomena of viscosity is due to internal diffusion, while 
i turbulent motion the tiansference of momentum is due to eddy 
Dnvection, so that it would appear that the mechanism giving rise 
) icsistance to flow is essentially the same as that giving rise to 
eat transmission, both m stieam-lme and turbulent motion 

The following geneial explanation of the Reynolds law of heat 
ansmission is due to Stanton *]* Neglecting the effect of conduc- 
vity compaied with that of viscosity, the ratio ot the momentum 
>st by skm fiiction between two sections Bx in apart, to the total 
lomentum of the fluid, will be the same as the ratio of the heat 
:tually supplied by the surface, to that which would have been 
ipplied if the whole of the fluid had been carried up to the surface. 

* Reynolds, Manchester Lit and Phil Soc., 1834 

f " Note on the Relation between Skin Friction and Suiface Cooling ", Tech, 
eport, Advisory Committee for Aeronautics, 1913-3. 



182 THE MECHANICAL PROPERTIES OF FLUIDS 

Thus in pipe flow: 

if 8p is the difference in pressure at the two sections; 
ST is the rise in temperature between the two sections; 
W is the weight of fluid passing per second; 
v m is the mean velocity of flow; 
T m is the mean temperature of the fluid; 
T s is the temperature of the surface; 
a is the radius of the pipe; 

the above relationship becomes 

QJ./ 9.\ T7frC' r n 

op(Tra*) Wol , v 

W ~ W(T,-TJ (I > 

T'" 

The heat gained per unit area of the pipe per second is 




wheie a is the specific heat. If R is the lesistance per unit area, 

T> __ 7m2 dp 



so thai if H is the heat transmitted per unit area per second, 

H = Ro < T .- T J (II) 

*'j 

Since, as pointed out by Reynolds, the heat ultimately passes horn 
the walls of the pipe to the fluid by conductivity, a correct expression 
for heat transmission should involve some function of the conduc- 
tivity, and for this reason expression (n) can only be expected to 
give approximate results. In spite of this it enables some lesults 
of extieme practical value to be deduced. Thus if R = kv n , and 
if a be assumed constant, 

Hoc ^-'(T.-TJ, (12) 

so that if the lesistance be pioportional to a 2 , and if T s T m be 
maintained constant, the heat transmitted per unit area will be 
proportional to v, and since the mass flow is also proportional to 
v , the change in temperature of the fluid during its passage through 



STREAM-LINE AND TURBULENT FLOW 183 

the pipe will be independent of the velocity. Otherwise, the heat 
transmitted will be directly proportional to the mass flow. 

The general truth of this was demonstrated experimentally by 
Reynolds,* who showed that when air was forced through a hot 
tube, the temperature of the issuing air was sensibly independent 
of the speed of flow. 

In the case of the flow of hot gases through the tubes of a boiler, 
or of the water through the tubes of a condenser, n is usually less 
than 2 and has a value of about 1-85. Moreover, in the former case 
any increase in the velocity of flow will be accompanied by an 
increase in the temperature of the metal surface, so that for both 
reasons the heat transmitted is not quite proportional to the mass 
flow, and the issuing gases are slightly hotter with a high velocity 
than with a low velocity of flow. The difference is, however, not 
great, and it appears that by increasing the velocity of flow of the 
fluid, the output of a steam boiler, or of a surface condenser, may 
be considerably increased without seriously affecting the efficiency. 
This is in general accordance with Nicholson's f investigations on 
boilers working under forced draught. These showed that by increas- 
ing the mass flow, the heat transmitted to the water was increased 
in almost the same proportion, while the tempeiature of the flue 
gases was only slightly increased. 

Numerous other observers J have verified the general truth of 
the relationship expiessed in equation (12), p 182, for the flow of 
liquids and gases through pipes. Its more general application to other 
cases of heat dissipation in a current still awaits experimental proof. 

Experiments on the heat dissipation from hot wires of small 
diameter in an air current, show that this is proportional to v s , 
which, if this relationship is correct, would indicate that the resistance 
should be proportional to v l 3 . This is not in agreement with the 
generally accepted result that the resistance is proportional to v z . 
An examination of the experimental data shows, however, that the 
product of vd in the wires on which the heat measurements were 
made, was small, and an examination of the curve showing R/pv z d 2 
plotted against vd/v (fig. 4, Chap. V), shows that in this part of the 

* Memoirs, Manchester Lit and Phil. Society, 1872 

f " Boiler Economics and the Use of High Gas Speeds ", Trans. Inst of Engi- 
neers and Shipbuilders in Scotland, 54; " The Laws of Heat Transmission in 
Steam Boilers ", J T Nicholson, D.Sc , Junior Institute of Engineers, 1909. 

JStanton, Phil. Trans, Roy Soc, A, 190, 1897, Jordan, Proc. Inst. Mech. 
Engmeeis, 1909, p 1317, Nusselt, Zeitschnft des Vereines deutscher Ingemeure, 
Z3rd and soth Oct , 1909. 



184 THE MECHANICAL PROPERTIES OF FLUIDS 

range the curve is very steep, indicating that the icsislance is pro- 
portional to a value of n much less than 2. Although the data are 
insufficient to indicate the exact value of n they do not, at all events, 
disprove the foregoing hypothesis 

The difficulty in foiming any definite decision as to the general 
validity of the hypothesis arises fiom the fact that in most icsistance 
experiments on cooling systems, it has been tacitly assumed that 
the resistance is propoitional to v 2 , and the published data usually 
give the average value of the coefficient of resistance based on this 
assumption. Thus the resistance of honeycomb radiatois is known 
to be neaily proportional to a 2 , while the heat tiansmission pci dogiee 
difference of temperature is approximately proportional to v &s . 

Experiments by Stanton and by Pannell* show that while 
equation (u), p. 182, gives modeiate icsults for air flow thiotigh 
pipes, the calculated results obtained with water as the fluid aie 
very different from those deduced expenmentally, as appeal s itom 
Table I. 

TABLE I 



Pipe 
Dia, 
Cm 



0736 
0736 
139 

488 
488 
488 
488 



Mean Tern- Flictlon HeatTi. ins- 
Mean peiatuie AOCLIUU, mittc( j 
Vel.Cm D ^ es Calones 
Surface, Fluid, ^rL,^ pei Sq Cm 

pci Sec 

443 
5-08 

3-28 

o 0162 
o 0205 
o 0300 
o 0369 



per Sec 


Surface, 
Deg C 


Fluid, 
Deg C 


per a 


296 


282 


1593 


298 


296 


5*65 


3965 


26 o 


123 2 


47-2 


20 96 


50 6 


69 o 


473 


21 21 


171 


940 


362 


227 


3 *' 


1180 


374 


225 


51. 


1480 


435 


235 


8 i. 


2188 


43 


26-2 


149 



RffCn-T,,,) Fluid 

V,W 



1235 

10 5 

10-8 

6-5 

o 0109 

00155 
0-0266 
0-0267 



Water 



Air. 



It will be seen that in the case of air flow the heat transmission 
calculated from equation (n) is about 76 per cent of that observed, 
while for water the calculated value is twice as great as that observed. 

*Phil Trans. Roy Soc A, 190, Tech Report, Advisory Committee for Aero- 
nautics, 1912-3 



STREAM-LINE AND TURBULENT FLOW 



185 



Mr. G. I. Taylor* suggests that equation (u), p. 183, may be 
modified to take into account the effect of conductivity by assuming 
that there is a surface layer of thickness t, having laminar motion, 
through which heat is conveyed by conduction; that the velocity 
it the inner boundary of this layer is U, and the temperature Tjj 
md that between the centre and this layer heat transmission is due 
L o eddy convection. 

The temperature drop in the laminar layer, of conductivity k, 
s given by 

rri r " 



f R is the resistance per unit area at the surface, this will be equal 
o the resistance to shear of the lamina. 



, R - 
so that T,-T, = 



(13) 



ly analogy with (u) the rate at which heat is transmitted from the 
lyer by eddies is 

H = R ^~P. .. ..(14) 

(v m - u ) 

ubstituting for in (14) from (13) gives 



Tf U 

If = r, 

*> 
.-U i-r 



rp rp 

* 1 * m 

T.-T, 



U 



U 



= i 



A 

OjU 



T 

x 

T 

* *i 



H = 



7 

r\k 



(is) 



his equation is identical with (n) if the quantity in brackets is 
* Advisory Committee for Aeronautics, Reports and Memoranda, No. 272, 1916. 




i86 THE MECHANICAL PROPERTIES OF FLUIDS 

unity, i.e if fco- = k. For air this is very nearly the case, since 
k i-6ju,C ft where C v is the specific heat at constant volume, and 

C f . = , so that ~ = 0-88. In the case of water, however, at 
i '4 k 

20 C., ft. = o-oi, k = 0-0014, cr i-o, and ^r = 7*i. 

K 

Stanton * has shown that the value of r necessary to bdng the 
results as found from equation (15), p. 185, into line with the expen- 
mental results for water quoted in Table I, p. 184, is 0-29, and that 
similar experiments by Soennekei f require a mean value of 0-34. 
Taylor, from an examination of data by Lorentz,J concludes that 
the ratio is approximately 0-38. Some idea of its value in the case 
of air may be deduced from direct measurements by Stanton, 
Marshall, and Bryant, of the velocities in the immediate vicinity of 
the pipe wall. These measurements would appear to indicate that 

true laminar flow is instituted at a point where is appioxi- 

v m 

mately 0-14. They show, however, that it is erroneous to assume 
that at this point the change to true turbulent flow is abrupt, but that 
the change is gradual over an appreciable ladial depth of the fluid. 
It follows that equation (15) has not a strictly rational basis, but that 
by assuming r = 0-30 it gives results which are, for piactical pur- 
poses, not senously in error. 
The ratio 

drop in temp in surface layer __ T s T x __ / r N/xa 
drop in temp, in rest of tube Tj T m \i r' k' 

Thus, if the effective value of r = 0-30, the ratio is 3-0 for watei at 
20 C., and 0-38 for air. 

Reynolds j| has shown by an application of the piinciple of 
dynamical similarity that in the case of pipe flow 

dp _ v z ~ n n B 71 
8^ ~ dF* Vm A ' 

* Dictiona.} y of Applied Physics, Vol I, p 401. 

j- Komg Tech. Hochschule, Munich, 1910 

I Abhandlung ilber theoretische Physic, Band, I, p. 343. 

In air flow R = o 002/3 Vm 2 approximately, and Stanton 's experiments (Proc. 
Roy Soc. A, 1920) indicate that t is approximately o 005 cm. when /" = 0-00018 
and V m 1850 cm. per second This makes U O'l^Vm- 

|J Chapter V. 



STREAM-LINE AND TURBULENT FLOW 187 

nd if this value of be used in (10), p. 182, on writing 

vX 

V = 7rr 2 pv, the expression becomes 

dl _ B" g V*-* 1 n- Z( _ 

-** (T *~" T)> ......... (I6) 



/7T* 

/here now T is the temperature of the fluid, and - is the tem- 
erature gradient along the pipe. ^ x 

If T 4 is sensibly constant along the pipe, integration of (16) 
ives 

, T, T! __ B" g v 2 ~ n n- 2 , 
g ~-T ~ A #="" Vm ' 



here / is the length of the pipe, and T x and T 2 are the temperatures 
' the fluid at inlet and outlet. 

Stanton,* from experiments on heat transmission from water 
> a cold tube and vice versa, deduced the expression, for small 
dues of T! T 2 , 

log *ri-Tl = *&=- ^" a/ {(I + aTs)(l + ^ TJ} ' ' (l8) 

here, in C G S units, a = o 004 and ft = o-oi It will be noted 
at this expression is identical in form with (17), except for the 
st two factors, which were intioduced to take into account the 
Feet of the variation m conductivity, with temperatuie, of the suiface 
rn of water. In those experiments in which the heat flow was from 
etal to water, k had a mean value of 0-0104 With flow in the other 
tection k was, however, distinctly less, having a mean value of 
proximately 0-0075. 

3plication of the Principle of Dimensional Homogeneity 
to Problems involving Heat Transmission 

The principle of dimensional homogeneity, Chap. V,f mayieadily 
extended to problems involving heat transmission. In this case, 
addition to the three fundamental mechanical units, a thermal 
it is needed to define all the quantities involved. Taking tempeia- 
e T as this unit, the new quantities, heat flow H, conductivity k, 

* Trans. Roy Soc. A, 1897 

f See also a note by Lord Rayleigh, Nature, 95, 1915, p, 66, 



i88 THE MECHANICAL PROPERTIES OF FLUIDS 

and specific heat o- which are now involved, may be expressed 
dimensionally as 



_ 

JTl - 



/heat flow perl _ /energy pei) _ MT 2 3 

\ , , / - \ I - 1VJLJU( 6 . 

\ unit time j ( unit time j 



k I ^ x k^* 1 \ _ H 
1 sectional area X Tj LT 

I heat per unit mass } Jrlt r n^nm i 

O 1 = { / == == Ju t JL . 

Inse in temperature] MT 

If attention be confined to the large class of problems of practical 
importance, involving the transmission of heat between a fluid and 
a surface moving with relative velocity, where temperature differ- 
ences are so small not exceeding a few hundred degrees that 
radiation is only of secondary importance, the only quantities in- 
volved are H T & n- w 7 n // 
11, J. , K, a, v, i, p, fj, 

We select /, v, p and a as the four independent quantities, and 
combine them with the other four H, T, k and p, in turn, so as to 
obtain 8 4 (=4) dimensionless quantities K, as explained at 
p 198; i e. we write PI = l x v v p z a n , and similarly with T, k and /x 
We thus find 

Kj = ; K 2 = ; K 3 = 



' ~ r~~ --4 j > 
v* ivpa Ivp 

i ,/ H O-T k LC \ 

whence M -^-5-, -, - , ) = o ; 
\l 2 v 3 p v z Jvpa Ivp' 

* 



or H = 1V 



, 
Ivpa Ivp/ 

which, by combining the last two terms, becomes 

H = P /V^I, A, JL) 
\v z aii Ivp/ 

At such speeds as are usual in the case of air flow over air-cooled 
engine cylinders, of the flow of gases through boiler flues, or of 
heating or cooling liquids through pipes, experiment shows that the 
heat flow is sensibly proportional to v n , where n is between 0-5 and 
i-o, its value depending on the type of flow and the foirn of surface. 
Expeiiment, moreover, indicates that if radiation be neglected the 
heat flow is directly proportional to the difference of temperature 



STREAM-LINE AND TURBULENT FLOW 189 

etween the fluid and the surface, in which case the function m (19) 

lust be of the form F( , -), and(io) becomes 
a 2 \fft Iv/' v w 



H _ opvaTp _?_ f* } ( 20 ) 

J. J. fJl' (s(J JL Jl I 5 ~ I * t** \^jZf\JJ 

\ (T/i /W/) / 

(, \ 1 W / k \ 
J } f( }, 
10; Decomes ^ ^ - 

H = p n l l + n v n (j?- n aTf(~\ (21) 

\ CTL6/ 

/ k \ 

' the fluid to which k, a and p belong is a gas, /( J is a constant, 

> cr/Lt/ T 

/ K \ 

f the kinetic theory of gases. In this case, we may take for /( ) 

7 7 1 \OXl' 

/v /A \ 

ther A or B( ) , these being constants; thus obtaining the 
cr/x \ a/x / 

ternative forms for H, 



or Hoc / 1|n (yp(r) w A 1 - n T ............. (22) 

F is the total resistance to the steady motion of the fluid, then 
ice the heat loss per degree difference of temperature, per unit 
>ecific heat, is of dimensions 



_ 
o-T Vt-tT- 1 T 

bile F is of dimensions ML/- 2 , the ratio F/- is of dimensions 

' crT 

It = v, so that the index n in H oc v n should be less by unity than 
e corresponding index m F oc v n ' From this it appears that with 
>w so turbulent as to give the n 2 law of resistance, n in equation 
2) becomes i, and 

H oc PvpaT, ............... (23) 

ule with stream-line flow (n' = i and n = o), 

Hoc&T .................. (24) 

>r example, in the case of flow through similar pipes, where the 
erm may be taken to represent the diameter, equation (23) indi- 
tes that in such circumstances H is independent of the conduc- 
>ity of the fluid. Also since the weight of fluid W passing 



190 THE MECHANICAL PROPERTIES OF FLUIDS 

a given section per second is propoitional to the product d z vp 

Hoc WaT 

It follows that in similar pipes the heat transmission per unit degree 
difference of temperature between wall and fluid is proportional 
to the weight of fluid passing, or in other woids, that with a given 
inlet temperature the outlet temperature is independent of the 
weight of fluid 

If, however, n is somewhat less than i, as is usually the case in 
practice, equation (22) shows that 

nin 

H " 



oc 

so that with a given pipe and fluid, the heat transmission does not 
increase quite so fast as the mass flow, and the outlet temperature 
will increase somewhat as the flow is increased. 



CHAPTER V 
Hydrodynamical Resistance 

A body in steady motion through any real fluid, or at rest in 
moving current, experiences a resistance whose magnitude depends 
Don the relative velocity, the physical properties of the fluid, the 
ze and form of the body, and, at velocities above the critical, also 
son its surface roughness. 

At velocities below the critical, where the flow is " stream line ", 
e resistance is due essentially to the viscous shear of adjacent layers 
the fluid. It is directly proportional to the velocity, to the vis- 
>sity, and, in bodies of similar form, to the length of corresponding 
mansions Thus the resistances to the motion of small spheres 
such velocities are proportional to their diameters * 
With stream-line motion there is no slip at the boundary of solid 
d fluid, and the physical characteristics of the surface do not 
"ect the resistance. 

At velocities above the critical, wheie the motion as a whole is 
finitely tuibulent, theie would still appear to be a layei of fluid 
contact with the surface in which the motion is non-turbulent f 
ic thickness of this layer is, however, very small, and any increase 
the roughness of the surface, by increasing the eddy formation, 
;reases the resistance. At such velocities the resistance is due in 
rt to the viscous shear in this surface layer, but mainly to eddy 
mation in the main body of fluid. This latter component of the 
sistance depends solely on the rate at which kinetic eneigy is being 
'en to the eddy system, and is proportional to the density of the 
id and to the square of the velocity. 

Although the viscosity of a fluid provides the mechanism by which 
dy formation becomes possible, and by which the energy of the 
dies, when formed, is dissipated in the form of heat, it has only 
rery small effect on the magnitude of the resistance in turbulent 
>tion, and, as will be shown later, it can have no direct effect in 

* See H. S Allen, Phil. Mag , September and November, 1900. 
t Stanton, Proc Roy. Soc. A, 97, 1930. 

191 



iga THE MECHANICAL PROPERTIES OF FLUIDS 

a system in which the resistance is wholly due to eddy formatioi 
and in which the resistance is, in consequence, pioportional to v'' 

Experiments carried out over a limited iange of velocities hav 
usually shown that with turbulent flow the resistance of any givei 
body is proportional to v n , where n is slightly less than 2, althougl 
experiments on flow in rough pipes, on the icsistance of cylindeis 
of inclined plane surfaces, and of air-ship bodies, show that in sucl 
cases the variation from the index 2 may be within the limits o 
experimental error. With smooth pipes, however, n may be as 
low as 1-75, and with ship-shaped bodies of lair form in water is 
usually about i 85. 

Moie recent experiments* indicate that no one constant value 
of n holds over a veiy wide range of velocities, but that n inci eases 
with the velocity, and that a formula of the type 

R = Av + Bv z 



where A and B and C are constants depending upon the form and 
roughness of the body and on the physical propeities of the medium, 
more nearly represents the actual results Over a moderate iange 
of velocities a single value of n can usually be obtained which gives 
the resistance, within the errois of observation, and in view of the 
convenience of such an exponential formula it is commonly adopted 
in practice. 

At velocities above the critical, the direct influence of viscosity 
increases with the depaiture of the index n fiom 2 When n = 2 
the resistance is proportional to the density of the fluid, and, in 
similar bodies, to the squaie of corresponding linear dimensions. 

Between the low velocities at which the motion is stieam-lme, 
and the high velocities at which it is definitely tuibulent, there is 
a range over which it is extremely unstable, and in which the 
resistance may be affected considerably by small modifications in 
the form, piesentation, or surface condition of the body. Thus 
the resistance of a sphere, at a certain velocity whose magnitude 
depends on the diameter, is actually increased instead of being 
diminished by reducing its loughness. 

In problems occurring in practice, however, velocities are in 
geneial well above the critical point. One noteworthy exception 
is to be found in the flow of oil through pipe lines in which, owing 
to the high viscosity of the fluid, the motion is usually non-tuibulent. 

In hydrodynamical problems it is usual to assume that the 
*N.P.L., Collected Researches, 11, 1914, p. 307. 



HYDRODYNAMICAL RESISTANCE 193 

esistance depends solely on the relative velocity of fluid and body, 
ad that it is immaterial whether the body is at rest in a current of 
uid, or is moving through fluid at rest. Although there is not much 
irect experimental evidence on this point, it is probable that while 
ith stream-line motion the resistance is identical in both cases, in 
irbulent motion it is not necessarily so, and that it may be sensibly 
reater when the fluid is in motion than when the body is in motion. 

This is to be expected when it is realized that in a fluid in motion 
ith a mean velocity v, many of its particles have a higher velocity, 
) that the kinetic energy is greater than that given by the product 
f the mass and the square of the mean velocity. Any difference 
ising from this effect will in general only be small, but compara- 
vely large differences may be expected over the range of velocities 
i which the motion is unstable, owing to the fact that, with a sta- 
onary body, the interaction occurs in a medium which has an initial 
ndency to instability owing to its motion. 

Thus the system of eddy formation in the rear of any solid body 
Ivancing into still water may reasonably be expected to differ 
om that behind the same body in a current of the same mean 
'locity, owing to the instability in the latter case of the medium 

which, and from which in part, it is being maintained. 

Except in the case of stream-line flow, the -laws of hydro- 
rnamical resistance can only be deduced experimentally Much 
formation can, however, be obtained regarding these laws from 
i application of the two allied principles of dynamical similarity 
id dimensional homogeneity. 

Dynamical Similarity 

Two systems, involving the motion of fluid relative to geometn- 

lly similar bodies, are said to be dynamically similar when the paths 

iced out by corresponding particles of the fluid are also geometrically 

"mlar and m the same scale ratio as is involved in the two bodies 

The densities of the fluids may be different, as also the velocities 

ith which corresponding particles describe their paths If the 

;nsities in the two systems are in a constant ratio, and the velocities 

corresponding particles are also in a constant ratio, then the ratio 

corresponding forces can be determined. In fact, the scale ratio 

velocities and that of lengths being both given, the scale ratio of 

nes is determined, and therefore also the scale ratio of accelerations 

f means of the fundamental relation, force = mass X acceleration, 

e ratio of corresponding forces can then be found. 

(D312) 8 



*c,4 THE MECHANICAL PROPERTIES OF FLUIDS 

In two systems, denoted by (i) and (2), if w be the weight of 
unit volume, p the density, v the velocity, / any definite lineai dimen- 
sion, and r the radius of curvature of the path, these forces are in 
the ratio 



F 2 T! 

El l l 

Pz 4 2 

It follows that for dynamical similarity corresponding velocities 
must be such as to make the corresponding forces due to each 
physical factor proportional to pl z v z . The velocities so related are 
teimed " Coiresponding Speeds ". 

Where the only physical factor involved is the weight of the 
fluid, since the force due to this is proportional to p/ 3 , the required 



condition will evidently be satisfied if = 

v* Y 4 
On the othei hand, if viscous forces are all important, since 

dv 

the force due to viscosity equals p, - pei unit area, wheie fi is the 

dl 

coefficient of viscosity, Fj ^ ^ /^ Vj / g 

F /w- 7 2 v 



M 2 4 v z 

p I a /v \ 2 
and for this to equal ~ - 1 - ( ) it is necessaiy that 

Pz 4 Vcy 2 y 



or that - 1 



where v is the " kinematic viscosity " ft /p. 

Generally speaking, whei ever the influence of gravity is involved 
in the interaction between a solid and a fluid, as is the case where 
surface waves or surface disturbances are produced, and where the 
direct influence of viscosity is negligible, corresponding speeds are 
proportional to the square roots of corresponding linear dimensions; 
while where gravitational forces are not involved and where the 
forces are due to viscous resistances, corresponding speeds are 



HYDRODYNAMICAL RESISTANCE 195 

inversely proportional to corresponding linear dimensions, and 
directly proportional to the kinematic viscosities 

The flow of water from a sharp-edged orifice under the action 
of gravity is an example of the first type of interaction, while the 
resistance of an air-ship, or of a submarine submerged to such a 
depth that no surface waves are produced, is representative of the 
second type. 

The lesistance of a surface vessel is one of a series of typical 
jases, of importance in practice, in which both gravity and viscosity 
are involved, and in which therefore no two corresponding speeds 
will satisfy all requirements. In other words, the speeds which give 
^eometucally similar wave formations around two similar ships will 
not give similar stream-lines in those parts of the systems subject 
to viscous flow If, however, the influence of one of these factors 
s much greater than that of the other, approximately similar results, 
iVhich may be of great value in practice, can be obtained by using 
corresponding speeds chosen with reference to the important factor. 
Thus in tank experiments on models of floating vessels the corre- 
sponding speeds are chosen with reference to the wave and eddy 
effects, and are proportional to the square root of coi responding 
mear dimensions. This involves a scale error for which a correction 
s made as explained on p 213. 

Dimensional Homogeneity 

In view of the value of the lesults which may be obtained by 
he use of the principle of dimensional homogeneity in problems 
nvolvmg fluid resistance, the method of its general application will 
low be outlined 

The principle of dimensional homogeneity states that all the 
eims of a correct physical equation must have the same dimensions 
That is, if the numerical value of any one term m the equation 
lepends on the size of one of the fundamental units, every other 
erm must depend upon it in the same way, so that if the size of 
he unit is changed, every term will be changed m the same ratio, 
ind the equation will still remain valid 

The quantities which occur m hydrodynamics may all be defined 
n teims of three fundamental units The most convenient units are 
isually those of mass (M), length (L) and time (t) 

Example i. Suppose some physical relationship to involve only 
our quantities, say a force R, a length /, a velocity v, and a density p. 



196 THE MECHANICAL PROPERTIES OF FLUIDS 

Let it be assumed piovisionally that the relationship is of the fornr 

R oc tWp*. 

Expressed dimensionally, this gives 

ML*- 2 = L" . L"t- v . M"L- 3a , 

and, on equating the indices of like quantities, 



whence z= i, y= 2, x ~ 2 

It follows that R oc I 2 v z p, provided the initial assumption as to the 
form of R is correct. 

It is possible, however, to obtain the result without making this 
assumption. What we have really proved, in fact, is that the quantity 



is dimensionless Also, since it is given that there is a relationship 
between R, /, v, p, the quantity R/l 2 v z p is certainly some function 
of/, v, p, say/(/, v, p). Now, since the units of /, v, p are independent 
we can, by changing the unit of /, say, change the numerical value 
of / without changing the numerical values of v or p This change 
does not change the value of /(/, v, p), since it does not change the 
value of the dimeiisionless number R/l 2 v z p, to which /(/, v, p) is 
equal. Hence the function / does not involve /, and similarly it does 
not involve v or p; it is therefore a mere numeiical constant, so 
that R oc Pv*p 

Example 2 If more than four quantities of diflerent kinds are 
involved, for example R, /, >, p, //,, wheie //, is a viscosity (dimensions 
ML,- 1 *" 1 , p. 28), the assumption R oc l a> v ja p z p? would not allow us 
to determine the values of x, y, z, p by considerations of dimensions, 
since there would be only three equations in four unknowns. It is 
possible, however, to obtain a new dimensionless number, not in- 
volving R, but of the form 



Equating the dimensions of //. to those of l a v b p, we find 

i = c, i = a + b 3^, i = 1)\ 
and c i, b = i, a = i. 

Thus nflvp is dimensionless. 



HYDRODYNAM1CAL RESISTANCE 197 

For brevity, we shall denote the two dimensionless numbers now 
found by Kj, K 2 ; i.e. 

R j^. fjt, -f,. , 

== "!> "; == -"-2 \I,) 

fo/) 



It will now be proved that if there is a relationship between R, /, v, 
p, //. it can be expressed in the form 

Ki=/(K.), (2) 

where the form of the function / remains undetermined. In fact, 
since R by assumption is some function of /, v, p, p, and since we 
:an substitute Ivp K 2 for p, it follows that R// 2 u 2 /? is some function 
)f 7, v, p, K 2 , or say 

Kj = <(/, 0, />, K 2 ). 

Then exactly the same argument as that given under Ex. i proves 
hat the function (j> does not involve I or v or p, but only K 2 , and 
his is what was to be proved The relation K t = /(K 2 ) may of 
ourse be written in other forms, as for example K 2 = F(Kj) or 
i(K,, K 2 ) = o. 

^he general theorem of dimensionless numbers 

The geneial theorem, of which the two preceding results are 
'articular cases, may be stated as follows * 

(1) Let it be assumed that n quantities Q,, Q 2 , , Q w which 
re involved in some physical phenomenon, are connected by a 
Nation, 

F(Q lt Q, . . . , Q n ) = o, (3) 

Dntaming these quantities and nothing else but pure numbers 

(2) Let k be the number of fundamental units (L, M, t, . . .) 
'quired to specify the units of the Q's 

(3) Let QJ, Q 2 , . . , Q 7c be any k of the Q's that are of independent 
nds, no one being derivable from the others, so that these k might, 
we so desired, be taken as the fundamental units 

(4) Let Qa, be any one of the remaining n k quantities Q, and 

t , which we denote by K x , be the dimensionless 

n on & oX 

^1 v ^2 ' ^/c 

lantity formed from Q a and powers of Q lf . . . , Q fc . 

*E Buckingham, Physical Review, IV, 1914, p 345, Phil Mag , Nov , 1931, 
696. 



198 THE MECHANICAL PROPERTIES OF FLUIDS 
(5) Then the theorem is that the equation 

F(Qi, Qa , Qn) = o 
is leduciblc to the foim 

Kx, K 2 , . , K n _ 7c ) = o, (4) 

or, alternatively, 



The pi oof follows exactly the same lines as m the two particulai 
examples already given 

The actual forms of the functions (f> and / can only be deduced 
fiom experiment 

Jn the most geneial case of a dynamical relationship between 
any numbei of quantities, say n, there will be n 3 quantities (K) 
of zeio dimensions, composed of poweis of the quantities, and 
deduced in the manner explained above The relationship between 
the n quantities originally considered reduces to a relationship be- 
tween the n 3 quantities K 1} K 2 , . , K n _ 3 Hence if all but 
one of the K's are known, the last one is deteimmed 

Some oi the K's may often be written down fiom inspection 
Suppose, ioi example, that a certain phenomenon involves a time i 
and an acceleration g, in addition to the R, /, v, p, p, of Ex 2 above. 
Then we see at once that the new K's are tv/l and gift* The re- 
lation between the seven quantities is therefore of the form 

./ R //, to gl\ , 

</'( ,., , -,-, -7, .,) ^ o, (6) 

\ rvp Lvp I V" ' 

01, say, 

T> 70 o ,/ V tV Pl 

t\ rr t **$}** r- "*' 

JIX *" ' b \s /-> 



T ) 



Tt is useful to remailc that any product of poweis of the K's is 
dimension less. Hence if we multiply the second and third of the 
aiguments of <}> in (7), we get a new dirnensionless number tgfv, 
which may peilectly well leplace one of the two, tv/l and gl/v z , in 
(6) and (7). 

If two 01 moic quantities of the same kind are involved, as, for 
example, in the case of the resistance of an anship body, where 
both the length and diameter of the body affect the result, these 
may be specified by the value of any one, and by the ratios of the 
others to this one. Thus, in the problem just considered, if besides 



HYDRODYNAMICAL RESISTANCE 199 

t, v, p, p, g, t and the length / of a body, there are also involved the 
ireadth b and the depth d of the body, the solution is 

* 



, , , ........ (8) 

lvp v v 2 I // 

It is clear from the above examples that the actual arithmetic 
ivolved m working out an application of the principle of dimensions 
i of the simplest possible kind. The real difficulty is m making 
jre that all the essential quantities concerned in the phenomenon are 
emg taken into account. If this preliminary condition is not 
ilfilled, the result obtained will be quite erroneous 



Resistance to the Uniform Flow of a Fluid 
through a Pipe 

An examination of the factors involved m the non-accelerated 
totion of a fluid through a pipe would indicate that the pressure 
rop_/>// per unit length of the pipe may depend in some way on thcdia- 
teter d, the velocity of flow, and the density and viscosity of the fluid 
i a liquid where the eJlect of elasticity is negligible, it is difficult 
* imagine any other factoi likely to aikct the pressure drop, except 
ie roughness of the pipe walls , and, so long as we only considei pipes 

the same degree ol toughness, the geneial um educed lelationship 

of the form 

F(d, v, p, p, p/l) = o (9) 

ere the number of dirnensionlcss quantities K is 5 3 = 2 
oiclly as m a former example (p 196) we find 

T<r _ P d K __ /* 

XXj , J-Vg . 

Ipv" avp 

icie is some advantage m working with v, the kinematic viscosity, 
nch is equal to /././/?, rather than with //, itself. Hence K 2 = v/dv 
The reduced relationship may therefore be written 



10 
d 



leie the form of the function c/> remains to be found from experi- 
mt. Note that the value of the function (f> for all values of its 
jument can be found by varying one only of v, d, v. 



200 THE MECHANICAL PROPERTIES OF FLUIDS 

With str cam-line flow, experiment shows that - is proportions 

/ v\ ki> 

to v. It follows that r/n y~ must equal ---,, and that 
\av/ 1 va 



P _ 

7 ~ 'd 1 ' 

where k is a numeiical coefficient. This is Poiseuille's expressio 
for the resistance to viscous flow, the coefficient k having the vain 



32.* 



If the flow is turbulent the piessure giadient is approximate! 
proportional to v n , where n is between 1-75 and 2-0 In this cas 

(V \ I V \ ( V \ 2 ~" 

- } must be such as to make <i(- -) = k'{ } , so that 
v' \Jv' -/ 



dv 

P __ , 

d 



P __ j,pv" /v\ 
~~ 



or 



where h is the diflercnce of head at two points distant / apar 
expressed as a length of a column ot the fluid This is tl 
Reynolds | foimula for pipe flow 

If the lou'going assumptions aic coirect, on plotting observe 

values ol /., against simultaneous values of '' , in any series < 
IV" p dv 

experiments in which chflerent liquids 01 pipe s of different diameter 
but ecjually lough, are used, the points should he on a single cutv 
That this is the case foi fluids so widely different as ai 
water, and oil, has been shown by vaiious observers, notab 
by Stanton and Pannell \ (sec fig 10 at p, 170, where R = pdltf 
An example is given m iig. i, where expenmental points fc 
both aii and water He evenly about the two ciuves showi 
The agieement m the case of air is only close where tl 
drop in prcssuir is so small that the effect of the change of densii 

* Gibson, Hydraulic 1 * and its Applications (doubtable & Co., 1913), p. 69. 
1 tfetfntifit. I'tipd 1 !, Osboinc Reynolds, Vol. 11, p. 97, 
y Phtl. Tians, Roy, floe. A, 214, 1914, p 199. 



HYDRODYNAMICAL RESISTANCE *>t 

is negligible. For large changes of pressure it may be shown* that 
formula (12) becomes 



cT 



Here T is the absolute temperature of the gas, C is the constant 
obtained from the relationship pV = CT,f and v m and p m are the 
mean velocity and pressure in the pipe. The results of experiments 
on the flow of air through pipes by several experimenters, with dia- 



r-a so 

~ C 



40 



so 

t Z 3456789 

Values of v" Scale t division - 2870 units 
Fig i 



meters ranging from 0-125 m to 9 1 **> anci at velocities from 10 to 
40 ft -sec , confirm the accuiacy of this formula 

Below the critical velocity, n = i, and the formula becomes 



showing that the pressure drop is now independent of the absolute 
pressure in the pipe, a result confirmed by experiment. 

Equations (10), p. 191, (n), p. 200, show that, with an incompres- 

sible fluid, if the resistance to flow varies as v z , $(-=-) degenerates 

\dv/ 

into a numerical constant. Viscosity ceases to have any direct 
influence on the resistance,! and true similarity of flow should 
be obtained in all pipes at all velocities. If n is less than 2, the 

Gibson, Phil Mag , March, 1909, p 389. 

f In the case of air, if the mass is unity, and if p be measured in pounds 
weight per square foot, the value of C is 53-18 X 32 z = 1710, while if p be 
measured in pounds per square inch C becomes 1 1 -9. 

J See footnote on p. 209 

(D812) 8 * 



202 THE MECHANICAL PROPERTIES OF FLUIDS 



equations show that for similarity of flow in two pipes of diffeiei 

diameters, or conveying different fluids, it is necessaiy that shoul 

vd 

be the same in both cases. 

This has been shown to be tiue over a moderate range < 
diameters by Stanton.* who measured the distribution of velocit 
with turbulent flow across the diameters of two rough pip< 
of different diameters and repeated the measurements for tw 
smooth pipes. In the rough pipes (n = 2) the velocity distribt 
tion (CC', fig. 2) was the same in both pipes and at a 



AA Velocity distribution In btfo smooth pipes in which l^d is constant 
One pfpe t diareter 4 93cm, Velocity at axis 1525cm parsecond 

. . T40CM} 1017 ... 



0-4 




A B Smooth p!ps diameter 4 33 cm 



cc' Velocity Distribution In hvo rough pipes soscm Sii-sscm 
atoli velocities 



Radius, In terms ofrapius of pipe 
o*t 06 

Fig a 



velocities. In the smooth pipes (n < 2) identical cuives wei 

obtained only when was the same in each pipe. Undei othe 
vd 

conditions the curves were sensibly identical from the centre u 
to a radius of about 0-8 times the radius of the pipe, but differs 
appieciably at larger radii (AA' and AB, fig. 2). 

In any type of pipe the " critical velocity " at which the type o 
motion changes from stream-line to turbulent is obtained with 

constant value of , and this is generally true for fluid motion unde 
vd 

other circumstances. 

As already indicated, experiment shows that the resistance ti 

* Proc. Roy. Soc. A, 85, 1911. 



HYDRODYNAMICAL RESISTANCE 203 

flow in a smooth pipe where n is less than 2 is not strictly propor- 
tional to any one power of the velocity, and Dr C. H. Lees* has shown 
that Stanton and PannelPs results for smooth pipes are very closely 
represented by the empirical relationship 



where a = 0*35 and a and b are constants, so that 



In the rough pipe the ratio of friction to v z increases with velocity, 
and Stanton and Pannell suggest, for both rough and smooth pipes, 
an expression of the form 

T> 9 f A V I T7- P T 

F = pa a ]A- + K+B 
( vd 

in which K depends only on the roughness of the pipe. It will 
be noted that this relationship is similar to the one obtained by 
Messrs Bairstow and Booth fiom expenments on the normal icsis- 
tance of flat plates (p 213) 

Skin Friction 

The lesistance to the endwise motion of a thin plane through a 
fluid is usually termed " skin friction " Expressing the resistance 
by fAv", where A is the wetted surface, the values of / for 
vanous sm faces in water weie detei mined by Mr Froude f In 
these expenments a series of flat boards was suspended vertically 
fiom a carnage diiven at a uniform speed and was towed through 
the still water m a large basin The boaids were -fV in thick, 19 
in deep, and varied m length fiom i ft to 50 ft The top edge was 
submerged to a depth of if in., and the boards were fitted with a 
cut- water, whose resistance was detei mined separately 

A short resume" of Mr Fioude's icsults is given on the follow- 
ing page, these paiticular figures lefemng to a velocity of 10 ft -sec 

Here A refeis to varnished surfaces 01 to the painted surfaces 
of iron ships, B to surfaces coated with paraffin wax, C to surfaces 

* Proc Roy Soc A, 91, 1914, p 46 
f Btitish Association Report, 1872. 



204 THE MECHANICAL PROPERTIES OF FLUIDS 





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HYDRODYNAMICAL RESISTANCE 205 

:oated with tinfoil, and D to surfaces coated with sand of medium 
:oarseness. 

The results show that n decreases down to a certain limit, with 
in increase m length, but is sensibly independent of the velocity, 
decreases with an increase in length, becoming approximately 
onstant when the length is large. Owing to viscous drag, those 
>aits of the surface near the prow communicate motion to the 
vater, so that the relative motion is smaller over the rear part of the 
uiface and its drag per square foot is consequently less, as indicated 
>y a comparison of lines 2 and 3 of the foregoing table 

For the case of air, the most reliable work appears to have been 
lone by Zahm,* who has made observations m an air tunnel 6 ft. 
quare, on smooth boards ranging from 2 to 16 ft. in length, at 
elocities from 5 to 25 miles per hour The results are similar to 
hose obtained by Froude m water, m that the resistance per square 
Dot diminishes with the length, and, for smooth surfaces, vanes as 
1 8s . The following icsults, corresponding to a velocity of 10 
t -sec., show that the resistances under similar conditions with 
hort boards are approximately proportional to the densities of the 

Length (feet) 2 4 8 12 16 

Mean resistance,] 



pounds pet 0000524 0000500 0000475 
square foot J 



o 000477 o 000457 



NO fluids. Thus for a smooth boaid (Froude 's surface A) the 
Distance is 790 times as gieat in water as in air 

Foi strict companson, experiments carried out with the same 

allies of should be consideied. Thus, taking the ratio of v for 

v 

11 to v foi water as 13 i, a velocity of 10 ft -sec with the 4-ft board 
i water would conespond to a velocity of 325 ft -sec with the 
6- it bojid in air Taking the lesistance m air as proportional to 
1 8s , the lesistance per square foot of the i6-ft. board at this speed is 

R = o 000457 X (3 <3 5) J 8s = 0-00402 Ib , 
and R <z; a = o 00000380 

'he value of R v z for the 4-ft. board in water is 0-00325, the 
* Phil Mag., 8, 1904, pp. 58-66. 



206 THE MECHANICAL PROPERTIES OF FLUIDS 

ratio of the two being 855, a value only about 4 per cent greatei th, 
the relative density o water and air at 60 F. 

In view of the fact that one set of experiments was earned o 
in still water, and the other in an air current whose flow was n 
perfectly uniform, the agieement between the two sets of resul 
is very close. 



Resistance of Wholly Submerged Bodies 

Where a body is submerged in a current to such a depth th 
no surface waves are formed, gravity has no effect on the resistanc 
This will happen with a deeply cubmeiged submarine, 01 wil 
an air-ship. If the speed is constant,, so that there is no accelen 
tion, and if the liquid is incompressible, or if in a compressib 
fluid the speedc do not approach the acoustic speed so that piessui 
changes arc so small that the compressibility may be neglectei 
the resistance R may evidently depend upon the iclative velocil 
of fluid and body, on the density and viscosity of the fluid, and on tl 
size and shape of the body. In a series of geometiically simil; 
bodies each is defined by a single linear dimension /, and the icsii 
tance R will be given by the relationship 

F(R, /, v, P , p) = o 



As in the previous examples 



K 2 



Inserting the dimensions of /, v, p, R, and /x, and detei mining th 
indices x y, s, a, b, c, so as to make K x and K 2 dimensionless, give 
the values of K.JL and K 2 These are 

R - lpv 



p 
or R - plW <l> jpv 



The value of the unknown function <j> might be found by plottin 



HYDRODYNAMICAL RESISTANCE 207 

R 7?) 

bserved values of - against simultaneous values of , and 
7 2 z> 2 /> v 

y finding an empirical equation to represent the curve joining the 
dotted points. Moreover, it should be noted that since the values 
f both terms in the function are dependent on v, the form of the 
unction, for any liquid, can be determined from experiments on 
single body at different speeds in the same medium. 

In the case of model experiments, if the medium is the same 
3r model and original, and if the suffix m denotes the model, then, 

F the speeds be chosen so that v m = , we shall have 



o that </>( j becomes a constant, and 
R / 2 v* 



The speeds thus related are " corresponding speeds ", and at these 
peeds the model and its original are " dynamically similar " In 
his case the corresponding speeds are inversely pioportional to the 
meai dimensions, and at these speeds the icsistance of the model 
nd of the original are equal 

Unfortunately this relationship would involve such high speeds 
Q the case of the model as to be of no practical value. If, however, 
he model cxpenments can be carried out in a medium whose lune- 
natic viscosity is less than that of the original, the corresponding 
peeds aie reduced Thus by using compressed air m a wind tunnel 
he corresponding speed is reduced in the same ratio as the density 
3 increased, since the kinematic viscosity of air varies inversely as 
ts density. Such a wind tunnel is in operation at Langley Field 
USA) 

Adopting a working pressure of 20 atmospheres 

?!S = 1-JL 
V 20 7 OT ' 

o that with a i/io scale model, the corresponding speed in the 
vmd tunnel would be one-half that of the onginal, and at these 
peeds 



2o8 THE MECHANICAL PROPERTIES OF FLUIDS 



__ 

R Pv z 20 



With bodies whose resistance is sensibly proportional to th 
square of the velocity, the form of the function must be such t 

to make <f>(\ a constant whose value depends only on the shape c 

the body, and the resistance is given by 

R 



It is now immaterial at what speed the model experiments are carrie' 
out, so long as this is above the " critical speed ". 

This discussion applies equally well to any case of motion of 
totally immersed body in a medium whose compressibility may b 
neglected. 

Resistance of Partially Submerged Bodies 

When a body is paitially submeiged, or, although submerged 
is so near the surface that surface waves are produced, part of tin 
resistance to motion is due to this wave formation. The influenc* 
of gravity must now be taken into account, and we have the relation 

shl p 



Since there are now 6 quantities involved, 6 3 (= 3) K's 
are required. Taking /, v, and /> as convenient independent quan 
tities and proceeding as before, 

K! = R/l<Wp* t 
K 



Inserting the dimensions of R, /, v, p, p, and , and deteimining the 
indices #, a, a, &c , necessary to make K a , K 2 , K 3 dimensionless 
gives the values of K 1$ K 2 , and K 3 . These are 

R K - ^ K - gl - 

~ 



l 



so that M =?-}. (~),(^}\ = o, 

Vwp/ 



or 



HYDRODYNAMICAL RESISTANCE 209 

[n the case of model experiments, it is necessary for dynamical 
similarity that each of the arguments of $ shall have the same value 
for both model and original. But if, as is usual in practice, both 
are to operate in water, v is sensibly constant, while g is also constant, 
so that for exact similarity both Iv and v z /l would require to be 
constant In other words, neither v nor I can vary It follows that 
the lines of flow and the wave formation around a ship and its 
model in the same fluid cannot simultaneously be made dynamically 
similar It remains to be seen whether any of the arguments in- 
volved in the function <f> may reasonably be neglected so as to give 
an approximation which is likely to be of use in practice 

Experiment shows that the resistance of a ship-shaped body at 
the speeds usual in piactice is proportional to v", where n is approxi- 
mately 1-83, and the nearness of this index to 2-0 indicates that 
the direct effect of viscosity is small If it be assumed that this 
influence of viscosity is negligible,* the argument v/lv may be 
omitted from the equation, which now becomes, 

R = 
If now be made the same both for the model and the original, 

<7j2 

R, Pm l m 4 Pn$J Pm D /n' 

where D is the displacement. 

In this case the " coiiesponding speeds " at which wave and eddy 
formation are the same for ship and model, are given by 

lm 

Model Experiments on Resistance of Ships 

In practice these corresponding speeds are used, but allow- 
ance is made for the diffeient effects of viscous resistances in the 
two cases by the well-known " Froude " method 

*This does not involve the assumption that skin friction is ummpoitant or that 
viscosity plays no pait in the phenomenon It is m effect assuming that skin 
friction, instead of being pioportional to v" where n is slightly less than 2, is pro- 
portional to w a . In this case the i esistance is due to the steady rate of formation 
of eddies at the surface of the body, and, once these have been formed and have 
left the immediate vicinity of the body, the rate at which they are damped out by 
viscosity has no effect on the drag. 



2io THE MECHANICAL PROPERTIES OF FLUIDS 



In determining the icsistance of any proposed ship, a scale model 
is made, usually of paraffin wax, and is towed through still watei, 
the lesistance corresponding to a number of speeds being measuied 



60 



55 



5 



90- 



-00 




30 

i 
Iff 



Speed of Model in knots 
4 o 50 

2O 25 

Corresponding speed of shfps In knots 
Fig. 3 



by dynamometer. A curve AA, fig 3, is plotted showing lesistance 
against speed. 

The length and area of the wetted surface being known, the skin 
friction (f^A^^ ) is calculated, the coefficient being taken from 
Froude's results on the resistance of flat planes towed endwise,* 

* See pp 203, 204 



HYDRODYNAMICAL RESISTANCE iti 

ri he curve BB of frictional resistance can now be drawn, and the 
itercept between AA and BB gives the eddy- and wave-making 
esistance of the model. If now the horizontal scale be increased 



i the ratio S : i, where S is the scale ratio of ship and model, and 
" the vertical scale be increased in the ratio D : i, this intercept 
ives the eddy- and wave-making resistance of the ship at the corre- 
ponding speed. If fresh water is used in the tank, the vertical 
cale is to be increased again in the ratio of the densities of salt and 
resh water The skin friction (fAv n ) of the ship is next calculated 
nd set down as an ordmate from BB to give the curve CC. The 
itercept between the curves AA and CC now gives, on the large 
cales, the total resistance of the ship 

The resistance at any speed v may be calculated from model 
bservations at the corresponding speed v m , as follows' 

Total resistance of modeh _ .. 
, , ,s \ = R OT Ib. 

(observed) . . . . J m 

Skin friction of modeh _ . . n 
(calculated) .. ../ '"*?> 1D ' 

.'. Wave- + eddy-iesist-j _ __ - - p Ib 

ance of model . J "~ '" J ^ V " - r 1D * 

.'. Wave |- eddy-iesist-^| , 

ance of ship in salt I = DF X -^~ Ib. 

62 4 

water . ' ^ 

Total resistance of 6 DF , fAv n lb 



_ 
ship .. ../ 62-4 



Scale Effects Resistance of Plane Surfaces of Wires 
and Cylinders of Strut Sections 

From what has already been said, it will be evident that in 
lost model experiments some one of the factors involved tends 
3 prevent exact similarity and introduces some scale effect, 
t is only when this effect is small, and when its general icsult 
i known, that the results of model experiments can be used 
dth confidence to predict the performance of a large-scale 
rototype. 

The resistance of square plates exposed normally to a current, 



THE MECHANICAL PROPERTIES OF FLUIDS 



affords a typical example of scale effects. Expressing the resistance 
of such plates as 

R = K'pa 2 in absolute units (British or C.G S ) 

= Kw a , where R is in pounds weight per squaie 
foot, and v is in feet per second, 

experiments show the following values of K' and of K. 

K' K Size of Plate. 



Dmes * 
Canovetti f 
Eiffel J 



Stanton 



56 
56 

55 
56 

59 
61 
62 

52 
62 
62 



00135 
00134 


i ft. squaie 
3 ft diametei (circular) 


00133 

00136 


10 m. square 
14 , 


00142 


20 , 




00147 


27 , 




00150 
00126 
00148 
00149 


39 . 

2 , 

5 ft squai 
10 ft squ 


e 
ire 



n 

Such experiments show that while is almost independent 



of v, it increases by about 18 per cent as the size of the plate is 
increased from 2 in. to 5 ft. square. 

As the compressibility of the air has been neglected in deducing 
expression (15), p. 206, it is impossible to say without further 
examination that this effect is not due to compressibility. Indeed 
if compressibility has any influence on R, a dimensional effect can 
occur which may be in accordance with a v 2 law of resistance, for, 
when this is taken into account, (15) becomes 

R = 



wheie C is the velocity of sound waves in the medium. This may 
be written 



* Proc. Roy /Soc , 48, p 252. 

f Socie'te' d 'Encouragement pour 1'Industne Nationale, Bulletin, 1903, 1, 
p. 189. 

I Eiffel, Resistance de I'Air 

N.P.L, Collected Researches, 1, p. 261. 



HYDRODYNAMICAL RESISTANCE 213 

ind since by hypothesis R is proportional to v z , this becomes 

T> 



\xi investigation of the possible effect of compressibility shows, 
lowever, that this is less than i per cent for speeds up to 
oo miles per hour, so that an explanation of the observed 
limensional effect based on this factor is not admissible. 

It has been suggested* that since, as shown by Mr. Hunsaker's 
ibservations on circular discs, there is a critical range of speed 
letermined by the form of the edge, and not dependent on the size 

n t> 

>f plate, the apparent discrepancy between = constant and = 

<D t 

triable may be due to the results of vaiious observers having 
>een affected by such ciitical phenomena, which were not, however, 
ufficiently marked to attiact attention. To determine whether this 
xplanation is valid would require an experimental investigation of 
he forms of edge which have been used. 

The more piobable explanation would appear to be that while 
xpenments on any one plate have been taken as showing the i esist- 
nce to be pioportional to v z , this is only appioximately true. 

An examination by Messis. Bairstow and Booth ,f of all reliable 
xpenments on squaie plates, leads to the conclusion that an empirical 
ormula of the type R = ^ /)2 + ^ /)3 (iy) 

jives a close appioximation to the experimental results. For values 
>f vl ranging from i to 350, a and b have the values 0-00126 and 
1-0000007 lespectivcly Here v is in feet per second and R m pounds 

Neglecting compicssibihty this indicates that (f>( } of equation 

1 6), p 212, equals m + nvl, wheie m and n are constants for any 
miticular fluid under given piessure and tempeiature conditions. 

It may be noted that experiments by StantonJ show that the 
>ressmes on the windward side of a square plate are not subject to 
dimensional effect, but that the whole variation can be traced to 
hanges in the negative pressure behind the plate. 

* By Mr. E Buckingham, Smithsonian Miscellaneous Collections, 62, No 4, 
an., 1916. 

f Technical Report, Advisory Committee for Aeronautics, 1910-1, p 21. 

%Proc. Inst. C. E., 171; also Collected Researches of the National Physical 
,aboratory, 5, p 192. 



2i 4 THE MECHANICAL PROPERTIES OF FLUIDS 

Resistance of Smooth Wires and Cylinders 

A somewhat similar scale effect is obtained from experi- 
ments on the resistance of smooth wires and cylinders. A series 
of such tests on a tange of diameters from 0-002 in. to 1-25 in., with 

R 

v ranging from 10 to 50 ft.-sec ,* shows that, on plotting 



3 J 

against or log , a narrow band of points is obtained which in- 
v v 




10 20 30 40 50 

Loq & 

-MO I/ 

Fig 4 

eludes all the experimental results (fig. 4). This shows that for a given 
value of the value of - is the same for all values of v and of 



v 

d. From this it appears that whereas experiments on a single wire or 
cylinder indicate that R is nearly proportional to v z , true similarity 

of flow is only obtained when is constant. 

v 

It becomes very necessary to satisfy this condition with low 

vd 
values of , owing to the changes which may occur in the type 

* Reports and Memoranda, Advisory Com. for Aeronautics, No. 40, March, 
1911; No. 74, March, 1913, No 102. 



HYDRODYNAMICAL RESISTANCE 215 

of flow around such bodies at comparatively low velocities or 
with small diarneteis. As in all other cases of flow, as this factor 
is reduced a critical value is ultimately reached where the type of 
flow undergoes a definite and rapid change, so that the function < 
ceases to be even approximately constant. 

For a given body in a given medium, this critical value of 
corresponds to a critical speed which may be calculated from the 

032 

28~ -x ,,, ,. 

\ s*~S s Ss S^S 

5rt - , 4' ^***^S * s J/ s r S s <- 
024 -* 

I 

S.020 
n 
S. 

y 
cc 

<Q \i ' ' 

3 

5 

008 - 

004 



2-4 6 8 10 12 14 16 18 2o 21 2* 

Values of v d in foot second units 
Fig 5 Resistance of Strut Sections 

values of d and v, li its value has once been experimentally deter- 
mined for bodies oi the given form by varying any one of the vari- 
ables d, v, and v. 

In some such bodies as spheies and cylinders the law of resistance 
may change widely with compaiatively small alteration in the con- 
ditions; thus, for example, at ceitam speeds the resistance of a sphere 
may actually be reduced by toughening the surface. In carrying 
out any such experiments, theiefore, it is of the greatest importance 
that the geometiical similarity between a model and its prototype 
should be as exact as possible, and that where possible vd should be 
kept constant. 

The variation in the type of flow at a definite critical velocity 
has been well shown in the case of flow past an inclined plate by 



ai6 THE MECHANICAL PROPERTIES OF FLUIDS 

C. G. Eden*. By the aid of colour bands in the case of water, anc 
smoke in the case of air, photographs of the eddy formation in th< 
rear of the plates of different sizes were obtained. These shov 





BLERIOT 





BETA 





BF, 34- 




B.F 35 




BABY 



Fig. 6 Strut Sections. 

that the types of flow were similar for both fluids and for all the 

plates so long as was maintained constant, and that the change 
v 

over from one type to the other took place at a critical velocity, 
defined by v ont oc - , in each case. 

/ 

* Tech. Report of Advisory Committee for Aeronautics > 1910-1, p. 48; also 
R. and M., No. 31, March, 1911. 



HYDRODYNAMICAL RESISTANCE 217 

ID 

The curve of fig. 5*, p. 315, shows the change in with a 

pdtr 

variation in vd in the case of a strut of fair stream-line form. Here 
R is the resistance per foot run of the strut. The curve shows that 
the resistance is very nearly proportional to v z for values of ^greater 
than 5, but that as vd is reduced below this value the law of resistance 
suffers a rapid change. Since, below the critical velocity, Roc v, the 
ordinates of the curve to the left of the critical point will be pro- 
portional to -, and this part of the curve will be hyperbolic. 
v 

The following tablef shows the resistance of typical strut sections 
of the types and sizes shown m fig. 6. 

Resistance of too Ft 

Type of Stiut of Strut m Pounds 

at 60 ft -sec 

Cucle, i in diametei 43 o 

Ellipse axes, i in X 2 in 22 2 

Ellipse axes, i m X 5 m 15-2 

De Havilland 25 5 



Farman 



22 9 



Blcnot A 23 7 

Blenot B 24 5 

Baby . 7 9 

Beta 6 9 

B F 34 72 

B F 35 .. 63 

B F. 35, tail foiemost 10 9 



* Applied Aerodynamics, Banstow (Longmans, Green, & Co , 1920), p. 392. 
f Tech Repot t of Advisory Commi tteefor Aeronautics, 1911-2, p 96 



CHAPTER VI 

Phenomena due to the Elasticity 
of a Fluid 

Compressibility 

Compressibility is defined (Chapter I, p 16) as the reciprocal 

of the bulk modulus, i e. by - ( ) 

v \op/T 

The compressibility of water varies with the temperature and 
the pressure, the values of the bulk modulus, obtained by different 
observers, being as follows * These values are in pounds per 
square inch. 



Temperature, Degrees C 
Authority 

o. 10 ao" 30 40 



318000 333000 
Grassi f 293000 303000 319000 322000 



TaitJ 



,, ( At low 

283000 301000 319000 334000 347000 352000^ piess ures 

292000 311000 328000 340000 347000 347000 | 

, i At 2 tons 

300000 321000 332000 346000 339000 339000-^ 



At i ton 



per sq in 



The bulk modulus K of sea water is about 9 per cent greater 
than that of fresh water. 

*See also Parsons and Cook, Proc. Roy Soc A, 85, 1911, p 343 At 4 C. 
Parsons finds K = 306,000 Ib. per square inch at 500 atmospheres, 346,000 Ib. at 
1000 atmospheres, and 397,000 at 2000 atmospheres. Results of experiments by 
Hyde, Proc R S A, 97, 1920, are in close agreement with these. 
f Annales de Chimie et Physique, 1851, 31, p 437 
%Math. and Phys Papers, Sir W. Thomson, Vol. Ill, 1890, p. 517. 

218 



PHENOMENA DUE TO ELASTICITY OF A FLUID 219 

For purposes of calculation at temperatures usual in practice, 
e modulus for fresh water may be taken as 300,000 Ib. per square 
ch, or 43-2 X io 6 Ib per square foot. 

The compressibility is so small that in questions involving water 
rest or in a state of steady flow it may be assumed to be an incom- 
essible fluid. In certain important phenomena, however, where a 
idden initiation or stoppage of motion is involved the compressi- 
hty becomes an important, and often the predominating factor. 

In such cases the true criterion of the compressibility or elasticity 

a fluid is measured by the ratio of its bulk modulus to its density, 
nee it is this ratio which governs the wave propagation on which 
ich phenomena depend. In this respect air is ooly about eighteen 
rates as compressible as water 

For olive oil the value of K at 20 C is 236,000 Ib. per square 
ch (Quincke) and for petroleum at 16-5 C is 214,000 Ib per 
[uaie inch (Martini). The following are values of K for lubri- 
itmg oils at 40 C * 



Tons per Castor Oil Sperm Oil 1X ?,H 1 P l1 
square inch 

1 291000 242000 287000 

2 302000 252000 291000 

5 330000 285000 315000 



Sudden Stoppage of Motion Ideal Case 

If a column of liquid, flowing with velocity v along a rigid pipe 
F unifoim diameter and of length / ft., has its motion checked by the 
LStantaneous closure of a ngid valve, the phenomena experienced 
e due to the elasticity of the column, and are analogous to those 
btaining in the case of the longitudinal impact of an elastic bar 
gainst a rigid wall. 

At the instant of closure the motion of the layer of water in 
mtact with the valve becomes zero, and its kinetic energy is con- 
srted into resilience or energy of strain, with a consequent sudden 
se in pressure. This checks the adjacent layer, with the result 
lat a state of zero velocity and of high pressure (this at any point 

* Hyde, Proc, R. S. A, 97, 1920. 



220 THE MECHANICAL PROPERTIES OF FLUIDS 

being p above the pressure obtaining at that point with stea* 
flow) is propagated as a wave along the pipe with velocity V # .* 
This wave reaches the open end of the pipe after t sec., whe 



Static. 



L 



a 



Aitruos ' Instant of Valve. Closure, 



Pressure 



Pressure 



,1-1. 
V P 



Static 



V P 



V 



Pressure 



Attnaf 



Tressure 



Fig i Ideal Case of Water-hammer in Pipe Line 



t = l-rV p . At this instant the column of fluid is at rest and in a 
state of uniform compression. 

This is not a state of equilibrium, since the pressure immediately 

* Vj) * s the velocit y of propagation of sound waves m the medium, and is equal 
t0 A/~of ' where is the weight in pounds per cubic foot, and K is in pounds per 
square foot. 



PHENOMENA DUE TO ELASTICITY OF A FLUID 221 

aside the open end of the pipe is p greater than that in the surround- 
tig medium. In consequence the strain energy of the end layer 
3 reconverted into kinetic energy, its pressure falls to that of the 
urrounding medium, and it rebounds with its original velocity v 
owards the open end of the pipe. This relieves the pressure on 
he adjacent layer, with the result that a state of normal pressure 
nd of velocity ( v) is propagated as a wave towards the valve, 
caching it after a second inteival / ~ V p sec. At this instant tru 
vhole of the column is at normal pressure, and is moving towards 
he open end with velocity v. The end of the column tends to leave 
he valve, but cannot do so unless the pressure drops to zero, or so 
iear to zero that any air in solution is liberated. Its motion is con- 
equently checked, and- its kinetic energy goes to reduce the strain 
nergy to a value below that corresponding to normal pressure. 
The pressure d ops suddenly by an amount equal to that through 
yhich it originally rose, and a wave of zero velocity and of pressure 
below normal is transmitted along the pipe, to be reflected from 
he open end as a wave of normal pressure and velocity towards the 
r alve When this wave reaches the valve 4/ V p sec aftei the 
QStant of closure, the conditions are the same as at the beginning 
>f the cycle, and the whole is repeated. 

Under such ideal conditions the state of affairs at the valve would 
>e represented by fig i a At any other point at a distance / x from 
he open end the pressure-time diagram would appear as m fig. i b 

If the velocity were such as to make p greater than the normal 
bsolute pressure in the pipe, the first reflux wave would tend to 
educe the pressure below zero. Since this is impossible, the pressure 
ould only fall to zero, and the subsequent rise in pressure would 
>e correspondingly reduced. Actually, at such low pressures any 
lissolved air is liberated and the motion rapidly breaks down. 

Effect of Friction in the Pipe Line 

The effect of friction in the pipe line modifies the phenomena 
n a complex manner. In the first place the pressure, with steady 
low, falls uniformly from the open end towards the valve, and the 
>ressure at the valve will be represented by such a line as AB (fig. 2). 
)n closure the pressure here will rise by an amount p as before. 

When the adjacent layer is checked its lise in pressure will also 
>e p, but since its original pressure was higher than that at the 
r alve, its new pressure will also be higher. It will therefore tend to 



J22 THE MECHANICAL PROPERTIES OF FLUIDS 

compress that portion of the column ahead of it, and will lose some 
of its strain energy in so doing. This will result in the pressure 
at the valve increasing as layer after layer is checked, but since this 
secondary effect travels back fiom each layer in turn with a velocity 
Vp, the full effect at the valve will not be felt until a time 2/ -r V^ 
after closure. At this instant the pressure will have risen by an 
amount which to a first approximation may be taken as dp Vz,* 
where dp is the pressure- difference at the ends of the pipe under 
steady flow. 

When reflux takes place the end layer, having transmitted part 




Static al 1- f f Pressure 

A 



Pressure 

Fig 2 Water-hammer as Modified by Friction 

of its energy along the pipe, can no longer rebound with the original 
velocity i). Moreover, since at the instant when the distuibance 
again reaches the valve, and the column is moving towards the open 
end, frictional losses necessitate the pressure at the valve being dp 
greater than at the open end, the pressure drop will be less than in 
the ideal case. 

* With steady flow the pressure distant x from the valve will be greater than 
that at the valve by an amount dpj. Therefoie excess strain energy of a layer 
of length Sx at this point, due to this pressure c dx(P. x .* j 

Assuming this excess energy to be distributed over the length x between this 

//}/! V V\ % 

layer and the valve, it will cause a use in pressuie p', where (p'Yx = <W -jr ) . 
Integrating to obtain the effect of all such layers from o to x gives \ * / 

^ fo1)\^x 11 i 3t> 

dv ~ I -f- 1 , and when x I, p = -*= 



This is only a first approximation, since the equalization of piessures will be 
accompanied by surges which will introduce additional frictional losses. 



PHENOMENA DUE TO ELASTICITY OF A FLUID 223 

If the velocity of the first reflux be kv t where k is somewhat less 
tan unity, the pressure diagram will be modified sensibly as in 

=' 2 '. 
Friction thus causes the pressure wave to die out rapidly, with- 

Jt affecting the periodicity appreciably. 



Magnitude of Rise in Pressure, following 
Sudden Closure 

Assuming a rigid pipe, on equating the loss of kinetic energy 
pound of fluid to the increase in its strain energy or resilience: 



zg 

p = /r 

V g 'g' 

here p is the rise in pressure, and v the velocity of flow in feet per 
'cond. 

Putting K = 43-2 X io 6 Ib -sq ft., 
,, w 62-4 Ib -c ft , 
g =- 32 2 ft.-sec. 2 , 
this becomes p = gi6ov Ib.-sq ft. 
= 63-72; Ib.-sq in 

Effect of Elasticity of Pipe Line 

Owing to the elasticity of the pipe walls, pait of the kinetic 
Lergy of the moving column is expended in stretching these, with 
resultant increase in the complexity of the phenomena, a reduction 

the maximum pressure attained, and an increase in the rate at 
bich the pressure waves die out. The state of affairs is then 
dicated in figs. 3 a and b, which are reproduced from pressure- 
ne diagrams taken from a cast-iron pipe line 3-75 in. diameter and 
;o ft. long f 

Fig. 3 a was obtained from behind the valve and fig. 3 b at a point 
; ft. from the open end of the pipe. 

The elasticity of the pipe line may modify the results in two 

* If a cube of unit side be subject to a pressure increasing from o to p, the 
ange in volume will be p K, and since the mean pressure during compres- 
in is p 2, the work done is p z aK 

f Gibson, Water Hammer in Hydraulic Pipe Lines (Constable & Co., 1908). 



224 THE MECHANICAL PROPERTIES OF FLUIDS 

ways. If the pipe is free to stretch longitudinally, at the insta 
of closure the valve end of the pipe and the water column w 
move together with a common velocity ,* less than v, and a wa' 
of longitudinal extension will travel along the pipe wall. Tl 
instantaneous rise in pressure at the valve will now be equal to 



(v 



S 



Since the velocity of propagation is much greater in metal tha 




Static 



Status 



Pressure 



Fig 3 Water-hammer in Experimental Pipe Line 

in water, this wave will be reflected from the open end of the pipe 
and will reach the closed end again before the reflected wave in the 
water column. At this instant the closed end of the pipe will 
rebound towards the open end with velocity u, and will produce 
an auxiliary wave of pressure equal to 

/Kw 

M-W 

v s 

* It may readily be shown that u = v\ w m a m V m \> where Wm, am, and Vm 



< JM/t V p ' f, 

refer to the weight per cubic foot, the cross-sectional area, and^ the velocity ot 
propagation in the metal, and w, a, Vp, to the corresponding quantities for water, 



PHENOMENA DUE TO ELASTICITY OF A FLUID 225 
in the water column. This will increase the pressure to the value 



8 

which would obtain if the pipe were anchored. So that the effect 
Dn the maximum pressure attained during this first period zl ~ "V p 
is zero. The effect on the subsequent history of the phenomenon 
s complex The net effect, however, is to superpose on the normal 
Dressure wave a subsidiary wave of high frequency (V OT 4/, 
tfhere V, M is the velocity of propagation of waves in metal) and of 
nagnitude 

/Kro 
8 

If, as is usual in practice, the pipe is anchored so that no appreci- 
ible movement of the end is possible, this effect will be small. 

The second effect of the elasticity of the pipe line is due to the 
act that, since the walls extend both longitudinally and circum- 
erentially under piessure, the appaient diminution of volume of 
he fluid under a given inclement of pressure is gi eater than in a 
igid pipe 

The effect of this is to reduce the virtual value of K to a value 
', where * 

i __ i , r I _ 4 
K 7 ~" K 2tE\ 5 a 

vhere r is the ladius of the pipe, t is the thickness of the pipe walls, 
i) is the modulus of elasticity of the material, i \a is Poisson's latio for 
he material (approximately 0-28 for iron or steel) 

If the pipe is so anchored that all longitudinal extension is pre- 
ented, but that circumferential extension is free, this becomes 

i _ i , zr 

K 7 ~~ K *E" 

""he use in pressure due to sudden stoppage of motion is now equal 



g 

* Hydraulics and its Applications, Gibson (Constable & Co , 1912), p. 235. 
CD 812) 9 



22 6 THE MECHANICAL PROPERTIES OF FLUIDS 

Valve Shut Suddenly but not Instantaneously 
If the time of closure t, while being finite, is so small thi 

t< - - * 
V ~ V~' 

the disturbance initiated at the valve has tiavelled a distance s 
and has not arrived at the open end when the valve i caches it 
seat. In this case, if each part of the column is subject to th 
same retardation (a), the relationship 

force = mass X acceleration 

wax 
gives p = , 

v vVs, 

and since a = - =2 ?, 

t x 

2/V 'w iK.zo 
this makes p = * = v\l Ib. per square foot, 

S V g 

the value obtained with instantaneous stoppage Whatever the la^ 
of valve closure then, if this is completed in a time less than / V^ 
the pressure rise will be the same as if closure were instantaneous 

Sudden Stoppage of Motion in a Pipe Line of 
non- Uniform Section 

In such a case the phenomena become veiy complicatec 
Let 4> / 2 , 4> &c., be the lengths of successive sections of a rigi 
pipe, of areas a lt a z , 3 . Following sudden closure of a valve a 
the extremity of the length 4> a wave of zero velocity and of pressur 
63-7 >! Ib. per square inch above normal is transmitted to the June 
tion of pipes i and 2. Here the pressure changes suddenly to 63-72 
above normal. This is maintained in the second pipe during th 
passage of the wave, and is followed by a change of pressmc to 63-72 
at the junction of 2 and 3, and so on to the end of the line. BL 
immediately the pressure at the junction of i and 2 attains its valu 
63'7# 2 > the wave in pipe i is leflected back to the valve as a wave c 
pressure 63-7^2 an d of velocity ^ v z , to be reflected from the valve 
a wave of zero velocity and pressure 63-7 {v z (2^ v z ) } above norma 

This wave then travels to and fro along pipe i , making a complel 
journey in time 4 -f- V # sec., until such time as the wave in pipe : 



PHENOMENA DUE TO ELASTICITY OF A FLUID 227 

lected from the junction of 2 and 3 with pressuie 63-7^3 above 
rmal and with velocity v 2 v &) again reaches the junction of 
nd 2 At this instant it takes up a velocity and pressure depending 
that at the junction end of pipe i, and as this depends on the latio 
the lengths of the branches I and 2, it is evident that after the 
it passage of the wave the phenomenon becomes very involved. 
Where a pipe is short the period of the oscillations of pressure 
my point becomes so small that the pencil of an ordinaiy indicator 
unable to record them, and simply records the mean pressure 
the pipe. Thus where a short branch of small diameter is used 
the outlet from a long pipe of larger bore, the pressure as recorded 
an indicator at the valve will be sensibly the same at any instant 
in the large pipe at the point of attachment of the outlet branch. 
Moreover, where a non-uniform pipe contains one section of 
>reciably greater length than the remainder, this will tend to 
Dose its own pressure- change on an indicator placed anywhere 
the pipe. 

These points are illustrated by the following results of expen- 
ats by S B Weston * In each case the outlet valve was on the 
i. length. 



Details of Pipe Line 

ii ft of 6-m pipe 
58 ,, 2 
99 4 
4 J " 

ii ft of 6-in. pipe 
58 2 

48 il 

3 3 
48 ij 

4 i 

82 ft of 6-m pipe 
66 , 4 
4 

I , 2 

7 



Calc 

154 
322 



Piessures, Pounds per Squaie Inch. 
Obs Calc Obs Calc 



73 
129 



ist i j-in pipe 



ij-in pipe 



6 9 

143 



71 
127 



3-in pipe 



2j-m pipe 



8 9 



and 



Obs 



H'5 



pipe 



715 


75 


180 


65 


71 5 


61 


142 


126 


355 


121 


H3 


114 


1 80 


150 


45 


150 


1 80 


139 


268 


203 


67 


207 


268 


196 



6-m pipe 



1 20 


49 


52 


22 


90 


4-8 


H9 


62 


645 


36 


II 2 


6-6 


223 


82 


97 


52 


16 7 


158 


466 


122 


2OI 


99 


35 


36-8 



* Am. Soc. C. E , 1 9th Nov, 1884 



228 THE MECHANICAL PROPERTIES OF FLUIDS 

Sudden Initiation of Motion 

If the valve at the lower end of a pipe line be suddenly opened, 
the pressure behind the valve falls by an amount p Ib. per square 
inch, and a wave of velocity v towards the valve 



and of pressuie p below statical pressure is propagated towards the 
pipe inlet. 

The magnitude of p depends on the speed and amount of opening 
of the valve, and if the latter could be thrown wide open instan- 
taneously the piessure would fall to that obtaining on the discharge 
side. In experiments by the writer* with a 2-|-m. globe valve on 
a 3f-in. main 450 ft. long, with the valve thiown open through o 5 
of a complete turn, the diop in pressure was 40 Ib per square inch, 
the statical pressure in the pipe being 45 Ib. per squaie inch, 
and on the discharge side zero With the valve opened through 
o-io of a turn the drop was 20 Ib. per squaie inch, while with 0-05 
of a turn it was n Ib. per squaie inch In each case the time oi 
opening was less than 0-13 sec (/ V p ) 

With a pipe so situated that the original statical picssmc is 
everywhere greater than p, this pressuie wave i caches the pipe inlei 
with approximately its original amplitude, and at this instant the 
column is moving towards the valve with velocity v and picssmcy 
below normal. 

The pressure surrounding the inlet is howcvci maintained noimal 
so that the wave letuins from this end with normal picssuie ant 
with velocity 2v relative to the pipe. At the valve the wave i 
reflected, wholly or in part, with a velocity which is the difieienc 
between 2v and the velocity of efflux at that instant, and since th 
velocity of efflux will now be greater than v, the wave velocity wil 
be less than v, and the use in pressure less than p above noimal 
This wave is reflected from the inlet to the valve and heie the cycl 
is repeated, the amplitude of the pressure wave diminishing lapidl 
until steady flow ensues. Fig. 4 shows a diagiam obtained undc 
these circumstances. 

Where the gradient of the pipe is such that beyond a ccitai 
point in its length the absolute statical piessuic is less than tl^ 
drop in pressure at the valve, the motion becomes partly discontinuoi 

* Gibson, Water Hammer in Hydraulic Pipe Lines (Constable & Co., 1908). 



PHENOMENA DUE TO ELASTICITY OF A FLUID 229 

this point on the passage of the first wave, which travels on to the 
let with gradually diminishing amplitude. The amplitude with 
hich it reaches the inlet determines the velocity of the icflected 



Valve , i. A yr=r?Y Closed 

Valve open, f \ / v ' Steady f lota 




Fig 4. Diagram of Ptesauie (per squaie inch) obtained on Sudden Opening of a Valve 

ave, which will be less than in the preceding case, and under such 

rcumstances the wave motion dies out rapidly. 
As the valve opening becomes greater, the efficiency of the valve 
a reflecting surface becomes less, so that with a moderate opening 

e pressure may never even attain that due to the statical head 

Valve opened 

Valve open. - -L . Steady flout 

4olhf-yj"u^ 




Fig 5 Sudden Opening of Valve 



his is shown in fig. 5, which is a diagram obtained from the experi 

ental pipe line when the valve was opened suddenly (time < -i- 
rough half a turn. /> 



Wave Transmission of Energy 

In the systems in common use for the hydraulic transmission 
energy, water under a pressure of about 1000 Ib. per square inch 
supplied from a pumping-station and is transmitted through pipe 
ics to the motor. This method involves a continuous flow of the 
orking fluid, which in effect serves the purpose of a flexible coupling 
;tween the pump and the motor. 



2 3 o THE MECHANICAL PROPERTIES OF FLUIDS 

It is, however, possible to supply energy to a column of flui 
enclosed in a pipe line, to transmit this in the form of longitudim 
vibrations through the column, and to utilize it to perform mechanics 
work at some lemote point. Such transmission is possible in virtu 
of the elasticity of the column.* 

If one end of a closed pipe line full of water under a mean pres 
sure p m be fitted with a reciprocating plunger, a wave of alternat 
compiession and rarefaction is produced, which is piopagated alon 
the pipe with velocity V r If the plunger has simple harmoni 
motion, the state of affairs in a pipe line so long that, at the give 
instant, the disturbance has not had time to be i effected from il 
further end, is represented in fig 6 The pressure at each poir 




Fig 6 

will oscillate between the values p m p, and the velocity betwee 
the values v, where v is the maximum velocity oi the pistoj 
At the instant in question, particles at A, C, E, and G aie oscillatir 
to and fro along the axis of the pipe through a distance r on eac 
side of their mean position, while paiticlcs at B, D, F, and H ai 
at test. If n be the number of revolutions of the ciank pci secom 
the wave-length X = V p n ft 

In a pipe closed at both ends such a state of vibiation is icilectc 
from end to end, forming a seiies of waves of pressure and veloci 
whose distribution, at any instant, depends on the latio of the lengi 
I of the pipe to A. 

In the cases where / is icspectivcly equal to A/4, A/2, and 
stationary waves are produced as indicated in fig y."j The exce 

* A number of applications of this method have been patented by Mi . < 
Constantmesco. 

"j. The pressure and velocity oscillate in a "statiomuy " manner, i e there t 
definite points called " nodes " wheie theic is no change m piessuie and hkewi 
points where the water does not move See any textbook on Sound, e.g. Datt 
Sound (Blackte), p 63, "Watson's Physics, Poyntmg and Thomson's Sound, & 
wheie the subject is fully explained for sound waves. 

This distribution, where I = A/4, is only possible wheie oscillation at the e 
A is possible, as where the pipe is fitted with a free plunger, 



PHENOMENA DUE TO ELASTICITY OF A FLUID 23 1 

^assure at a given point oscillates between equal positive and 
egative values, the range of pressure being given by the intercept 
etween the two curves. The velocity at the points of maximum 
nd minimum pressure, as at A, D, and B in fig. jc, is zero, while 
t the points C and E, where the variation of pressure is zero, the 
elocity varies from -\-v to v. 
In the case where 7 = A/4, the plunger, if free, would continue 



B (a,) 




A 




B (I) 




Fig 7 Stationary Waves in a Closed Pipe 

) oscillate in contact with the end of the column without the appli- 
ition of any exteinal force. 

In a pipe fitted with a reciprocating plunger at one end and 
[osed at the other, the wave system initiated by the plunger will be 
iperposed on this reflected system. Thus if / = A/2 or A, the 
r ave initiated by the plunger will be reflected, and will reach the 
lunger as a zone of maximum pressuie at the instant the latter is 
Dmpletmg its m-stroke and is producing a new state of maximum 
ressure. The pressure due to the reflected wave is superposed 
n. that due to the direct compression, with the result that the pres- 
ire is doubled. The next revolution will again increase the pres- 
j.re, and so on until the pipe either bursts or until the rate of dis- 
pation of energy due to friction, and to the imperfect elasticity of 
ic pipe walls, becomes equal to the rate of input of energy by the 
lunger. 



232 THE MECHANICAL PROPERTIES OF FLUIDS 

On the other hand, if the length of the pipe be any odd multiple 
of A/4, the pressure at the plunger at any instant, due to the reflected 
wave, will be equal in magnitude but opposite in sign to that primarily 
due to the displacement of the plunger, and the pressure on the 
latter will be constant and equal to the mean pressure m the pipe 
Except for the effect of losses in the pipe walls and in the fluic 
column, reciprocation may now be maintained indefinitely withou 
the expenditure of any further energy. For any intermediat< 
lengths of pipe, the conditions will also be inteimediate and th< 
wave distribution complex. 

Instead of closing the end of the pipe at B (fig. 8), suppose 5 
piston to be fitted to a crank rotating at the same angular velocity 
in the same direction, and in the same phase as the ciank at A 
If the column were continued beyond B, the movement of thi 




Fig 8 



piston would evidently produce in the column a sciics of wave 
forming an exact continuation of the wave system between A am 
B. There will now be no reflection from the surface of the piston 
and if the latter drives its crank and if the resistance is, at ever 
instant, equal to the force exerted on the piston by the wave system 
it will take up the whole energy of the waves pioduced by piston A 
It is to be noted that the piston B may be placed at any point in th 
pipe so long as its phase is the same as that of the liquid at the poin 
of connection. 

If more energy is put in by the piston A than is absoibed by F 
reflected waves will be formed, and the continuation oi the motio 
will accumulate energy in the system, increasing the maximur 
pressure until, as in the case of the closed pipe, the pipe will buisi 

This may be avoided by fitting a closed vessel, filled with liquic 
having a volume large in comparison with the displacement of tb 
piston, in communication with the pipe near to the piston. Altei 
natively this may be replaced by a spring-loaded plunger. In eithe 
case the contrivance acts as a reservoir of energy. If the piston 
is not absorbing the whole of the energy supplied from A, the liqui 
in this chamber is compressed on each instroke of the piston t 



PHENOMENA DUE TO ELASTICITY OF A FLUID 233 

re-expand on the outstroke, and by giving to it a suitable volume, 

e maximum pressures, even when the piston B is stationary, 

ay be reduced to any requned limits If perfectly elastic, the 

servoir will return as much energy during expansion as it absorbed 

iring compression, so that the net input to the driving piston is 

ly equivalent to that absorbed by piston B. 

In the case of a pipe (fig. 7 c) whose length is one wave-length, 

d which is provided with branches at C, D, E, and B, respec- 

ely one-quarter, one-half, thiee-quarteis, and one wave-length 

>m A, if all the branches are closed, stationary waves will be 

oduced in the pipe as pieviously described. 

If now a motor running at the synchronous speed be coupled 

the branch at D, this will be able to take up all the energy given 

the column The stationary half-wave between A and D will 

tiish, being icplaced by a wave of motion, while the stationary 

ve will still persist between D and B 

Since there is no pressure vanation at C and E, motors coupled 
these points, with the remaining branches closed, can develop 
energy. 

If a motor be connected at any inteimediate point, part of the 
>ut of energy can be taken up by the motor The stationary 
ve will then persist, but be of reduced amplitude between A and 
motor, the wave motion ovci this region being compounded of 
=> stationary wave and of a travelling wave conveying eneigy 
With a motor at the end B of the line, not absoibmg all the energy 
en by the generator A, there is, in the pipe, a system of stationary 
vcs superposed on a system of waves travelling along the pipe, 
that there will be no point in the pipe at which the variation of 
ssuie is always zero It follows that under these conditions a 
tor connected at any point of the pipe will be able to take some 
rgy and to do useful work 

In practice a three-phase system is usually employed, as giving 
re uniform torque and ease of starting A three-cyhndei gener- 
% having cranks at 120, gives vibrations to the fluid in thiee 
es, which are received by the pistons of a three-cylinder hydraulic 
tor having the same crank angles The mean pressure within 
system is maintained by a pump, which returns any fluid leaking 
t the pistons. 



(D312) 



234 THE MECHANICAL PROPERTIES OF FLUIDS 

Theory of Wave Transmission of Energy 

The simple theory of the process is outlined below, on th< 
assumption that the friction loss due to the oscillation of the columr 
in the pipe is dnectly proportional to the velocity. Where such 5 
viscous fluid as oil is used this is true, but where water is used i 
may or may not be true, depending upon the velocities involved 
If the resistance is equal to kv z per unit length as with tuibulen 
motion, an approximation to the true result may be attained b 
choosing such a frictional coefficient k' as will make k'v = kv z a 



the mean velocity. At velocities below the critical, k' = ~- ii 

pounds per squaie foot of the cioss section (Poiseuille) per urn 
length of the pipe. 

Consider the fluid normally in a plane at x } displaced fiom tha 

f\ 

plane through a small distance u, so that its velocity v = - Th 

at 

difference of pressure on the ends of an element of length S#, du 
to the variation in compression along the axis oi the pipe, is equal t 

K! 

dx 
and the equation of motion becomes 

dt 2 4 9# 2 4 <tf a 4 9^ ' 

^ 0,0 == ^ o v "~ , > (i) 

or 9r oiC 

where a = A/ an< ^ ^ - ^ > 

v p pd z 

If 6 is small compared with spm, wheie ?z is the ficquency of t] 
vibration, a solution of equation (i) is 

u = u Q e * sm2-rrn(t -Y (2) 

which represents an axial vibration throughout the column, of maj 
mum amplitude % at the end where x = o. 

-J 6 A 

At any other point the maximum amplitude is e ? , grad 



PHENOMENA DUE TO ELASTICITY OF A FLUID 235 

ally diminishing along the pipe owing to the friction term repre- 
sented by the term b. 

The excess pressure p, at any instant and at any point, is equal 

j-rdu 
to K -, i.e. 

ax b 

T ~^x _ 2777Z / X\ 

p = Kw e X cos27r#( t - - ) approx., 
a \ a/ 

ind the maximum excess pressure, p max , at any point, 

27rwKw ~i- f N 

= - * a ................. (3) 

a 

The velocity of a particle at x is equal to 

du -t-* /_. si 

= 2rrmi e a cos27rw( -- 
9i \ a 

ind the raaximum velocity, 



The eneigy transmitted by the excess pressure the mean 
>ressure conveys no energy on the average across a given section 
f the pipe in time Bt is equal to 



-fr ~ 

-~~ K -- 
4 4 dx at 



The mean rate of transmission of eneigy per second over each stroke 
tf the plungei is thus given by 

rf K 


4 

There r is the duration of a stroke, i e. of a half-cycle. 

da 



K r du 8u ^ 
1 __ __ -dt t 

TJ ox ot 



/ x\ 

Wilting 2irn(t ) = a, 

\ a] 



dt 



27m 2,7m 



- . , Kd z r du 'du, 

this becomes / - da 

4 J ox ot 

TrK * 



(midu^fe * f <;) 



236 THE MECHANICAL PROPERTIES OF FLUIDS 

and the horse-power transmitted, if the foot be the unit of length, 
is obtained by dividing expression (5) by 550. 

The loss of energy per unit length of the pipe, due to friction s 
and converted into heat, is 

9E b 

~5~ ~*^> 

ox a 

and the efficiency of transmission, through a pipe line of length / is 

bi 
E z ~ EO = i - . 

It should be noted that in any application of these results, if th< 
calculations are in English units, 

w _ 62-4 

P g ~ 3^2 

for water, while the value of p, is to be taken in pounds per squar 
foot, and the pipe diameter in feet. 

For a more detailed investigation of the theory, which become 
complex when a complicated pipe system is used, Mi Constant 
nesco's original papers should be consulted*. 

There is an exceedingly close analogy between wave it am 
mission by Constantmesco's system and alternating-cm rent electr 
power transmission; in fact, in the " three-pipe system " tl 
known facts of three-phase electrical cngmcciing can be appln 
with scarcely any except verbal changes. 

* The Theory of Somes (The Piopnetois of Patents Conti oiling Wave Tuir 
mission, 132 Salisbury Square, E.C , 1930) 

NOTE The foiegomg theory of wave tiansraission is due to H Moss, D.J 
See also Proc Insf Mech Eng , 1923. 



CHAPTER VII 

The Determination of Stresses by Means 
of Soap Films 

When a straight bar of uniform cross section is twisted by the 
application of equal and opposite couples applied at its two ends, 
it twists in such a way that any two sections which are separated by 
the same distance are lotated relative to one another through the 
same angle. The angle through which sections separated by a unit 
length of the bar are twisted relatively to one another is called the 
" twist ", and it will be denoted by the symbol 5 throughout this 
chapter. If the section is circular, particles of the bar which originally 
lay in a plane perpendicular to the axis continue to do so aitei the 
couple has been applied 

The couple is tiansmitted thiough the bar by means of the shearing 
force exeited by each plane section on its neighbour. The shearing 
stress at any point is, in elastic materials, propoitional to the shear 
strain, or shear In the case where a bar of circular cross section 
is given a twist 5, the shear evidently increases from zero at the 
axis to a maximum at the outer surface of the bar; at a distance r 
from the axis it is in fact r% If two series of lines had been drawn 
on the surface of the untwisted bar so as to be parallel and peipen- 
dicular to the axis, these lines would have cut one another at light 
angles. After the twist, however, these lines cut at an angle which 
difteis from a right angle by the angle r5, which measures the shear 
at the point in question. The shearing strain at the surface of any 
twisted bar can in fact be conceived as the difference between a 
right angle and the angle between lines of particles which were ongm- 
ally parallel and peipendicular to the axis. 

In the case of bars whose sections are not circular, the particles 
which originally lay in a plane perpendicular to the axis do not 
continue to do so after the twisting couple has been applied; the 
cross sections are warped in such a way that the shear is increased 

837 



238 THE MECHANICAL PROPERTIES OF FLUIDS 

in some parts and decreased in others. In the case of a bar of elliptic 
section, for instance, the point on the suiface of the bar where the 
shear is a maximum is at the end of the minor axis, while the point 
where it is a minimum is at the end of the major axis If the sections 
had remained plane, so that the shear at any point was proportional 
to the distance of that point from the axis of twist, the leverse would 
have been the case. 

The warping of sections which were originally plane is of funda- 
mental importance m discussing the distribution of stiess in bent 01 
twisted bars. It may give rise to very large increases in stress In 
the case where the section has a sharp internal corner, for instance, it 
gives rise to a stress there which is, theoretically, infinitely great. 

The method which has been used to discuss mathematically the 
effect of this warping is due to St. Venant.* If co-ordinate axes 
Ox, Oy be chosen in a plane perpendicular to the axis of the bai , 
and if <j> represents the displacement of a paiticle from this plane 
owing to the warping which occuis when the bar is twisted, then 
St. Venant showed that </> satisfies the equation 



and that it must also satisfy the boundaiy condition 

_i = y cos(xn] x cos(yti), . (2) 

on 
fit 

wheie represents the rate of change of ^ m a dnection perpendiculai 

9/2 

to the boundary of the section, and (xn), (yn) icpiescnt the angles 
between the axes of x and y respectively and the normal to the 
boundary at the point (x, y). 

Functions which satisfy equation (i) always occur in paiis. II 
ift is the function conjugate to </>, i e. the other membei ol the pan 
ijj is related to </> by the equations 

d A = ?, ^ = -, ........ (3) 

dx 9y' "dy fix 

and ijj also satisfies (i). In the case under consideiation it turns oui 
that it is simpler to determine t/r and then to deduce <j> than to attempl 
to determine c/> directly. From (i), (2), and (3) it will be seen thai 
to determine ifj it is necessary to find a function ift which satisfies 

* See Love, Mathematical Theory o/ Elasticity , second edition, Chap. XIV. 



DETERMINATION OF STRESS BY SOAP FILMS 239 
i) at all points of the cross section, and satisfies the equation 
~ y cos(xn) x cos(yn) ........... (4) 

C/u 

t points on the boundary, where -^- represents the rate of variation 

OS 

f ?/( lound the boundary. 

(j*\} fj *V 

Now cos(xn) = ~- and cos(yn) = , so that (4) reduces to 

a 



ir / \ . 

- ~ Ij-f^ 2 H~ y z ), that is to say the boundary condition reduces 

i t/O \ / 

_j_ constant ........... (5) 



""he advantage in using i/j instead of < is that the boundary condition 
5) is more easy to satisfy than (2). 

The problem of the torsion of the bar of any section is therefore 

educed to that of finding a function ^ which satisfies -f- ^~ = o 

coc*"* ^/y*" 1 

nd (5) There is an alternative, however If a function W be de- 
ned by the relation W ip %(x 2 -f- y 2 ), then ^F evidently must 
atisfy the equation 



t all points of the section, and 

1 P = constant .......... (7) 

t the boundary 

This function 1 P, besides having a very simple boundary condition, 
ias also the advantage that it is simply related to the shear, in fact 
he shearing strain at any point is proportional to the rate of change 
i W at the point in question in the direction in which it is a maximum. 

Prandtl's Analogy 

It has only been possible to obtain mathematical expressions for 
>, i/, and W in very few cases The stresses in bars whose sections 
ic rectangles, ellipses, equilateral triangles, and a few other special 
hapes have been lound, but these special shapes are of little inteiest 
o engineers. There is no general way in which the stresses in 
wisted bars of any section can be reduced to mathematical terms 

The usefulness of equations (6) and (7) does not cease, however 
vhen W cannot be represented by a mathematical expression. It has 



240 THE MECHANICAL PROPERTIES OF FLUIDS 

been pointed out by various writers that certain other physica 
phenomena can be represented by the same equations In some 
cases these phenomena can be measured experimentally fai moic 
easily than direct measurements of the stresses and strains in i 
twisted bar can be made. Under these ciicumstances it may b< 
useful to devise experiments in which these phenomena are mcasuiec 
in such a way that 1 F is evaluated at all points of the section Th< 
values thus found for W can then be used to dcteimine the stiessei 
in a twisted bai . 

Probably the most useful of these " analogies " is that of Piandtl 

Considei the equations which icpicscnt the smiace of a soa] 

film stretched over a hole in a flat plate of the same size and shap 

as the cioss section of the twisted bar, the film being slightly dis 

placed from the plane of the plate by a small picssme p. 

If y be the surface tension of the soap solution, the equation o 
the surface of the film is 

P52,v )2 W A 

1 _L ~ JL P. o (K\ 

o o n ~ r, r - v^, ., i o i 

cx l fly 2y 

wheie % is the displacement oi the film horn the plane ot ,vy and 
and 3; are the same co-oidmates as beioie. Round the bouncUn 
i e the edge of the hole, % = o. 

It will be seen that if z is measiued on such a scale that 1 F ~ ^yz/j 
then equations (6) and (8) aic identical The boundaiy conihtioi 
are also the same. It appeals thcreloie that the value oi 1 F, coin 
spending with any values of x and y, can be lound by mcasuiing ll 
quantities pjy and % on the soap film. 

In other woids the soap film is a guiphical icpicscntation of tl 
function *Ffor the given cioss section. Actual values ol 1 F can I 
obtained from it by multiplying the oidimites by ^y//> 

To complete the analogy it is necessary to bung out the due 
connection between the measurable quantities connected with tl 
film and the elastic properties of the twisted bar 

If N is the modulus of rigidity of the matei ml and % the twi 
of the bai, the shear stress at any point of the cross secti( 
can be found by multiplying the slope of the W surface at tl 
point by N, so that, if a is the inclination oi the bubble to tl 
plane of the plate, the stress is 

(9) 



DETERMINATION OF STRESS BY SOAP FILMS 241 
The torque T q on the bar is given by 

, dy, 



or T, = N5CV, ................... (10) 

where V is the volume enclosed between the film surface and the 
plane of the plate. 

The contour lines of the soap film in planes paiallel to the plate 
correspond to the " lines of shearing stress " in the twisted bar, that 
is, they run parallel to the dnection of the maximum shear stress at 
every point of the section. 

It is evident that the torque and stresses m a twisted bar of any 
section whatever may be obtained by measuring soap films in these 
respects. 

In order to obtain quantitative results, it is necessary to find the 

value of in each expenment. This might be done by measuring 

P 
y and p directly, but a much simpler plan is to determine the curva- 

ture of a film, made with the same soap solution, stretched over a 
circular hole and subjected to the same piessure difference, p, be- 
tween its two surfaces as the test film. 

The curvature of the circular film may be measured by obseivmg 
the maximum inclination of the film to the plane of its boundary 

If this angle be called /? then 



_ TT 

T ~~ -- 5' .......... ' 

p snip 

where h is the radius of the circular boundary 

The most convenient way of ensuring that the two films shall 
be under the same pressure, is to make the circular hole m the same 
plate as the expeiimental hole. 

It is evident that, since the value of ^.y/p for two films is the 
same, we may, by comparing inclinations at any desired points, find 
the ratio of the stresses at the corresponding points of the cross 
section of the bar under investigation to the stresses m a circular 
shaft of radius h under the same twist. Equally, we can find the 
ratio of the torques on the two bars by comparing the displaced 
volumes of the soap films. This is, in fact, the form which the 
investigations usually take. 



24 3 THE MECHANICAL PROPERTIES OF FLUIDS 

As a matter of fact, the value of can be found fiom the test- 

* 

film itself by integrating round the boundary, a, its inclination to the 

plane of the plate If A be the area of the cross section, then the 
equilibrium of the film requires that 



1 2y smack = pA (12) 



This equation may be written in the foim 
4.v area of cross section 

TV ___ 2 V ~ 

p (perimeter of cioss section) X (mean value of sma) 

By measuring a all round the boundaiy the mean value of sma 

can be found, and hence may be deteimined This is, however, 

P 
more laborious in practice than the use of the cnculai stanclaid 

It is evident that if the radius of the cncular hole be made equal 

zA. 
to the value of , wheie A is the area and P the peumeter of the test 

hole, then sinjS = mean value of sino, It is convenient to choose 
the radius of the ciicular hole so that it satisfies this condition, in 
Older that the quantities measuied on the two films may be of the 
same oidei of magnitude. 



Experimental Methods 

It is seen fiom the mathematical discussion given above that, 
in ordei that full advantage may be taken of the mfoi matron on 
toision which soap films are capable of furnishing, apparatus is ic- 
quired with which three kinds of measuiements can be made, namely: 

(a) Measurements of the inclination of the film to the plane of 
the plate at any point, for the determination of stresses. 

(b) Determination of the contour lines of the film. 

(c) Comparison of the displaced volumes of the test film and 
circular standard for finding the corresponding torque latio. 

The earliest appaiatus designed by Dr. A. A Griffith and G. I. 
Taylor for making these measurements is shown in fig. i (see 
Plate).* The films are formed on holes cut in flat aluminium plates 

* From Proc. Inst. Mcch, Eng, t I4th December, 1917. 



o 





I'ai ing p<.if>e 2(3 



DETERMINATION OF STRESS BY SOAP FILMS 243 

the icquired shape. These plates are clamped between two 
Ives of the cast-iron box A (fig. i). The lower part of the box 
:es the form of a shallow tray J in. deep blackened inside and 
pported on levelling screws, while the upper portion is simply 
iquare frame, the upper and lower surfaces of which are machined 
rallel. Arrangements are made so that air can be blown into the 
ver part of the box in order to establish a difference in pressure 
tween the two sides of the film. 

In older to map out the contour lines of the film, i.e. lines of 
ual z, or lines of equal }F in the twisted bar, a steel point wetted 
th soap solution is moved parallel to the plane of the hole till it 
5t touches the film. The point being at a known distance from the 
me of the hole marks a point on the film where z has the known 
lue. The requiied motion is attained by fixing the point (shown 

C in fig. i) to a piece of plate glass which slides on top of the 
3t-non box. The height of the point C above the plate is adjusted 

fixing it to a micrometer screw B 

In older to record the position of the point C when contact with 
e film is made, the micrometer carries a iccordmg point D, which 
mts upwards and is placed exactly over C The record is made 

a sheet of paper fixed to the boaid E, which can swing about a 
nzontal axis at the same height as D To maik any position ot 
c scicw it is meiely necessary to puck a dot on the paper by 
mgmg it down on the lecoidmg point The piocess is repeated a 
gc number of times, moving the point to difleient positions on the 
m but keeping the setting of the rmciometer B constant In this 
ly a contour is pucked out on the papci. To puck out another 
ntour the setting of B is altered. The photogiaph shows an actual 
2ord in which foui contouis traced in this way have been filled 

with a pencil. The section shown is that of an aeroplane pro- 
ller blade. 

To measure the inclination of the film to the plane of the plate 
e " auto-collimatoi " shown in figs ia and 2b was devised. Light 
)m a small electiic bulb A is reflected fiom the surface of the 
tn thiough a V-neck B and a pin-hole eyepiece C placed close 

the bulb. 

Dnect light fiom the bulb was kept away from the eye by 
small screen. The inclinometer D, which measuies the angle 
nch the line of sight makes with the vertical, consists of a 
nit level fixed to an aim which moves over a quadrant graduated 

degiees. In using the auto-collimator the soap-film box is 



244 THE MECHANICAL PROPERTIES OP FLUIDS 

adjusted till the plane of the hole acioss which the film is sti etched 
is horizontal. 

The volume contained between the film and the plane of the hole 
can be measured m a variety of ways. One of the most simple is to 
lay a flat glass plate wetted with soap solution over the test hole in 
such a way that all the air is expelled from it. The volume contained 
between the spherical film and plane of the circular hole is then in- 
creased by an amount equal to the volume lequired. This increase 
in volume can. be determined in a variety of ways, one of the simplest 
being to make measuiements with the auto-collimator of the inclina- 
tion of the spherical film at a point on its edge. 



Accuracy of the Method 

Strictly speaking, the soap-film surface can only be taken to 
represent the torsion function if its inclination a is everywhere so 
small that sina = tana to the required older of accuiacy This 
would mean, however, that the quantities measuied would be so 
small as to render excessive experimental eriois unavoidable A 
compromise must therefore be effected. In point of fact, it has 
been found from expeiiments on sections for which the toision 
function can be calculated, that the ratio of the stiess at a point m 
any section to the stress at a point in a circular shaft, whose ladius 

2A 

equals the value of for the section, is given quite satistactonly by 

the value of where a and B are the respective inclinations of the 

smjS ^ F 

corresponding films, even when a is as much as 35 Similaily, the 
volume ratio of the films has been found to be a sufficiently good 
approximation to the corresponding torque ratio, foi a like amount 
of displacement. 

In contour mapping, the greatest accuracy is obtained, with the 
apparatus shown in fig. i, when J3 is about 20. That is to say, 
the displacement should be rather less than for the other two methods 
of experiment 

In all soap-film measurements the experimental eiror is natuially 

aA 
greater the smaller the value of . Reliable results cannot be 

zA. 
obtained, in general, if is less than about half an inch, so that a 

shape such as a rolled I beam section could not be treated satis- 



DETERMINATION OF STRESS BY SOAP FILMS 



245 



;tonly in an appaiatus of convenient size. As a matter of fact, 
wever, the shape of a symmetrical soap film is unaltered if it be 
?ided by a septum or flat plate which passes through an axis of 
nmetry and is normal to the plane of the boundary. It is there- 
e only necessary to cut half the section in the test-plate and to 
ice a normal septum of sheet metal at the line of division. An 
beam, for instance, might be treated by dividing the web at a 
stance from the flange equal to two or three times the thickness of 
3 web It has been found advisable to carry the septum down 
rough the hole so that it projects about J in. below the under side 
the plate, as otherwise solution collects in the corners and spoils 
5 shape of the film. 

TABLE I 

SHOWING EXPERIMENTAL ERROR IN SOLVING STRESS EQUATIONS 
BY MEANS OF SOAP FILMS 






Radius 



Sin 



Error ' Error > 



& 



Square side, 3 in 



I" D* Deg 

I0 33 5S 2119 IS36 I4 QO : 500 + 24 -07 

15 29 u at 34 i 364 r 337 i -550 -|-i o i o 

I2 ? 6 37t 2432 1263 1240 i 234 +24 +05 

i 410 31 10 24 oo i 296 i 270 i 276 +i 6 o 5 

, Ellipse 4 X o 8 in i 196 35 35 26 58 i 331 i 293 i 286 +3 5 +o 5 

i Rectangle 4 X 2 in i 333 31 70 22 36 i 418 i 380 i 395 +i 6 i i 

' Rectangle 8 X 2 in i 60 34 83 27 23 i 279 i 247 i 245 +27 +02 

! Infinitely long rec-j 6 6 6 OQO +o6 + 

tangle i in wide J j ^ o v o 



Ellipse 3 X i ni 



The values set down in Table I indicate the degree of accuracy 
3tamable with the auto-colhmator in the determination of the 
aximum stresses in sections for which the toision function is known 
hey also give an idea of the sizes of holes which have been found 
tost convenient in practice. The angles given are a, the maximum 
iclination at the edge of the test film, and /3, the inclination at the 



* On 4-m. length. 



246 THE MECHANICAL PROPERTIES OF FLUIDS 

''A 
edge of the circular film of radius - . They are usually the means 

of about five observations and are expressed in decimals of a 
degree. 

The last two columns show the errors due to taking the ratio of 
angles and the ratio of sines respectively as giving the stress ratio. 

The error is always positive for a//?, and its mean value is 1-98 

per cent. In the case of the average error is only 0-62 per cent. 

smj8 

In only two instances does the error reach i per cent, and m both 
it is negative The presence of sharp corners seems to introduce a 
negative error which is naturally gieatest when the corneis aic neaiest 
to the observation point Otheiwise, theie is no evidence that the 
error depends to any great extent on the shape Nos 4, 5, 7, and 8 
in the table are examples of the application of the method of noimal 
septa described above in which the film is bounded by a plane per- 
pendicular to the hole at a plane of symmetiy. 

Table II shows the results of volume determinations made on 
each of the sections i to 8 given in the picvious table. 

TABLE II 

SHOWING EXPERIMENTAL ERROR IN DETERMINING TORQUES BY MEANS 

OF SOAP FILMS 



No. 



*8. 



Section 



Equilateral triangle 

height, 3 in. 
Square side, 3 m 
Ellipse semi - axes\ 

2 in X i m / 

Ellipse- 3 m X i in 
Ellipse' 4 in X o 8m 
Rectangle- sides, 4") 

m. X 2 m . . / 
Reactangle 8m. X \ 

2 in. . . / 

Infinitely long rec-\ 

tangle . . . / 



Maximum Observed Calculated 
Inclma- Volume Toiquc 



Knot 



tion 
Deg 


Ratio 


Ratio 


I'ci cent 






32 06 


1953 


I 985 


-i 6 


3039 


I 416 


I 43 2 


i i 


3050 


I 143 


I 133 


-1 09 


31 oi 

36 12 


2147 
3 041 


2 147 

3 020 


-j-o 7 


3133 


1456 


M75 


-13 


3528 


1-749 


174-1 


+03 


36 oo 


0858 


0-848 


+ 12 



* On 4-m length. 



DETERMINATION OF STRESS BV SOAP FILMS 24? 

The average error is o 89 per cent. In four of the eight cases con- 
idered the error is greater than i per cent and in three of these it 
i negative. One may conclude that the probable error is somewhat 
reater than it is for the stress measurements, and that it tends to 
e negative. Its upper limit is probably not much in excess of 2 
er cent. The remarks already made regarding the dependence of 
ccuracy on the shape of the section apply equally to torque measure- 
lents 

When contour lines have been mapped, the torque may be found 
Lorn them by integration. If the graphical work is carefully done, 
tie value found in this way is rather more accurate than the one 
btamed by the volumetric method. Contours may also be used 
3 find stresses by differentiation, that is, by measuring the distance 
part of the neighbouring contour lines; but here the comparison 
5 decidedly m favour of the direct process, owing to the difficulties 
iseparable from graphical differentiation. The contour map is, 
eveitheless, a very useful means of showing the general nature of 
tie stress distribution throughout the section in a clear and com- 
>act manner. The highly stressed parts show many lines bunched 
Dgether, while few traverse the regions of low stress, and the direc- 
Lon of the maximum stress is shown by that of the contouis at eveiy 
iomt of the section. Furtheimore, the map solves the torsion pro- 
ilem, not only for the boundary, but also for every section having 
tie same shape as a contour line 

Example of the Uses of the Method 

The example which follows serves to illustrate the use of the 
oap-film apparatus in solving typical problems in engineeiing 
tesign. 

It is well known that the stiess at a sharp internal corner of a 
wisted bar is infinite or, rather, would be infinite if the elastic equa- 
tons did not cease to hold when the stress becomes veiy high. If 
he internal corner is rounded off the stress is reduced; but so far 
LO method has been devised by which the amount of reduction in 
tram due to a given amount of rounding can be estimated. This 
iroblem has been solved by the use of soap films. 

An L-shaped hole was cut in a plate. Its arms were 5 in. long 
>y i m wide, and small pieces of sheet metal were fixed at each end, 
lerpendicular to the shape of the hole, so as to form normal septa. 
The section was then practically equivalent to an angle with arms of 



248 THE MECHANICAL PROPERTIES OF FLUIDS 

infinite length. The radius in the internal corner was enlarged step 
by step, observations of the maximum inclination of the film at the 
internal corner being taken on each occasion. 

The inclination of the film at a point 3-5 in. from the corner 
was also observed, and was taken to represent the mean boundary 
stress in the arm, which is the same as the boundary stress at a point 
far fiom the corner The ratio of the maximum stress at the internal 
corner to the mean stress in the arm was tabulated for each radius 
on the internal corner. 

The results are given in Table III. 

TABLE III 

SHOWING THE EFFECT OF ROUNDING THE INTERNAL CORNER ON THE 
STRENGTH OF A TWISTED L-SHAPED ANGLE BEAM 

Radius of Internal R atlo Maximum Stress 
Cornei Stress in Arm 



Inches 

o 10 I 890 

o 20 I 540 

o 30 I 480 

040 1-445 

0-50 I 430 

0-60 I 420 

070 t'^S 

80 i 416 
r oo i 422 

1 50 I 500 

2 OO I 660 



It will be seen that the maximum stiess in the internal corner 
does not begin to increase to any great extent till the ladms of the 
corner becomes less than one-fifth of the thickness of the aims 
A curious point which will be noticed in connection with the table 
is the minimum value of the ratio of the maximum stress to the stiess 
in the arm, which occurs when the radius of the corner is about o 7 of 
the thickness of the aim. 

In fig. 3 is shown a diagram representing the appearance of these 
sections of angle-irons. 

No. i is the angle-iron for which the radius of the corner is one- 
tenth of the thickness of the arm. This angle is distinctly weak at 
the corner. 



DETERMINATION OF STRESS BY SOAP FILMS 249 

In No. 2 the radius is one-fifth of the thickness. This angle-iron 
is nearly as strong as it can ? 

be. Very little increase in 
strength is effected by round- 
ing off the corner more than i 
this. No. 3 is the angle with 
minimum ratio of stress in 
corner to stress in arm. 

A further experiment was 2 

made to determine the extent 

of the region of high stress in /" ' 

angle-iron No. i For this 3 

purpose contour lines were 

mapped, and from these the 

slope of the bubble was found 

at a number of points on the F IS 3 

line of symmetry of the 

angle-iron. Hence the sti esses at these points were deduced, 

The results are given in Table IV. 

TABLE IV 

SHOWING THE RATE OF FALLING-OFF OF THE STRESS IN THE 
INIERNAL CORNER OF THE ANGLE-IRON 

Distance fiom Ratio Stt ess at Point 

Boundary * Boundary Sti ess m Arm" 



Inches 

000 1-89 

O 05 i 36 

O 10 I 12 

O'2O 077 

0-30 o 49 

o 40 0-24 

o 50 o oo 



It will be seen that the stress falls off so rapidly that its maximum 
value is to all intents and purposes a matter of no importance, if 
the material is capable of yielding. If the material is brittle and not 
ductile a crack would, of course, start at the point of maximum 
stress and penetrate the section. 



250 



Comparison of Soap -film Results with those obtained 

in Direct Torsion Experiments 

As an example of the order of accuracy with which the soap- 
film method can predict the toisional stiffness of bais and girders 
of types used in engineering, a comparison has been made with 
the experimental results of Mr. E. G. Ritchie.* The toisional 
stiffness of any section can be represented by a quantity C such that 
torque = CN, wheie N is the modulus of rigidity. C has dimen- 
sions (length) 4 . In Table V column 2 is given the value of C found by 
soap-film methods, while m column 3 is given the conesponding 
experimental results taken from Mr. Ritchie's papei . 



Section 



Angle: 
Angle. 



1-175 X 1-175 in. 

i-oo X i oo in. 
Tee- 1-58 X 1-58 m. _ 
I-beam: 5-01 X 8-02 in. 
I-beam. 3-01 X 3-00 in. 
I-beam: 1-75 X 4-78 in 
Channel 1 o 97 X 2 oo m. 



TABLE V 

C (Soap Film) 

0-01234111 4 
o 0044 m 4 

01451 m 4 

1 160 in 4 
o 1179 m 4 
0-0702 m. 4 
0-0175 m 4 



C ('Dnect Toision 
Experiments) 

o 01284 m 4 
0-00455 m 4 
o 01481 in l 
1-140 in 4 
o 1082 in a 
o 0635 m 4 
o 0139 m 4 



Torsion of Hollow Shafts 

The method descubed above must be modified when it is desned 
to find the torsion function loi a hollow shaft In this case the lunc- 
tion satisfies the equation (6) and the boundaiy conditions aic W = 
constant on each boundary, but the constant is not necessanly the 
same for each boundary. In order to make use of the soap-film 
analogy it is therefore necessaiy to cut a hole m a flat sheet of metal 
to lepiesent the outer boundary, and to cut a metal plate to leprc- 
sent the inner boundary. These are placed in the conect relative 
positions in the appaiatus shown in fig. i, and they arc set so that 
they lie in parallel planes. The soap film is then stretched acioss the 
gap between them. 

The planes containing the two boundaries must be parallel, 

*A Study of the Circular Arc Bow Girder, by Gibson and Ritchie (Constable & 
Company, 1914). 



it they may be at any given distance apart and yet satisfy the con- 
tion that IP" = constant round each boundary. On the other hand 
ie contour lines of the film, and hence the value of IP, will vary 
eatly according to what particular distance apart is chosen. The 
ilution of the torsion problem must be quite definite, so that 
ust be possible to fix on the particular distance apart at which the 
anes of the boundaries must be set in order that the soap film 
retched on them may represent the requned torsion function. 
o do this it is necessary to consider again the function <, which 
presents the displacement of a particle from its original position 
ving to the warping of plane cross sections of the twisted material. 
his function < is evidently a single-valued function of x and y, 
;. it can have only one value at every point of the material. In 
neral the values of W found by means of the soap-film apparatus 
) not correspond with single-valued functions <j). On the other hand, 
ere is one particular distance apart at which the planes of the boun- 
ties can be placed so that the W function does correspond with 
single-valued function 0. To solve the torsion problem we must 
id this distance. 

If <j> is single- valued, I ds --= o when the integral is taken lound 

her boundary; and since = --, this condition reduces to 

9* 8 * 

~ds ~ o. Substituting W = ift ~l(x z + y z ) and remem- 

7tl 

ring that 

3 t n . 9X dy 3 /r 2 -f- y\ . dx 3 /# 2 -f y z \ dx dy 

;. ftf -j- y*\ = JL ( ' J \ -j --- I .. - !_^_ \ = y x~, 

dn 8/2 dy \ 2 ] dn dx\ 2 / 9i 9^ 

will be seen that 

dtft, 

fds= ~- 

on J on 

lere A represents the area of the boundary. The condition that 
shall be single-valued is therefore 



f&Fj , . . 

~-ds-2&, ......... (14) 

J 



05) 



if erring again to the soap-film analogy, and putting W 
will be seen that (15) is equivalent to 



'/' 



= A.p (16) 



552 THE MECHANICAL PROPERTIES OF FLUIDS 

Equation (16) applies to either boundary; it may be compared with 
equation (12), which there applies only to a solid shaft. Taking the 
case of the inner boundary, it will be noticed that Ap is the total 
pressure exerted by the air on the flat plate which constitutes the 

inner boundary, zylsmads* on the other hand is the vertical 

component of the force exerted by the tension of the film on the 
inner boundary. Hence the condition that < shall be single-valued 
gives rise to the following possible method of determining the posi- 
tion of the inner boundary. The plate representing it might be 
attached to one arm of a balance. The film would then be stretched 
across the space between the boundaries, and if the outer boundary 
was at a lower level than the inner one the tension in the film would 
drag the balance down. The pressure of the air under the film 
would then be laised till the balance was again in equilibimm. The 
film so produced would satisfy condition (16). 

As a matter of fact this method is inconvenient, and another 
method based on the same theoretical principles is used in practice, 
but for this and further developments of the method to such 
questions as the flexure of solid and hollow bars the reader is 
referred to Mr. Griffith's and Mr. Taylor's papers published in 
1916, 1917, and 1918 in the Reports of the Advisoiy Committee 
for Aeronautics. 



Example of the Application of the Soap -film Method 
to Hollow Shafts 

As an example of the type of research to which the soap-film 
method can conveniently be applied, a brief description will be given 
of some work undertaken to determine how to cut a keyway in the 
hollow propeller shaft of an aeroplane engine, so that its strength 
may be reduced as little as possible. These shafts used to be cut 
with sharp re-entrant angles at the bottom of the keyway, and they 
frequently failed owing to cracks due to torsion which started at 
the re-entrant corners. It was proposed to mitigate this evil by 
putting radii or fillets at these corners, and it was requited to know 
what amount of rounding would make the shafts safe. 

The shafts investigated were 10 in external and 5-8 in. internal 
diameter. This was not the size of the actual shafts used in aero- 

* The factor 2 comes in owing to the fact that y is the surface tension of one 
surface md the film has two surfaces. 



DETERMINATION OF STRESS BY SOAP FILMS 253 

)lanes, but it was found to be the size which gave most accurate 
esults with the soap-film apparatus. 

Some of the results of the experiments are shown graphically in 
he curve in fig. 4.* In this curve the ordinates represent the maximum 



SO I Outside diam of shaft to", 

5 v Viam of hol&(concentHc) 5 $ 

Depth of keyway / o., 

VMft/i afkeunau z 5 



3 O 




20 



The maximum stress is given as a multiple cf the 
maximum stress ma similar shaft without hey way 

(A) When the two shafts tti e twisted through the same angle 

(B) When they are subjected to the same torque 
The radius of the filkt isgtvenas a fraction of 

the depth oftKe key way 

The dotted curves show the respective stresses 
in the middle of the key way 



01 02 03 04 05 O6 O 7 

RADIUS OF FILLET IN 
CORNER OF KEYWAY 

Fig 4 Toraional Strength of Hollow Shaft with Keyway 

shear stress, on an arbitrary scale, while the abscissse lepresent the 
adms of the fillet, in which the internal corners of the keyway were 
-ounded off It will be seen that the shaft begins to weaken rapidly 
.vhen the radius is less than about 0-3 m . 

* This diagram and also that shown in fig 5 aie taken from Messrs. Griffith 
md Taylor's report to the Advisory Committee for Aeronautics, 1918. 



254 THE MECHANICAL PROPERTIES OF FLUIDS 

The lines of shearing stress, i.e. the contour lines of the soap film, 
are shown in fig. 5 for the case when the radius of the fillet is 0.2 in. 




S Lines of Shearing Stress m the Torsion of a 
Hollow Shaft with Keyway 



It will be seen that the lines of shearing stress ai c crowded together 
near the rounded corner of the keyway. 



CHAPTER VIII 
Wind Structure 

During the present century great advances have been made in 
5 field of aviation, and problems, some of them entirely new, others 
der a new guise, have presented themselves. Among the latter 
iy be included the problem of wind structure Slight changes of 
5 wind, both in direction and in magnitude, are of little account 
some problems where only the average effect of the wind is of 
f moment On the other hand, for the aviator these small changes 
often of far greater moment than the general drift, especially 
en his machine is either leaving or approaching the ground. Now 
s just at this point that the irregularities are often greatest. 
Before proceeding to deal with the cause of these vanous irregu- 
ities, let us consider what are the governing factors in the move- 
nt of a mass of air over the surface of the globe. 
Apart from the difficulties of dynamics, the general problem is 
2 of much complexity. In the first place, the surface of the earth 
not at all uniform. It consists of land and water surfaces, and a 
d suiface and a water surface behave quiet differently towards 
ar radiation, so that air over one area becomes more warmed up 
n that over another. Further, the land areas are divided into 
icrts and regions rich in vegetation, flat plains, and mountain ranges, 
am, water vapour, to whose presence in the atmosphere nearly all 
teorological phenomena are due, while being added at one place, 
not subtracted simultaneously at another, so that the amount 
sent in the atmosphere varies veiy inegulaily. These and other 
tors tend to render an exact mathematical solution of the problem 
ctically impossible. 

An approximate determination, however, of the effect of the 
th's rotation on the horizontal distribution of pressure when the 
moves over the surface of the globe in a simple specified manner, 
i be found. 
To obtain this approximate solution of the problem, we shall 

255 



256 THE MECHANICAL PROPERTIES OF FLUIDS 

assume that the air is moving horizontally* with constant linear 
velocity v, i.e. that a steady state has been reached. The forces 
acting on a particle of air, in consequence of its motion, under these 
conditions at a place P in latitude < arise from two causes, (i) the 
rotation of the earth, and (2) the curvature of the path in which the 
particle is moving at the instant, relative to the earth. The problem 
before us therefore is (i) to find the magnitude of the accelerations 
arising from these causes and (2) to show how the forces required 
for these accelerations in the steady state are provided by the pressure 
gradient. 

Consider first the effect of the rotation of the earth on great-circle 
motion. We shall suppose the particle is constrained, by properly 
adjusted pressure gradients, to move in a great circle through P with 
uniform velocity. The particle is therefore supposed to move in a 
path which is rotating in space about an axis passing through its 
centre. 

The rotation of the earth takes places about its axis NS, fig. i. 
The great circle Q'PQ is the specified path of the paiticle. The 
earth's rotation may be resolved by the parallelogram of rotations into 
two rotations about any two directions in a plane containing NS 
Let these two directions be the two perpendicular lines OP and OW, 
where O is the centre of the earth and P the point in latitude ^ 
referred to above. If the angular velocity of the earth about SN be 
co, then the component angular velocities are co cos</> about OW and 
CD sm< about OP. As the two axes are mutually perpendicular, it 
follows that any particle in the neighbourhood of P is in the same 
relation to OW as a particle on the equator is to ON. But a particle 
on the equator moving with uniform horizontal velocity has an 
acceleration directed only perpendicular to the axis ON, and therefore 
its horizontal velocity is not affected by the rotation about ON. 
Similarly the horizontal velocity of a paiticle near P is affected only 
by the component co sin< about OP, and not by the perpendicular 
component co cos< about OW. We need consider therefoie only the 
effect of the component co sm<ji. 

When the particle crosses the point P, it will travel a distance 
PA = vdt (see fig. id) in time dt, as the velocity is v. In the same 
interval of time, the line along which the particle started will have 
moved into the position PA', so that the element of arc ds = AA 
= PAco s'w.(j)dt. 

*Ie.m a plane perpendicular to the direction of the force compounded of the 
force of gravity and the centrifugal force. 



WIND STRUCTURE 



257 



Also ds or AA', which is described in a direction perpendicular 
to PA, may, by the ordinary formula, be expressed in the form 



North 

t 

Horizontal 
Plane at- P 



tacit' 




e t-Eaaf 




&/W 2 > where /is an acceleration in the direction perpendicular to the 
direction of motion Hence 

\f(dt} z = PAw sin^d? = 
since PA = vdt; 

i.e. / = 



258 THE MECHANICAL PROPERTIES OF FLUIDS 

Hence the transverse force (F) necessary to keep a mass (m] of ai 
moving along a great circle, in spite of the rotation of the eaith, i 
given by 

F = mf = zmvai sin^, (i) 

acting, in the northein hemisphere, towards the left, in the southerr 
towards the right, when looking along the direction of th( 
wind. 

This expression, which is very nearly correct, shows that th 
deflective force due to the rotation of the eaith* on a mass of moving 
air is (i) directly proportional to the mass, to the horizontal velocity 
to the earth's angular velocity, and to the sine of the latitude of the 
place; (2) independent of the direction of the great circle, i.e. of 6 
(fig. i); (3) always perpendicular to the instantaneous direction oi 
motion of the air and therefore without influence on the velocity 
with reference to the surface; (4) opposite to the direction of the 
earth's rotation. 

When the air moves, as specified, in a great circle, the acceleiation 
zvo) sin^ is the only transverse acceleration in the hoiizontal plane, 
for the acceleration arising from the curvature of the path relative to 
the earth (which exists even if co were zero) is radial and theiefoie 
has no appreciable component in the horizontal plane. 

Now suppose that the path is not a great circle but a small one, 
R'P (fig. i) In addition to the term 2,va> sm^ there will now be a 
term arising from the curvature of the path This term is inde- 
pendent of oj. Let fig. 2 be a section of the sphei e through a diameter 
of the small circle, PR' being the diameter. The path of the air at 
P is now curved, and if r is the radius of curvature of the path at P, 

2,3 

in the horizontal plane, is the acceleration in the horizontal plane 

r 

arising from the curvature of the path. But this acceleration is also 

the horizontal component of where / is PM, i.e. the radius of 

r 

curvature of the small circle. If a is the angular radius of the small 
circle it is also the inclination of the horizontal plane to the plane 
of the small circle (see fig. 2), hence 

= cosa: 
r r 

.*. r' = r cosa, 
* I.e the force F, reversed. 



WIND STRUCTURE 



2 59 



e. N (fig. 2) is the centre of curvature of the path in the horizontal 
lane. 
It is also clear from fig. 2 that R sina = r f , hence 



= cosa 
r r' 



. cosa = - cota. 
R sina R 




Fig 2 Relation between Radn of Curvature of the Path on the Earth 
and the Path in the Horizontal Plane PM = r' PN = r'/cosa = > PG 



the air is moving freely in space, i e if the barometric pressure 
umfoim, the resultant horizontal acceleiation is zero, i.e. 



2VO) sm<j> -f- cota = o. 



[ence " free " motion is only possible when out of these accelera- 
ons has the opposite direction to the other, and 



= ~ cota 



260 THE MECHANICAL PROPERTIES OF FLUIDS 

numerically, i.e. when the acceleration due to path curvature 
balances that due to the earth's rotation. 

When the barometric pressure is not uniform, we proceed 
thus: 

If the particle or element of air at P occupies the volume of a 
small cylinder, of length S in the direction of the outward drawn 
normal at P to the path of the air in the horizontal plane, and of unit 
cross-sectional area, the force on the air in the inward direction 

due to variation of pressure is ( ^n ) . The mass of air is p8?2 where 

\dn I 
p is the density at P, hence 

8p s fi f . , . w 2 cotal 

-on = pt)n\ 2vaj sm< + 

dp ; i prf cota ., s 

" in. = p ZJa>sm ^+ R ............ (*) 



The formula is true for positive and negative values of v, lemem- 

8i> 
bering that -j- is the gradient of pressure in the outward direction 

of the normal, and that the rotational term in the acceleration is 
towards the left hand when looking along the direction of the wind in 
the northern hemisphere. The two cases of cyclonic and anticyclomc 
wind (i.e. -f and ~v) are shown in fig. 3. The forces indicated 
in this figure are those required to keep the air in its assumed path, 
relative to the earth. These forces are provided by the pressure 
gradient. If we take the numerical value of the pressure gradient 
and the wind speed, then 



, , 

n^ + T5- cota 
on K 



for the cyclone, where ~ is the rate of rise of pressure outwards. 
For the anticyclone, " n 

, pv* 

n^ - - cota, ........... (30) 



t 
where ^- is the rate of rise of pressure inwards. Both cases are 

included in (2) without ambiguity. 



WIND STRUCTURE 



261 



These expressions give a value of the wind velocity called the 
f adient wind velocity. The direction of this gradient wind according 
> the previous reasoning is along the isobars, and is such that to one 
loving with it in the northern hemisphere, the lower pressure is 
i the left hand. It must be distinctly understood that in the above 
cpressions for the gradient wind a steady state has been reached; 
id further, it is assumed in arriving at these expressions that there 



Cyclone 



Anticyclone 





Fig 3 

Pa = apvta sm</> 

= term ausing from rotation of the earth 

Pb = P."-" cota 
"~R~ 
= term arising from the angul it radius of the path 



no friction between the air and the surface of the earth ovei which 
s passing. 

Under actual conditions the relation cannot be satisfied exactly 
there is always a certain amount of momentum absorbed irom the 
earn of air by the friction at the surface. This absorption of 
^rgy is manifested by the production of waves and similar effects 
water surfaces, on forests, and on deserts. Yet under the most 
favourable conditions this relation between wind and pressure can 
recognized, and therefore it must be an important principle in the 
ncture of the atmosphere. Also when we ascend into the at- 
sphere beyond the limits where the influence of surface friction 
ikely to be felt, we find very little difference in the velocity of the 



262 THE MECHANICAL PROPERTIES OF FLUIDS 

wind for hours on end. According to Shaw,* " pressure distribu- 
tion seems to adjust itself to the motion of the air rather than to speed 
it or stop it. So it will be more profitable to consider the strophic 
balance between the flow of air and the distribution of pressure as 
an axiom or principle of atmospheric motion." This axiom he has 
enunciated as follows :f " In the upper layers of the atmosphere the 
steady horizontal motion of the air at any level is along the horizontal 
section of the isobaric surface at that level, and the velocity is in- 
versely proportional to the separation of the isobadc lines in the level 
of the section." 

Throughout this short study of wind structure we shall follow 
Shaw therefore, and regard the wind as balancing the pressure 
gradient. It may be argued that this assumption strikes at the root 
of the processes and changes in pressure distribution one may desire 
to study. The results of investigation appear to indicate, however, 
that in the free atmosphere, at all events, the balance is sufficiently 
good under ordinary conditions for us to take the risk and accept the 
assumption. Under special circumstances and in special localities 
there may occur singular points where the facts are not in agreement 
with the assumption, but the amount of light which can be thrown 
upon many hitherto hidden atmospheric processes, appears to justify 
our acceptance of it. 

In the expression for the calculation of the gradient wind the 
right-hand side consists of two terms The first term, 2pvw sm^, 
is due as we have seen to the rotation of the earth, and in consequence 
has been called the geostrophic component of the pressure gradient. 

7) O 

The other part, - cota, arises from the circulation in the small 
R 

circle of angular radius a, and so has been termed the cydostrophic 
component. With decrease in <f>, i.e. the nearer we approach the 
equator, a remaining constant, the first component therefore be- 
comes less and less important, the balance being maintained by the 
second term alone practically. On the other hand, with increase 
in a, i.e. the nearer we approach to the condition of the air 
moving in a great circle, <j> remaining constant, the second term 
becomes less and less important, until finally with the air mov- 
ing on a great circle the gradient and the geostrophic wind are 
one and the same. Consequently in the pressure distributions in 
mean latitudes where the radius of curvature of the path is 

* Manual of Meteorology, Part IV, p. 90 
t Proc. Roy Soc. Edin., 34, p. 78 (1913). 



WIND STRUCTURE 263 

lerally very large, the geostrophic wind is commonly taken as 
5 gradient wind. 

Having obtained expressions indicating the connection between 
j pressure gradient and the theoretical velocity of the wind, we 
ill now consider some of the reasons for the variations of the wind 
ocity from this theoretical value. 

In the equation for the gradient wind and in the statements made 
;arding the effects of friction on the wind, there is nothing to 
licate that the flow of air is not steady But it is a perfectly well 




Scales Ihf-em JoyS- S7Jln. lOmi/kr 6in, IOft/j, . tOSln, Bwufort tinutb at 03, 21 IS 
1 U (2 1^, 14 15 16 f 13 19 20 



| E __ 

N ' N Lowor lnM/n - 2 in +J 

Fig <L Anemometer Record at Aberdeen Observatory, a6th September, 1923 

>wn fact that the air, at all events near the surface, does not flow 
h a constant velocity even for a very short interval of time. This 
iteadiness in the wind velocity is exhibited very well by the records 
self-recording anemometers Fig. 4, which is part of the record 
26th September, 1922, at Aberdeen, exhibits this moment- to- 
ment vanation. Not only does the velocity vary but the direction 
> shows a similar variation, as indicated by the lower trace in the 
ire. As a rule the greater the variation in velocity, the greater also 
variation in direction. 

The variations both m velocity and direction are very largely 
undent upon the nature of the surface over which the air is 
sing, i.e. the nature of the records is greatly affected by the 
osure of the anemometer. A comparison of figs. 4 and 4*2 reveals 
i very plainly The first, as stated above, is a record from the 



264 THE MECHANICAL PROPERTIES OF FLUIDS 

anemometer at King's College Observatory, Aberdeen. The head of 
the instrument is 40 ft. above the ground, the instrument itself being 
housed in a small hut * in the middle of cultivated fields and placed 
about | mile from the sea. The second is a record from an ane- 
mometer situated about 5 miles inland from the first, at Parkhill 
Dyce, and belonging to Dr. J E. Crombie. The exposure in this 
case is over a plantation of trees, and though the head of the instru- 
ment is 75 ft. above the ground, it is only 15 ft above the level of 




Lower margin a Z //I, 

Fig 4 Anemometer Record at Parkhill, Dyce, aGth September, 1922 

the tree-tops. The two records refer to the same day, and the 
anemometers are situated comparatively close the one to' the other, 
yet the " gustiness " as indicated on the second is much greatei 
than that on the first. At the same time the average velocity of 
the wind in the second case is considerably reduced by the geneial 
effects of the nature of the exposure. 

Other factors which affect this variation of the wind are to be 
found in the undisturbed velocity of the wind in the upper air 
and in the temperature of the surface of the ground, i.e. in the time 
of day and in the season of the year. A great deal of light has been 
thrown on these variations of the wind near the surface by G. I. 
Taylor through his investigations of eddy motion in the atmosphere.f 

*The position of this hut is about a quarter of a mile directly eastwards fiom 
the position of the anemometer shown in fig 6 Fig 6 is compiled from records 
taken in the old position 

t ''Phenomena connected with Turbulence m the Lower Atmosphere ". Proa. 
Roy Soc. A, 94, p 137 (1918). * ' 



WIND STRUCTURE 



265 



From the aviator's point of view these variations are often oi 
prime importance. Beginning therefore at the surface, we shall 
endeavour to asceitain how the actual wind is related to the 
geostrophic or gradient wind for various exposures, and after- 
wards determine how these relations alter as we ascend higher into 
the atmosphere. 

When the hourly mean values of the surface wind velocities 
are examined for any land station, it is found that there is a diurnal 

6 ^ f. 




k O 2 4 fe 8 1O 12, 14 16 ifi 2X) 22 24 hour 
Aberdeen- R&UJ 

Fig 5 Diurnal Variation in Wind Velocity for January and July, 
for the period 1881-1910 

and also a seasonal variation in the velocities. Fig 5 represents 
this diurnal variation for the two stations, Aberdeen and Kew, for 
the months of January and July. On the other hand, no correspond- 
ing diurnal variation of the barometric gradient is to be found for 
these stations. The diurnal variation of the wind is evidently 

(D312) 10* 



2 66 THE MECHANICAL PROPERTIES OF FLUIDS 

dependent upon the diurnal arid seasonal variations of temperature, 
and therefore the i elation of the suiface wind to the geostrophic 
wind is also dependent on these quantities. The curves in fig 5 
show a maximum corresponding closely with the time of maximum 




Fig. 6 Relation between Geostropluc and Observed Surface Winds of Force 4 at Aberdeen 

Central circle shows position of the c,ty of Aberdeen with reference to the two 

river valleys and the sea Stippled area shows high ground Hatch nj ar^a 

shows town buildings 

The outer circle represents the grad ent wind The inner circle represents 43 per cent 
of the gradient wind The dotted line represents the observed wind 

temperature. Therefore if W represent the surface wind and G the 
geostrophic wind, the ratio W/G increases with increase of tem- 
perature, and vice versa. If then the surface layers be warmed 
or cooled from any cause whatsoever, we always find this effect on 
the latio W/G. 

The exposure of a station also has its effect on the ratio W/G. 







UB3TAT 


CO 


00 





OX 


oo 


CO CO IO C 


50 VO 








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267 



2 68 THE MECHANICAL PROPERTIES OF FLUIDS 

Fig. 6 shows this effect on winds of force 4 at Aberdeen. The 
geostrophic wind is represented by the outer circle, while the 
dotted irregular curve gives the percentage which the suiface wind 
is of the geostrophic wind. An idea of the exposure of the station 
is afforded by the circular poition of the ordnance map of the dis- 
tiict placed at the centre of the figure. On the west side there is 
land, on the east, sea. To the south-west of the station lies the 
city, and we find that in this direction the surface wind has the 
lowest percentage, while in the north-easterly directions the pei- 
centages are largest. Towards the north-west lies the valley of 
the Don, and a fairly open exposure, the effects of which aie also 
well brought out in the figure. 

The effect of different exposures on winds of the same geo- 
strophic magnitude will be understood readily from an examination 
of Table I. 

It is evident, therefore, that no general lule can be given with 
regard to the value of the ratio W/G. It may have a wide lange 
from approximately unity downwards, depending on the time of day, 
the season of the year, and the exposure of the station. In the 
same way the deviation a of the surface wind fiom the dnection of 
the geostrophic wind is found to vary over a wide range. 

An example of this is afforded by Table II, wherein are set out 
the values for Pyrton Hill and Southport, as given by J S Dines in 
the Fourth Report on Wind Structure to the Advisoiy Committee 
on Aeronautics. 

It is necessary, therefore, in giving an estimate from the baiometnc 
gradient of the probable surface wind, as regaids either direction or 
velocity, that due attention be paid to the details mentioned above 

Occasionally the surface wind is found to be in excess of the 
gradient. This probably anses from a combination of a katabatic * 
effect with the effect of the pressure distribution, the katabatic effect 
more than compensating for the loss of momentum in the normal 
wind due to friction at the earth's surface. 

Data for the purpose of examining the ratio W/G over the sea 
are very limited. The following table, as given in the Meteoro- 
logical Office report for moderate or strong winds over the Noith 
Sea, will serve to show the deviation of the surface wind from the 
gradient wind, both in velocity and direction. 

* I e Katabatic or Gravity Wind when the suiface au ovci a slope cools at 
night or from any othei cause it tends to flow down the slope, this is especially 
pronounced on clear nights In ravmes, if snow-coveied and devoid of forests, 
this wind often reaches gale force. Such a wind is known as a katabatic wind. 



WIND STRUCTURE 



269 



w 

PQ 



MNN 



MNM 



MSM 



$* * 



as i 



3N3 



3NN 



3 a 

Q.T3 O 



& a. 

p <* 



SI* 
El 

PnW 



I 



270 THE MECHANICAL PROPERTIES OF FLUIDS 



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WIND STRUCTURE 271 

Here again we see that no definite rule can be given for estimating 
ie surface wind from the geostrophic wind. It will be observed, 
)wever, that each quadrant exhibits certain dominant features and 
ust be considered therefore by itself. In this way a considerable 
nount of guidance is obtained by the forecaster in estimating the 
md over the sea from a given pressure distribution. 

We now pass to consider the actual wind in relation to the 
:ostrophic wind in the first half-kilometre above the surface. 
r H. Dines, in his investigation on the relations between pressure 
id temperature in the upper atmosphere, has found a high correla- 
m between the variations of these elements from their normal 
lues for heights from 2 Km upwards. Below the 2-Km. level 
e coi relation coefficients gradually diminish until at the surface 
actically no connection at all is found. Above the 2-Km. level 
2 may regard the air as in an " undisturbed " condition, i e free 
Dm the effect of the friction at the earth's surface In this un- 
sturbed region the velocity and direction of the wind at any given 
ight are governed by the pressure and the temperature gradients 
ling at that height, while in the lower layers we find considerable 
viation from this law, evidently due to the effects of the surface 
the earth on the air m contact with it 

Several empirical foimulas have been given whereby the velocity 
the wind at any height in the lower layers of the atmosphere may 
calculated ftom that at a definite height, say 10 m , above the 
rth's surface Fiom observations, up to 32 m , over meadow- 
id at Nauen, Hellmann* confirmed an empirical formula v =/e/z^, 
nch agrees very nearly with a formula v = ktf suggested by 
-chibaldf from kite observations in 1888. The results of observa- 
ms up to 500 m., carried out m 1912 with two theodolites, aie 
yen by J. S. Dines in the Fourth Report on Wind Structure already 
fened to Here he has represented his conclusions by a series 

curves, and m doing so has grouped the winds into three sets: 
) very light, wheie the velocity at 500 m. is less than 4 m. per 
cond, (2) light, with velocity between 4 m per second and 10 m. 
r second; and (3) strong, with velocity greater than 10 m per 
cond The curve for very light winds (see fig 7) shows that in 
is class the surface wind approaches the geostrophic value, which 
also marked for each group at the top of the diagram, much more 
osely than m any of the other groups. Cuives of this type enable 

* Meteor Zeitschnft, 1915. 
[ Nature, 27, p. 243. 



272 THE MECHANICAL PROPERTIES OF FLUIDS 

one to judge of the aveiage behaviour of the wind in the lowest 
half-kilometre according to the pressure gradient at the suiface, 
When, however, curves are drawn for different hours of the day, 
7 hr., 13 hr., 18 hr , these show differences among themselves even 
for the same surface gradient. A whole series of curves for various 
hours of the day and different seasons of the year would be necessary, 
therefore, before a complete solution of the problem could be obtained. 
As these formulas and curves just referred to are applicable 
under certain conditions only, and as the constants used differ for 



Lig 



rv 
ght 



Strom, 



400 



300 



2.00 



1OO 





16 



2.0 



2 4 6 8 1O 12 14 

V&L.QCITY IN METRES PER SECOND 

Fig 7 Change of Wind Velocity with Height within 500 metres above the suiface 

different times of the day and different seasons of the year, though 
they supply a rough working rule, yet a more exact solution of the 
problem is desirable. This has been supplied by the investigations 
of G. I. Taylor.* In his solution he regards the wind m the un- 
disturbed layer as equivalent to the geostrophic wind at the surface, 
while the region between the surface and the undisturbed layer is 
considered as a slab through which the momentum of the undis- 
turbed layer is propagated, as heat is conducted through a slab of 
material the two faces of which are kept at different temperatures. 
The momentum is propagated, according to the theory, by eddy 
motion, the surface of the earth acting as a boundary at which the 
momentum is absorbed The equation representing the propaga- 
tion is given as g . 9 , 

<"\ / i V I UH \ t \ 

p3tt /a ( = _(_), (4) 

* " Eddy Motion in the Atmosphere ", Phil, Trans. A, 215, p. i (1915) 



WIND STRUCTURE 273 

iere p = the density and K the "eddy conductivity" of the 
. For small heights up to i Km or thereby, p and K are approxi- 
ttely constant. Therefore the equation, representing the distri- 
tion of velocities with height and time within this region, may 
written as 

pdufdt - Kpd 2 u/dz z ................. (5) 



The value of K is, according to Taylor,* roughly ^wd where w 
the mean vertical component of the velocity due to the turbulence, 
cl d represents approximately the diameter of a circular eddy. 

The value of AC differs, however, according to (i) the nature of 
5 surface over which the air current is passing, (2) the season 
the year, and (3) the time of day. As both heat and momentum 
j conducted by the eddies, the value of K will be the same for 
th Values of K have accordingly been determined by Taylor as 
lows. 

'i) Over the sea (determined fiom temperature ob-\ 3 X io 3 C.G S 
seivations over the Banks of Newfoundland)/ units. 

'2) Over grassy land (determined from velocity ob-^ 4 CCS 

servations by pilot balloons over Salisbury H , ' 

YM \ I U.I111S* 

Plain) . . . . . . . . . .) 

3) Over land obstructed by buildings (detei mined } o 4 C C S 

from the daily range of temperature observa- j- 
tions at diffeienl levels on the Eiffel Tower)J 

The effect of the season of the year on the value of K is seen by 
sparing the values obtained from the Eiffel Tower observations 
January and June. 

Whole range, 18 to 302 m.: 

(1) January . . . 4 3 X io 4 C G S. units 

(2) June . . . . . 18 3 x 10* 

(3) Whole year . . . . io X io 4 ,, 

That K is also to a certain extent dependent upon the height 
ly be understood by comparing the values for the first stage, 
to 123 m., with those for the last, 197 to 302 m 

Mean value for the whole year. 

(1) Lowest stage . . . . . 15 x io 4 C G.S units 

(2) Highest stage . .. u X io 4 

The reason for this variation is to be found in the method used 
calculating K. The nearer the ground the greater the daily 

*Proc Roy. Soc A, 94, p 137 (1917) 



274 THE MECHANICAL PROPERTIES OF FLUIDS 

variation of temperature, and therefore the error arising from the 
method used in the calculation is proportionately greater for the 
lower stages than for those higher up. The value of K for the lowest 
stage is therefore not likely to be so accurate as that for the highest 

From, wind measurements Akeiblom* has deduced the value 
of K for the whole range of the Eiffel Tower The value found, 
7-6 X io 4 C.G.S. units, is in fairly good agieement with Taylor's 
mean value, and the agreement is sufficient to show that K is the 
same both for heat and for momentum 

This theory of eddy conductivity has been applied by Taylor 
in order to furnish an explanation of the diurnal variation of the 
velocity of the wind at the surface and in the lowei layeis of the 
atmosphere. Two important conclusions have been reached He 
has shownf that when once the steady state has been reached, a state 
which previous theories claiming to explain this diurnal variation 
took no account of, a relation can be found between the undisturbed 
wind (i e. the geostrophic wind), the surface wind, and the angle 
between the direction of the isobars and that of the suiface wind. 
This relation takes the form 

W/G = cos a sin a, . (6) 

where W represents the suiface wind, G the geostrophic wind, and 
a the angle between their directions The accuracy of this i elation 
has been tested by comparing values of a observed by G. M. B. 
DobsonJ with the calculated values for certain winds over Salisbury 
Plain. Some of these results are given in Table IV 

TABLE IV. 

W/G = cos a sin a. 



Light Moderate Stiong 

Winds Winds Winds 

Observed value of W/G 0-72 o 65 o 61 

a observed . 13 deg 21^ deg. 20 deg 

a calculated . 14 18 20 ,, 



* " Recherches sur les courants les plus bas de 1'atmospheie au-dessus de Pans ", 
Upsala Soc. Sclent Acta , 2 (Ser 4), 1908, No a. 

f" Eddy Motion in the Atmosphere ",Phil Trans A,215(i9i5). See note, p. 285. 
\Quar^our Roy Met. Soc,, 40, p 123 (1914), 



WIND STRUCTURE 275 

The table shows that the agreement between observed and cal- 
lated values is very close; with a greater than 45, the equation, 
wever, no longer holds. 

The other conclusion, as shown by Taylor,* on the assumption 
at the lag in the variation in wind velocity behind the variation 

turbulence which gives rise to it is small, is that the daily 
nation in turbulence is sufficient to explain qualitatively, and to 
certain extent quantitatively, the characteristics of the daily varia- 
n in the wind velocity If the geostrophic wind G be reduced 

surface friction so that the direction of the surface wind is inclined 
an angle a to the "undisturbed" wind, then it is found that the 
-ce of the surface friction, or the rate of loss of momentum to the 
rface, is given by 2/cpBG sma, where B = *v/a>sin0//<:. As before, 
is the angular velocity of the earth and (f> the latitude The 
ation between this force of friction F and the velocity of the 
rface wind has also been examined by Taylor,f and found to be 

F = 0-0023 pW 2 . 
/. o 0023 W 2 = 2/cBG sma. 

If now numerical values be given to at and j> in K cosing /B 2 
find that 

I 2O'4 / \o / \ 

^- (cosa sin a) 2 , .(7) 

BG sma 

at = 0000073, and siriA = 077, since for Salisbury Plain 
= 50 N The values of i/BG can therefore be found for a series 
values of a. Also fiom the same equation we see that /c/G 2 is 
unction of a If we tabulate the values of /c/G 2 foi the same sei ics 
values of a, we can find then the relations between a and K These 
nous values are given in Table V (p. 276). 

Basing his discussion upon these values of the constants, 
lylor has constructed the curves given in fig 8. The abscissae 
Dresent the latio of the wind velocity at any height to the geo- 
ophic wind, while the ordmates give the ratio of the height to 
5 geostrophic wind. If the geostrophic wind be 10 m per second, 
m the numbers for the ordinates will give the heights in dekameties, 
d those for the abscissae the velocities in dekametres per second, 
le shape of each curve is determined by the value of a chosen, 
:h curve having its a value attached to it. Consequently where 

* Proc Roy Soc. A, 94, p 137 (1917) 
\Proc. Roy. Soc. A, 92, p. 198 (1916). 



276 THE MECHANICAL PROPERTIES OF FLUIDS 

TABLE V 



rs , , 

a, Degrees C G.S. uAits C.G.S. Units. 



4 252 3*54 

6 155 *'35 

8 106 0-635 

10 77-5 0-338 

12 58-5 0-192 

14 44 -8 0-116 

16 34-9 o 069 

18 27-3 0-042 

20 21-9 0-027 

22 167 0-0156 

24 12 9 0-0094 

26 99 o 0055 

28 7-4 00031 

30 55 o 0017 

32 3-7 0-00085 

34 2-6 o 00038 

36 1-7 o 00016 



the geostrophic wind and the deviation at the surface are known, 
the curves enable us to deteimine the wind velocity at any desired 
height. 

The curves may also be used to find the variation in velocity 
at a particular height under varying conditions of K The value 
of K for the open sea, for Salisbuiy Plain, and for Pans we saw to 
be 3 X io 3 , 5 X io 4 , and 10 X io 4 C G S. units respectively. If 
we take a = 10, then /c/G 2 = 0-338, which means that undei these 
three conditions G must have the values o 9, 3-8, and 5-4 m per second 
respectively Therefore the same geostrophic wind will suit dif- 
ferent curves if the value of K be altered, which it will be accoidmg 
to the exposure of the station, the season of the year, and the time 
of day. 

From the foregoing we see how the wind at the surface and in 
the lowest layers differs considerably from the geostrophic value. 
As we ascend above the surface, a nearer approach is made to the 
geostrophic values, for the effect of surface friction diminishes with 
height. Turbulence also diminishes as we ascend, its influence 
being on an average very little felt at 1000 m., though on occasions 
it may reach to 2000 m. 



WIND STRUCTURE 



277 



10. 



70 



12 



60 



Si 

ul EC 



UlUJ 



3? 

u^ 

IQ 
H2 



Oh 

^ 
o 
too; 
mu. 




3 -4 -5 6 -7 -8 -9 1-0 

RATIO OF WIND VELOCITY TO GEOSTROPHIC VELOCITY(V/fci) 



Fig 8' 



-Curves showing Variation of Wind Velocity with Height according to the Theory 
of the Diffusion of Eddy-motion. (Taylor) 



278 THE MECHANICAL PROPERTIES OF FLUIDS 

The spiral of turbulence affords another method of representing 
the variation with height of wind velocity in magnitude and diiec- 
tion In this method, first introduced by Hesselberg and Sverdrup * 
m 1915, when lines representing the wind velocity are drawn from 
the point at which the wind is measured, then their extremities He 
on an equiangular spiral having its pole at the extremity of the line 
which represents the geostrophic wind. Thus in fig 8a, if O be 
taken as the origin and OgX. the direction of the A;-axis, O^ represents 
the geostrophic wind G, OS the surface wind and L SOg the angle 
a between the two. The wind at any height Z is represented by 
OP, and is the resultant of the geostrophic wind G and of another 
component represented by gP of magnitude -\/2G smae~ B * and 

acting in a direction which makes an angle (a + Ite) with the 




X 

Fig 8a Equiangular spiral representing velocity and direction of wind at any height 

geostrophic wind. B has the same meaning as previously. An 
analysis of this method has been given by Brunt,f on the assumption 
that the coefficient K is constant and that the geostrophic wind is 
the same at all levels. In a note added to the paper, Brunt also 
deals with the case where K vanes inversely as the height or in- 
versely as the square of the height. The problem of K varying as 
a linear function of the height has been considered by S. Takaya J 
in a paper e< On the coefficient of eddy- viscosity in the lower atmo- 
sphere ". The solution enables the components of the wind to be 
calculated. The relations are equivalent to those found by Taylor 
(see Note i). It must not be concluded, though the mathematical 
analysis appears to indicate it, that whenever a test is made on the 
wind that the results will produce an equiangular spiral. The 
gustmess of the wind prevents this, so that only when the mean 
of a large number of ascents is dealt with may one expect the wind 
values to form the equiangular spiral 

* " Die Reibung in der Atmosphare " Veroff d Geophys Inst d Umv Leipzig, 
Heft 10, 1915 -\Q J Roy. Meteor Soc , 46, 1920, p 175 

J Memoirs of the Imperial Marine Observatory, Kobe, Japan, Vol IV, No. i, 1930 



WIND STRUCTURE 



279 



The next region to be considered stretches from the surface to 
leight of approximately 8000 m. 

Observations with pilot balloons indicate that the geostrophic 
ocity is -reached on an average below 500 m., while the direction 
lot attained until about 800 m. above the ground.* Each quadrant 
)ws its own peculiarities, however. Thus Dobsonf finds that for 
rth-east winds the gradient velocity is reached at 915 m., for 
ith-east below 300 m , for south-west about 500 m., and for 



22 1912 

29 1912 

13 1913 

5 1913 

5 1913 

21 1912 

26 1913 




7 5 

O +5 +10 +15 +20 +25 

VELOCITY IN METRES PER SECOND 

Fig 9 W to E Component of Wind Velocity on East Coast 



+30 



+35 



th-west below 300 m Also the winds in the north-east and 
ith-east quadrants show little or no increase after reaching the 
(Strophic value, those in the north-east often showing a decrease, 
lie those in south-west and north-west quadrants are marked by 
ontmual increase beyond the geostrophic value, the velocity in 

north-west being at 2500 m., 145 per cent of this value. The 
iation in direction also differs according to the quadrant. In 

north-east quadrant, even at 2500 m., Dobson finds that the 
ection is 6 short of the direction of the isobars On the other 
id, m the south-east quadrant the direction of the isobars is 
ched at 600 m , and above this level the wind veers still farther, 
e south-west winds behave somewhat similarly, only the gradient 

* For theoietical tieatment, see Note II, p 2850 
fp.y R Met *$to,40,p 123(1914). 



28o THE MECHANICAL PROPERTIES OF FLUIDS 



direction is not attained until 800 m. is reached, while in the north- 
west quadrant the direction follows the isobars at 600 m. In this 
last quadrant, however, no further veer occurs until 1200 m. is 
reached, when a further veer begins. On the average the deviation 
of the surface wind from the gradient decreases from north-east to 
north-west, passing clockwise, Dobson's mean values being 27, 
24, 19, and 11 respectively. 

These results refer to an inland station. When we come to 




-20 




5 2 1 34 7 
-15 -1O -5 +O +5 +10 

VELOCITY IN METRES PER SECOND 
Fig. 10 S to N Component of Wind Velocity on East Coast 



+ 15 



deal with a station on the coast we find even greater complications. 
Figs. 9 and 10, which represent the component velocities of a number 
of observations canied out with the aid of two theodolites at Aber- 
deen by the author,* serve to show the irregularities of these veloci- 
ties. A greater variation is shown in the west-east component than 
in the south-north, as is to be expected from the exposure of the 
station. On the whole, the west-east component shows a tendency 
to increase with height, while any east-west velocity gradually dies 
out. The south-north diagram gives mainly negative values, 

*Q.J. R. Met. Soe., 41, p. 133 



WIND STRUCTURE 281 

e. the winds observed had mainly a north-south component. This 
)mponent changes comparatively little in the first 4000 m., though 
igher up there is a tendency to increase indicated both for south- 
orth and north-south winds. This is in general agreement with 
ic results arrived at by Dobson. 

Cave, in his Structure of the Atmosphere in Clear Weather, has 
iven the results of observations carried out at Ditcham Park, 
/hen we consider his icsults for heights between 2500 m. and 
500 m., we find that there is a decided increase with height in 
ic westerly components, the easterly components tending to die 
nt. On the other hand, both southerly and northerly components 
low an increase, the range at 7500 m being much greater than 

2500 m., the actual values being from 20 m per second to 
-20 m. per second at the higher level, to 6 m. per second to 
-9 m. per second at the lower. 

As the result of investigation, Cave has divided his soundings 
i the troposphere * into five different groups, and has added a sixth 
>r winds in the stratosphere f These are: 

(a) i. " Solid " current; little change in velocity or direction. 

2. No current up to great heights 
(&) Considerable increase in velocity. 

(c) Decrease of velocity m the upper layers. 

(d) Reveisals 01 gieat changes m direction 

(e} Upper wind blowing outward fiom centres of low pressure: 

frequently reversals at a lowei layer. 
(/) Winds m the stratosphere. 

In gioup (a) the giadient diiection and velocity are reached 
irly, and thereafter the wind remains nearly constant. There is 
Tactically no temperature gradient, and the pressure distribution 
t different heights is similar to that at the surface. 

Group (b) is mainly due to a westerly or south-westerly type, 
md represents the average conditions where depressions are passing 
istwards over the British Isles. There is here a marked tempera- 
ire gradient over the area. 

In group (c) are included mainly easterly winds, the pressure 

* Troposphere, i e the part of the atmosphere in which the temperature falls off 
ith inci easing altitude In latitude 55 it extends from the surface up to about 
Km , m the tropics it extends to about 17 Km. 

f Stratosphere, i.e the external layer of the atmosphere in which theie is no 
>nvection It lies on the top of the troposphere, and the height of its base above 
ie suiface varies from equator to poles (see troposphere). The temperature 
langes within it are m a horizontal direction. 



282 THE MECHANICAL PROPERTIES OF FLUIDS 

distribution showing an anticyclone to the north. The gradient 
velocity is reached at about 500 m., the gradient direction at a point 
a little higher. Thereafter decrease in velocity takes place and 
occasionally a backing of the wind, though the latter does not invari- 
ably occur. 

With " reversals ", which are placed in group (d), the surface 
wind is almost always easterly; the upper, westerly or south-westerly 

Here we have a warm current passing over a colder, and the 
result is generally rain. Very often in summer theie is found 
a south-west current passing over a south-east, the two being 
associated with shallow thunderstorm depressions, the south-west 
current supplying the moisture to form the cumulo-nimbus clouds. 

From an examination of the winds in group (e), it is almost 
always found that the depressions from which the winds come 
advance in the direction of the upper air current. This is par- 
ticularly the case with north-westerly upper winds. With south- 
westerly upper winds we have very often conditions similar to 
those mentioned under (J), with corresponding results. 

Observations within the stratosphere are comparatively few, but, 
m general, they show that the wind within this region tends to fall 
off with increase in height, and that the direction is almost mvaiiably 
from some point on the west side of the north-south line. 

Several models, which show at a glance how the air currents 
change with height, have been constructed by Cave For a desciip- 
tion of these and a full account of his investigation the reader is 
referred to his book already mentioned. 

Let us now examine the wind structure in these upper legions 
of the atmosphere from the theoretical standpoint. 

We have already noted that the variations in the distribution of 
pressure in the upper atmosphere are closely correlated with the 
variations in the temperature distribution. Starting with the 
ordinary equation for the diminution of pressure with height and 
combining it with the characteristic equation for a permanent gas, 
we are able to find an equation giving the variation of pressure 
gradient with height. These equations are: 



and p/T = R/>, 

the latter giving - = -^ - - .................. (9) 

P P T 



WIND STRUCTURE 283 

Iso if the horizontal pressure and temperature gradients be written 

o . <5np 

3 JL = s, and -= = <?, respectively, then we have 
ox ox 

ds d z p 
9# dxdz* 
e. from equation (8) 

ds dp , N 

= -^ r .................. ( I0 ) 

9# ox 



'herefore combining (9) and (10) we have for the change with 
eight in pressure gradient, 



= _ 

9* ~" SP \p dx T dx 



To find numerical values we must substitute for p its value 
/RT. For dry air R = 2-869 X io 6 C.G S. units, while for air 
iturated with water vapour at 273 a its value is, 2 876 X io 6 C.G S. 
nits This differs only slightly from the value for dry air Also 
le uncertainties which arise in connection with the determination 
f the wind velocity in the upper air are greater than the variations 
i R, and therefore the value for dry air may be used on all occasions 
ithout any appreciable eiror With this value, and with g as 
8 1 cm /sec. 2 , we have 



f now we express the variation in pressure in millibars per metre 
f height and take the giadients in pressure and temperature over 
oo Km., the rate of increase of pressure gradient per metre of 
eight in millibars per 100 Km. is 

3M.Xxo-.Jg-J>, 

and S being expressed in millibars, T and Q in degrees absolute 

The variation in pressure gradient depends therefore on the 
ifference of the quantities Q/T and S/P. Now T falls from about 
8o<7 at the surface to approximately 220<z at 9 Km., whereas P 
banges trom 1010 millibars to nearly 300 millibars within the 



284 THE MECHANICAL PROPERTIES OF FLUIDS 

same lange We see, therefore, that S/P runs tlnough a consider- 
able range of values, while Q/T remains comparatively constant, 
The variation in pressure difference is therefore not constant within 
the region considered, but is likely to show positive values at first, 
then change through zero to negative values higher up. If the 
pressure difference remained constant up to 9 Km., then V/> would 
be constant, and the velocity of the wind would increase in inverse 
proportion to the density of the air as we ascend. Now Egnell 
believed that he found by observation of clouds that Vp actually 
was constant, and in consequence this law, Vp = a constant, has 
been termed Egnell's Law. We have seen, however, that the 
observations by pilot balloons do not confirm the law. The wind 
very often shows an increase in velocity with increase in height, 
especially winds with a westerly component, but this increase is 
generally less, even in the latter case, than in accordance with a 
uniform gradient. Equation (u) is theiefore much more m agiee- 
ment with the behaviour of the actual winds than a constant piessure 
gradient would be. 

The variation of wind with height can now be obtained by 
combining equation (u) with the relation 

s = zvpio sm<, or, vp = s/zco sin</> (12) 



Let -v be the component of the wind velocity paiallel to thejy-axis 
drawn towards the north, the x-axis being drawn towards the east. 
Then 

dv 

p s-* 

i.e. - 1 ^ 4 

v dz 

Q]^ _ . _ , _ I __ O __!_ 2. ' 

s dx p dz 
9T\ / 1 dp __ 8T 
dx/ \p dz T 8^ 

_- ^ SP i 1 8T 
s \p ~ 



s dv 
v dz 



8p 


ds, . , 




- iy_L = 


~ /2co siiiu) 




8^ 


3^ 




i 8p 


i ds 




p 8^ 


s dz' 




i 8w 


i ds i dp 




v dz 


s dz p dz 






_gP( ty__ 


i 




s \p dx 


T 



WIND STRUCTURE - 28 



and as s/v = 

. dv i J _ $p _, p 9T 

" 9* " 



|_^ + /Z\ 

I 9^ J^J 

r9* 9T dp dT} , . 

\i ^r - / -*-} ' - (13) 

I nv* /i*y H*y /Sv I 

l^t/i^/ i/iia C/<^3 t/i-v J 



>ng the tf-axis the corresponding value will be 

du i (dp 9T dp 8T 

_ . j _ , _ _ *!, , _ 



re the negative sign must be used because if the piessure 
rease towards the north then the wind will be from the east. 
If then the wind be observed at vanous levels it is possible from 
se equations to calculate the separation of the isobars and iso- 
rms at the different levels. For this purpose it is necessary to 
>w the values of p and T for each level considered. In any parti- 
ar case the normal values of these quantities for the month in 
ich the observation takes place may be taken without any very 
LOUS error. We may then proceed to calculate the separation of 
isobars and isotherms at intervals of a kilometre m the following 

f 

The change of piessure difference we already expressed in the 

m 

d =- 343 Xio-{!-f}inCGS units 

'hen the pressure be expressed in millibars and the temperature 
degrees absolute, the change of pressure-difference per kilometre 
icight may be written as 

A P/AT AP\ , . 

A* = 34'^r(-^r - -p-J, ........... (14) 

sre AP and AT are the horizontal changes per 100 Km m 
ssure and temperature respectively. The wind velocity W due 
this pressure difference AP can be found from the relation 

R T T 

W = K i AP = K^AP . ... (15) 
P P v ' 



U and V be the components of W from west to east and 
n south to north respectively, then the components of pressure 



2 8b THE MECHANICAL PROPERTIES OF FLUIDS 

difference at any level as deduced from the wind observations ai 

i P " 

ANP== KT U 



and A W P = 

Similarly the components of temperature difference can fc 
expressed from equation (14) in the form 



r* i^ v t j 

and A w = T^-~ X T + A W P) 

Table VI (p. 287) is an example of the application of thes 
equations. Of the last four columns the first two give the separa 
tion in kilometres between the component isobars where th 
difference is I millibar, the second two between the componen 
isotherms where the difference is i C When the duection of th 
resultant isobars and isotherms for the various levels are calculate* 
we find the following directions 

Height in Km 01234 
Isobar from 370 272 271 214 263 

Isotherm from 15 271 210 334 



This appears to indicate the approach of a waimer current fron 
the south-west at a height over 3 Km, The pressure distnbutioi 
at 7 hr. on the zSth gave a depression over Iceland with a smfaa 
temperature of 50 F. The increase of temperature indicated a 
the 3ooo-m. level is apparently due therefore to the warm air fion 
this depression pushing its way across the colder northerly current 
This seems to be in agreement with Bjerknes' theory* of the circu 
lation of air within a cyclone. 

The observations we have been considering hitherto refer to on< 
station only, so that we have obtained only a very small section of the 
isobars and isotherms for the different levels. If a number of obser- 
vations be made simultaneously at different stations over the Bntisl: 

*Q.y. Roy. Met, Soc , 46, p. 119. 



WIND STRUCTURE 



Tt- 

M 

ON 



M 10 
ui CO 

+ I 



CO 

o 



o 

10 
M 



ON 
O 
ON 



Pi. 



10 ^O 

1O CO 



ON 

VO 



CO ON 
N O 



O 

> * 



w 2 
w I 



ex, 



O CO 

o o 



ON cO 

O H 



+ 4 



ON M 
O O 



fin 



O 

o 

+ 



ON 

o 



O vp 00 
O M N 



-0 00 

o o 

+ + 



xf H 



oj C 

H 2 



H OO 
ON OO 



o r^. 

O H 



O ON 

r^. cb 



wg 



288 THE MECHANICAL PROPERTIES OF FLUIDS 

Isles, say, then a series of maps may be drawn showing the air flo 1 

at each level. These will afford an indication of the distiibution c 

pressure and temperature at the vai lous levels. In fig. 1 1 the piessm 

distribution at the surface at 18 hr. on yth September, 1922, is give 

in (a). The following four members of the series show approximate! 

the run of the isobars at the levels indicated as deduced fiom pile 

observations made at 17 hr., while the last of the series depicts th 

pressure distribution at the surface at 7 hr. on the following morning 

The separation of the isobars is 2 millibars m eveiy case. At th 

suiface the isobars run from north-east to south-west, but higher u 

the direction changes towards a noith to south direction. Thi 

appears to indicate a mass of rather waimer air towaids the wes 

01 south-west, especially about the 6ooo-Jt. level. The velocities 

however, are compaiatively small and therefore a break-up of th 

system is not to be expected. Instead, as the 7 hr. chart of th 

following morning shows, there has taken place a fmthei development 

and the direction of the isobars at the suiface has now become mucl 

moie in accordance with the uppei air isobais of the pievious evening 

We must now consider the case of curved isobars. In th< 

expression foi the gradient wind determined neat the beginning o 

om suivey there were found to be two paits, one dependent upor 

the rotation of the earth, the other on the cuivaluie of the path 

Hitherto we have dealt only with the fiist part, but now we shal 

consider buefly the eflect of the cuivatuie ol the path upon the 

relation of the wind to the distiibution oi piessurc. In discussing 

the cii dilation of air m temperate latitudes, Shaw ai rives at the 

following conclusion * " Thus out of the kaleidoscopic featuies oi 

the circulation of an in temp ei ate latitudes two definite states soit 

themselves, each having its own stability. The fust rcpicsents air 

moving like a poition of a belt lound an axis thiough the earth's 

centre. It is dependent upon the earth's spin, and the gcostiophic 

component of the giadicnt is the important feature; the cuivatuie of 

the isobais is of small importance. The second repiesents air lotat- 

mg round a point not very far away: it is dependent upon the local 

spin, and the curvature of the isobars with the corresponding cyclo- 

strophic component of the giadient is the dominant consideration." 

Up to the present we have been considering only one point in 

the path of the air, and the lines of flow of the air at that point we 

have regarded as coincident with the isobars m the upper air and 

making a definite angle with them near the surface owing to the 

* Manual of Meteorology, Part IV, p. 236: 



WIND STRUCTURE 



289 



3,000 FEET / *\ 




1O.OOO FEET 



Fig ii. Map showing the Pressure Distribution and Wind Direction at 
the Surface at 18 hr on 7th September (a), at 7 hr on 8th September (/), 
and (b to e) the app-oximate Direction of the Isobais at 1000 ft , 3000 ft , 
6000 ft , and 10,000 ft , as deduced from Pilot Balloon Observations at 
17 hr. on 7th September, 1933. 

(D312) 



11 



290 THE MECHANICAL PROPERTIES OF FLUIDS 

tuibulence in the atmosphere. When we come to consider a sue 
cession of states, however, we see that the paths of the air are no 
necessarily coincident with the lines of flow or the latter with th< 
isobars. In the case of the first state mentioned above by Shaw 
the isobars are straight and the paths of the air are coincident witl 
the lines of flow; but when the two states are superposed and a series 
of maps drawn giving the pressure distribution at definite intervals 
it is seen that the paths of the air are no longer coincident with the 
lines of flow or with the 
isobars. What then are the 
paths of air m a cyclone ? 
A partial solution of 
the problem may be 
reached after the follow- 
ing manner. It is a well- 
known fact of experience 
that one of the character- 
istics of a cyclone is that 
it travels across the map, 
and when the isobars are 
circular that the velocity 
of translation is rapid. 
We shall here confine 
ourselves theiefore to the 
examination of a circular, 
rapidly moving storm, 
termed the " normal " cyclone or cartwheel depression. 

If two horizontal plane sections of a normal cyclone be taken, 
including between them a thin lamina or disc of rotating air, and 
if this disc travel unchanged in a horizontal direction, then to obtain 
the actual velocities of any point on the disc the velocity of trans- 
lation must be combined with the velocity of rotation. If the 
velocity of translation be V and the angular velocity n, then the centre 
of instantaneous rotation will be distant from the centre of the 
disc a distance V/w. This centre of instantaneous rotation will 
travel in a line parallel to the line of motion of the centre of the 
disc The actual paths of the air particles are traced out by points 
attached to a circle which rolls along the line of instantaneous 
centres and whose radius = V/n Fig. 12 represents these trajec- 
tories. The figure shows, in one position, the circle of radius V/w, 
its centre 0, and the instantaneous centre O'. The circle rolls on 




I<ig 12 Trajectories of Air for a Normal Cyclone 
(from Shaw's Manual of Mai o o'oi>y) 



WIND STRUCTURE 291 

line through O' perpendicular to OO'. The path of a particle 
a cusp, a loop, or neither, according as the tracing point is on, 
lout, or within the circle. 

From this we see that in the normal cyclone there are two centres, 
one, O (fig. 12), the actual centre of the rotating disc which is 
led the tornado centre, the other, O', the centre of instantaneous 
tion termed the kinematic centre. They lie on a line perpendi- 
r to the path of the cyclone, and are distant from each other by 
length V/n. 

[n such a system as this the isobars will not coincide exactly 
L the lines of flow. For the present neglecting the variation in 
ude and m density, and also the curvature of the earth's surface, 
shall regard the cyclone as moving along a horizontal plane. 
hat case the system of isobars will be obtained by compounding 
stem of circular isobars embedded in a field of straight isobars. 

centre of the circular system will not coincide with either of 
centres already referred to, but will be at a distance from the 
matic centre = V/(soj sm< + ri) and lie on the line joining the 
matic and tornado centres. This centre has been termed the 
'mic centre, and is the centre of the isobanc system as drawn 

map It is therefore quite easily identified, but one must bear 
imd that it is not the only centre m a normal cyclone 
f we take the centie of the rotating disc as origin, with x and y 

towards the east and north respectively, then for an eastward 
:ity of translation V, the pressure will dimmish uniformly towards 
north at the rate 2pVo> sm</>, i.e. the field of pressure will be 
rented by 



r*' c y 

I dp I 2pVco sm<f)dy t 

J '* Jo 



e p' = pressure at any point, and p' Q = pressure at any point 
te #-axis; 

i.e. p' p f = 2/>Vte> sin^y ---- . .(16) 

'or the circular field with its centre at the origin we have 

n^ -f- v z p cota/R)^r, 



f* f r 

I dp = I 

J P, Jo 



e p is the pressure at any point distant r from the origin, and 
e pressure at the origin. 



292 THE MECHANICAL PROPERTIES OF FLUIDS 

If we neglect the curvature of the earth, then v z cotct/R = v z l 

Also v = rn. 

/P /-r 

dp = pn I (20) sin< -f- n) rdr, 
P Jo 

2 2 

By combining the two equations (16) and (17), we have for tl 
resultant field 



P P = ( 2 o) sm< + n) (x z + y z ) 2/>Vco siny.. .(18) 

2* 

This represents a circular field of pressure round a point 

__ 2o) sin<^V 

n(2") sm<jk + w)' 

and P is the pressuie at the centie of this field and not at the origi 
Now the distance of the kinematic centre from the tornac 
centre which was chosen as oiigm is equal to V/n Theiefore tl 
distance of the kinematic centie from the dynamic centre is 

TT/ 20) sind> V TT / / ; i \ 

V/ _ . __ .[ x - == V/ (2cu sin^ + ). 

20) sm.<f> -\- n n 

We see, therefore, that this combination of a field of straight isobai 
with a circular system embedded in it is sufficient to give the fid 
of picssure necessary to keep the disc lotating. 

In the normal cyclone it follows that the centie of low piessui 
being not the centre of the lines of flow, the wind possesses a defini 
counter-clockwise velocity at the centre of low pressure. When 
actual example of a lapidly moving circular storm such as that 
loth-nth September, 1903, is examined, we find that the syste 
actually does possess such a wind, and consequently this diveige 
wind, which has often been regarded as accidental, is in reality 
perfect agreement with the pressure system. Another feature whi< 
the normal cyclone possesses in common with an actual circul 
cyclone is the greater incurvature in the rear of the cyclone as cor 
pared with that in front. 

" From these considerations," says Shaw,* " we are led to acce 
the conclusion to be drawn from the conditions of the normal cyclon 
* Manual of Meteorology, Pait IV, p. 343 



WIND STRUCTURE 



293 



mely, that the wind calculated from the gradient by the full formula 
mg the curvature of the isobars, gives the true wind in the free air 
t at the point at which the gradient is taken but at a point distant 
m it along a line at right angles to the path and on the left of it by 
3 amount V/(2,zo sin^ + )." 
The calculated trajectories in the case of the normal cyclone 




K 



Fig 13. Trajectories of Air in Circular Storm, leth to i ith September, 1903 

ve already been referred to and given in fig. 12. A comparison 
these with the actual trajectories for the storm of xoth-iith 
ptember, 1903 (fig. 13), shows at once the remarkable similarity 
tween the two sets, indicating still further that this actual cyclone 
i the normal cyclone are very close akin the one to the other. 
te trajectories of the September cyclone are repioduced from 
e Life History of Surface Air Currents* 

Here we have considered in a very fragmentary way only one 
m of stable rotation, namely the circular. Even then no account 
s taken of the discontinuity of velocity which must occur at the 

* Life History of Surface Air Currents, by W. N. Shaw and R. G. K. Lempfert, 
D., No. 174, London, 1906. 



294 THE MECHANICAL PROPERTIES OF FLUIDS 

edge of the rotating disc in the normal cyclone, but it is possible 
show that this discontinuity can be accommodated by mchidn 
in the revolving column of air an outer region represented by tl 
law of the simple vortex with vr constant. The regions beyor 
what have hitherto been included in the revolving disc will al; 
form part of the cyclone, therefore, and not simply belong to tl 
environment. 

For further treatment of this subject the reader is referred i 
treatises on dynamical meteorology, such as Shaw's Manual < 
Meteorology, as in this brief study some of the intricacies only of tl 
problems of wind structure, rather than their solutions, have bee 
placed before him. 

NOTE I 

The equations of motion of air over the surface of the eart 
when the effect of eddy viscosity is taken into account aie, whe 
the steady state has been reached and the motion is honzonta 

dhi 
<j) + K ................... (0 

ax* 



i j /-i j i v , \ 

o = + 2o>M snip ~ 2o>G sm<p + /c-~ ....... (2) 

CIS* 

On eliminating u from the above we find that the equation fo 
v becomes 

d*v . 4eo 2 sinV> 

4- ^ v = o, 

<fe* ^ K Z 

d*v , -DA i, T> 2 <*> sin< 

or - + 4B% == o, where B a = r . 



Now v does not become infinite for infinite values of z, and th 
solution of the equation is therefore 

v A 2 e~^ smBx -f A 4 e~ Ba cosB^ .......... (3) 

On differentiating this value of v twice with respect to z, an< 
substituting in (2), we get 

u = G + A 2 e~ B;s cosEs 1 A 4 e~ B * smite ........ (4) 



Now G = value of the gradient wind velocity, 
and therefore for great heights u = G, i.e. the gradient wind velocity 
and v =s o. 



WIND STRUCTURE 295 

The values of A 2 , A 4 are found by imposing suitable boundary 
iditions. 
Now at # = o these are 



r&Afe-i r*/ 

L u J*=o L *> 



*=o 



ere a = angle between the observed wind and the gradient 
id. 
From these conditions 

. _ tana(i -f- tana).-, A tana(i tana)^ 

A 2 , 9 . ^; A 4 . o , ^ 

tan 2 a + i tan^a -f- i 

The surface wind W = */[> 2 -f z; 2 ] 2==0 

= s/V"+ (A 2 + G) 2 

C 1 

= . v tan 2 a(i tana) 2 -f- (i tana) s 
I + tan 2 a v / ' \ / 

.^(i tana) , , 

= G^ ---- ' = G(cos a srna) 
seca 

>r W/G = (cosa sma). 

NOTE II 

The theory of eddy motion also accounts for the observed fact 
t the magnitude of the gradient wind is reached at a level lower 
n that at which the gradient direction is attained 

The height at which the gradient direction is reached is found 
m equation (3) of Note I by putting v = o If Hj. be the 
ght, then 

o = A 2 smBHi f A 4 cosBH 1} 

A 
1 therefore tanBH = -* 

A 2 

Substitute the values of A 4 and A 2 already found, and we obtain 



tanBIL = tan (a 

I -f- tana \ 4 



296 THE MECHANICAL PROPERTIES OF FLUIDS 



7T 



Since a is positive and less than -, the smallest value of H x is 
got from 4 



BH t = 5 
4 



The height H 3 , at which the value of gradient velocity is 
reached, is given by n z + fl 2 = G 2 , which, on substitution, 
becomes 

-BI-I,, _ (i + tana) cosBH 2 (i tana) sinBH 2 . . 

6 . . 12) 

tana v 

From this equation BH 2 can be found in terms of tana 
The following table, given by Taylor, shows the values of BH T , 
BH 2> and Hi/Kg as a goes from o to 45. 

a. EHj BH 2 Ili/IIa 



o 


235 


078 


30 


20 


2-70 


1-04 


26 


45 


3 IS 


144 


2'2 



Now for Salisbury Plain Dobson found that the deviation a 

was, in a laige number of cases, 20; also that i for this devia- 

800 metres ,, 2 

tion was = 2 66. 

300 metres 

This is in good agreement with the theoietical value 2-6, 



(D312) II* 



CHAPTER IX 

ubmarine Signalling and the Transmission 
of Sound through Water 

Although practically every other branch of science has had 
siderable technical application, that of acoustics has until the 

few years remained practically in the academic stage, and few 
n among scientific men gave seiious attention to it Bells, 
gs, whistles, sirens, and musical instruments have indeed been 
d from remote times both for enjoyment and foi signalling pm- 
es, but their development has mainly been on cnipnical lines, 
h but little assistance from the physicist. 
The Great War has, however, bi ought about a sinking change 
this as in many other directions, and acoustics is now becoming 

only an important branch of technology, but shows signs even 
developing into the engineering stage and giving us a new and 
verful method of powei transmission, to judge by the pioneci 
k of M. Constantinesco, who has already developed it for the 
ration of lock drills and riveting machines, and shown how it 
f be applied to motors and other machines. Few blanches of 
:nce now offer such possibilities to the inventor. 
Acoustic signalling is of especial impoitancc m connection with 
igation, as sound is the only foim of eneigy which can be tians- 
ted through water without great loss by absorption. The lela- 
sly high electrical conductivity of water renders it almost opaque 
light and to electromagnetic waves. 

The present article deals principally with acoustic signalling 
ier water, but certain allied problems, such as sound langing, 
ith sounding, and other applications to navigation, will also be 
;fly referied to. 

As is well known, sound consists of a vibratory disturbance of 
latenal medium, such as a gas, solid, or liquid, and its phenomena 

208 



SUBMARINE SIGNALLING 299 

e almost of a purely mechanical nature. When a bell or tuning 
>rk is struck it is thrown into vibration, and as any part moves 
rward it compresses the medium in front of it and also gives it 
forward velocity. As the vibration reverses so that the move- 
icnt is in the opposite direction, the mass or inertia of the medium 
jeps it moving forward, and a partial vacuum or rarefaction is 
oduced, until the vibration again reverses and forms a fresh 
unpression. These regions of compression and rarefaction there- 
re travel forward as a series of pulses or waves away from the 
urce in much the same way as ripples are formed on the surface 
a pond when a stone is dropped into it. In the case of the 
rface ripples, however, the leal motion of the water is partly up 
id down, or transverse to the direction of movement of the waves, 
iereas in the case of sound the motion of each particle of the 
edmm is mainly forwards and backwards along the line of propa- 
tion of the sound. Sound vibrations are therefore spoken of as 
igitudinal or in the direction of transmission, which differentiates 
em from all other kinds of vibrations, such as those of ordinary 
ives, light, or electromagnetic waves, which are said to be trans- 
rse. It at once follows from this that although many of the 
entific principles of optics can be and indeed have been success- 
ly applied to sound, there can be nothing in acoustics correspond- 
l to polarization in light. This point is made clcai at the outset, 
the phenomena of light are fairly generally known and are most 
tnulating to acoustic development. 

Fundamental Scientific Principles 

In older to understand the operation of modern acoustic trans- 
ttmg and receiving instruments properly, it will be well to start 
h a brief statement of certain scientific principles and definitions, 
ne of these are well known, but others requiie a few words of 
ilanation. 

Sounds are divided into musical notes and noises, and musical 
es are spoken of as differentiated by intensity, pitch, and timbre 
quality. A musical note is produced whenever the vibrations are 
i legular character, so that each wave is similar to the previous 
. The intensity or loudness of the note depends on the strengtL 
implitude of the vibration, its pitch on the number of vibrations 
second or frequency, and its timbre or quality on the form 
the vibration. The purest musical note is given by uniform 



300 



THE MECHANICAL PROPERTIES OF FLUIDS 



vibrations of a simple harmonic character, and if the wave-form 
is saw-toothed or shows any other variation from the sine form, 
the note is more or less piercing in quality, due to the presence of 
overtones or higher harmonics besides the fundamental pure tone. 
Noises are differentiated from musical tones by having no 
regular character, and are made up of a number of vibrations of 
different intensity and pitch. Speech may be described as a noise 
from the point of view of acoustic transmission and reception, on 
account of the variable nature of the vibrations; and also the sound 
from machinery, ships, &c. This is a serious difficulty as regards 
the detection and recognition of such sounds, as nearly all trans- 
mitting and receiving devices are more or less " selective " in char- 
acter, i.e. they respond better to certain definite frequencies and 
are relatively insensitive to others. Everyone knows that telephones 
or gramophones reproduce certain sounds better than others, and 
acoustic signalling, like wireless transmission, is far moie effective 
with " tuned " devices, which are, however, very insensitive to 
other frequencies. 

Velocity of Propagation 

An accurate knowledge of the velocity of propagation of sound 
is of great importance in connection with acoustic signalling, espe- 
cially as regards determination of range or position as in sound 
ranging. The velocity is very diffeient in different substances, as 
it depends on the elasticity and density of the matenal, and thcre- 
foie on its composition, pressure, and temperatuic We are here 
concerned chiefly with the velocities in air and in sea water, although 
the acoustic properties of other substances require consideiation 
when the transmitting and receiving devices aie being dealt with. 

For air at temperature f C. the velocity v = 1087 + i-Sit ft. 
per second. 

For sea water v = 4756 + i3'8* O'i2t z ft per second, accoid- 
mg to the latest determination of Dr. A. B. Wood, for sea 
water having a salinity of 35 parts per thousand; the velocity being 
increased by about 3-7 ft. per second for each additional pait per 
thousand in salinity. This gives a velocity of 1123 ft. per second 
in air, and 4984 ft. per second in normal sea water at a temperature 
of 20 C., so that the velocity in the sea is about four and a half 
times as great as in air. 



SUBMARINE SIGNALLING 301 

Wave -length 

As above mentioned, acoustic waves from a vibrating source 
msist of a number of compressions and rarefactions following 
le another and all travelling with a velocity given above. The 
stance between one compression or one rarefaction and the next 
called the wave-length of the sound, and it is evident that if the 
equency of vibration is n cycles per second there will be a train 

n waves in a distance equal to the velocity, so that the wave- 

m 

ngth A = - . For example, if we take a frequency n of 500 ~- , the 

>rresponding wave-length in air and in sea water respectively 
20 C. will be: 



In air = 2 246 ft., and in sea water _ = g.g68 ft. 

500 T 500 yy 

The gi eater wave-length in water introduces somewhat serious 
fficulties as regards directional transmission and reception, as will 
>peai later 

Transmission of Sound through Various 
Substances 

As was mst shown by Newton, the velocity of sound in any sub- 
ance can be calculated fiom a knowledge of its elasticity of volume 
id its density. If K is the elasticity and p the density, it is easy to 

IK 
ove that the velocity of propagation of sound v = A/-. It must 

> 

' lemembered, however, that with the rapid vibrations of audible 
unds the heating and cooling resulting from compression and 
refaction have no time to die away, and we must therefore take 
e adiabatic elasticity instead of the constant temperature or iso- 
eimal elasticity in the above formula. For gases the isothermal 
isticity is equal to the pressure, or about io 8 dynes per square 
ntimetre in the case of ordinary atmospheric pressure, and the 
labatic elasticity of air is 1-41 times this amount, while the density 
air p = 0-00129 8 m< P er cu bi c centimetre, so that the velocity 

= A/ 4 = 33,000 cm. per second, or about 1085 ft. per 

" 0-00129 

cond, agreeing closely with the value obtained by direct experiment. 
The mathematical theory also enables us to calculate the amount 



302 THE MECHANICAL PROPERTIES OF FLUIDS 

of acoustic powei transmitted by sinusoidal waves, and the pheno- 
mena which lesult when the sound passes from one medium mti 
another matters of considerable importance in connection wit] 
submarine signalling. It can be shown that the relation betweej 
the pressure P due to vibiation (i e. the alternating excess ovei th 
mean pressure), and the velocity V of a moving particle, at an 
point of the medium in the case of a plane wave of laige aiea com 
pared with the wave length, is given by the relation P = RV 
where R = v /c/o. This iclation being analogous to Ohm's Law i] 
Electiicity, the quantity R has been called by Bullie the " acousti 
resistance " * of the medium. The powei transmitted (w) per uni 
area of the wave front is^P raax V max , i e. P 2 max /zR, 01 RV 2 max /2 
For a plane wave sinusoidal distuibance of frequency n peiiod 
per second, and writing co for 27772, we have V mvv = a>a, wher 
a is the amplitude of the displacement, so that w = axiP max / 
= Rw 2 # 2 /2 ergs pei square centimetre pei second 

For ordinary sea watei in which K 2-2 X io 10 dynes pei squar 
centimetre, and p 1-028, R = v itp = 14 X io 4 , so that foi 
frequency of 500 ~ and a displacement of o i mm., the powe 
transmitted would be 7 watts per squaie ccntimctie 

When sound passes from one medium into anothei, it can b 
shown that unless the two media have the same acoustic icsistanc 
there will be a certain amount of reflection at the interface II 
is the ratio of the acoustic lesistance of the second medium t 
that in the first, and the wave fronts aie paiallel to the inteifac 
which is large in compaiison with the wave length, 

Pa , ^ P/ = ^ Va = -L-V,, and V/ = ;;-!v, 

\vheie P t is the pressuie and V x the velocity in the oiigmal wave 
P 2 > V 2 liansmittcd 

PI V/ reflected 

If the second medium is highly resistant compared with th 
fiist, so that r is very large, P 2 = aPj, P/ = P : , V 2 = o, and V/ ~ V 3 
so that the pressuie at the interface is double that in the origins 
wave and the velocity, being equal to (V x V/), is zero, since th 
movements in the direct and reflected waves aie equal and ii 

*H. BnlliS, Le G&ue Civil, z-^id and 3oth August, 2:919; "Modem Mann 
Problems in War and Peace", nth Kelvin lecluto to Institution of Electnci 
Engineers, by Dr. C V Drysdale. your. Inst Elect. Eng., 58, No. 293, Jub 
1933, PP 591-3. 



SUBMARINE SIGNALLING 303 

iposite directions. The wave is therefore totally reflected back in 
e first medium and there is no transmission. 

On the other hand, if the second medium has a very small 
oustic resistance compaied with the first, so that r is very small, 

P 2 = o, P/ = -P 15 V 2 = 2V 1} and V/ = - V x . 

this case the total piessure (P 1 -[- P/) at the interface is zero, 
d the velocity (Vj V/) is aVj, so that the velocity is doubled, 
it there is again no transmitted wave since P 2 = o, and the wave 
totally reflected with a leveisal of the velocity Vj. In the first 
se the surface is called a " fixed " end, and m the second a " free " 
id. 

If, finally, the two media have the same acoustic resistance, so 
at r i, 

P 2 = P 1? P/ = o, V 2 = V lf and V/ = o, 

d the wave passes on without any reflection This is the ideal 
ndition to be secured m transmitters and receivers 

When the two acoustic icsistances are not equal, it is easily shown 
at the ratio of the eneigy in the tiansmitted wave to that of the 
igmal wave, which we may call the efficiency of transmission 

(\ 9 
T ' I \ 
) . Now 
r+ij 

i watei we have seen that Rj = 14 X io 4 , and for air R 2 = 40 

2e also table on p. 292), so that r = ~ 2 = 2 86 X io~ 4 , and 

l\i 

= ^ oooii, so that only a little ovei o-i per cent of 

(r -|- i) 2 

e eneigy is tiansmitted This at once illustrates the difficulty in 
[ underwater listening, as the sound passing through the water must 
nerally pass into the air before falling on the drum of the ear. 

Again, the value ol R foi steel is about 395 X io 4 , so that on 
issing from water to steel r = 28 approximately, and the efficiency 

transmission is about 13 per cent, while from steel to air it is 
ily o 004 per cent. Hence for sound to pass from water thiough 
e side of a ship to the air inside, the efficiency would be only 
, per cent of 0-004 P er cent > or 0-00052 per cent, were it not for the 
ct that the plates of a ship are sufficiently thin to act as diaphragm, 
id thus allow a greater transmission than if they were very thick. 
L any case, however, the loss of energy is extremely great, and this 
is led to the practice of mounting inboard listening devices, either 



304 THE MECHANICAL PROPERTIES OF FLUIDS 

directly on the sides of the ship or in tanks of water in contact with 
the hull, as will be described later. 

The following table of the acoustic properties of various media 
has been given by Bnllie\ 



Medium 



Steel . 
Cast iron 
Brass . . 
Bronze 
Lead 

("Teak 
Wood I Fir 
i I Beech 

Water 

Rubber . . 
Air .. 



Vaiue of K 
(kg per sq mm ) 



2 X io 4 
o 95 X io 4 
065 X io 4 
o 32 X io 4 
o 06 X io 4 
0-16 X io 4 
o 09 X io 4 

06 X io 4 

2 X I0 a 

Below i (vanable ac- 
cording to the nature 
of the rubber) 

1 40 X io~ a 



Value of o 
(COS) 


Value of V - 
p 
(velocity metres 
per second) 


Values of 
R = V/cp in 
C G S units. 


7-8 


CJIOO 


395 X"io* 


70 


3680 


258 X io 4 


84 


2780 


234 X io 4 


88 


1910 


168 X 10* 


114 


735 


82 5 X io< 


086 


4300 


37 X io* 


o 45 


4470 


20 X 10* 


08 


2740 


22 X 10* 


I 


1410 


14 X io 4 


>i (appioxi- 
rnately) 


j Below 100 


/ Below 
I i X io 4 


o 0013 


328 


o 004 x io 1 



It should be noticed that the values of R for pine or beech wooc 
are not greatly different fiom that for water, so that sound shoulc 
pass from water to wood or vice veisa without great icflection 



Pressure and Displacement Receivers 

From what has been said concerning the theory of acoustic 
transmission, it is evident that sound may be detected eithci by the 
variations of pressure in the medium or by the displacements thej 
produce, in the same way as the existence of an electncal supply 
may be detected by the electiical pressure or by the cunem 
it produces. Acoustic receivers may thercfoie be classed as pies 
sure receivers, analogous to electrical voltmeteis, and displacemen 
leceivers corresponding to ammeters; but this classification is no 
a rigidly scientific one, as a receiver cannot be operated by picssuu 
or by displacement alone. We have seen that the power per uni 
area of wave front is |P max V max- , so that unless the receiver makes 
use of both the pressure and velocity of displacement it receives 
no energy and can give no indication. A perfect pressure receive: 
would, in fact, constitute a fixed-end reflector, and a perfect dis 
placement receiver a free-end reflector, in both of which cases w< 
have seen no -energy is transmitted. 



SUBMARINE SIGNALLING 305 

The distinction between pressure and displacement receivers is, 
>wever, a useful one, just like that between a voltmeter and am- 
eter. A voltmeter is predominantly an electrical pressure-mea- 
ning device although it takes a small current, and an ammeter a 
irrent-measuring device although it requires a small P.D across 
3 coils. Similarly a pressure receiver is one in which the dia- 
iragm is comparatively i igid and yields very little to the vibiations, 
hile a displacement receivei is one with a very yielding diaphragm, 
he distinction is of importance directly we consider directional 
ceivers, as the piessme in a uniform medium is the same in all 
rections while the displacements are in the line of propagation, 
i that a pressure receiver will give no difference of intensity on 
ling rotated into diffeient diiections if it is so small that it does 
3t distoit the waves, wheieas a displacement receiver will give 
maximum when facing the source 

As regaids sensitiveness, however, it is evident that the best 
suits should be obtained when the receiver absorbs the whole 

the eneigy which falls upon it, which will only be the case when 
given alternating pressme on the diaphragm pioduces the same 
splacement as it does in the medium, so that the eneigy is 
impletely tiansmittcd into the leccivmg device without reflection, 
his will only be the case if the diaphicigm is in icsonance with 
e vibrations and the icceiving mechanism absoibs so much 
lergy as to give cutical damping 

In the case of a piopeily designed receivei which is small 

compaiison with the wave length, it will draw off energy 
om a giedtcr aiea of the wave fiont than its own area, just as 
wnelcss aenal may absoib eneigy fiom a fanly large region 
ound it. 

The piactical constiuction of undei water receivei s and hydio- 
loncs will be dealt with Litei , but it will be well at this point to 
ve some idea ol their essential features. The simplest form of 
ich a leceiver, which is analogous to the simple trumpet for air 
ception, is what is called the Broca tube, which consists of a 
ngth of metal tube with a diaphragm over its lower end. When 
is is clipped into the water, the sound from the water is com- 
unicatcd through the diaphragm to the air inside the tube, and 
e observer listens at the fiee end This is moderately effective, 
it not veiy sensitive or convenient, as it makes no provision for 
apHfymg the sound, and it is not easy to listen through long bent 
bes, so that the observer must generally listen only a few feet 



3o6 THE MECHANICAL PROPERTIES OF FLUIDS 

above the water. Modern hydrophones are therefore neaily all of 
an electrical character, containing microphones or magnetophones 
from which electrical connections are taken to ordinary telephone 
receivers at the listening point. 

Microphones are generally used, as they are more sensitive, and 
there are two types of microphone which conespond approximately 
to pressure or displacement receivers lespectively. 
The former is termed the " solid back " type, in which 
a number of carbon granules are enclosed between a 
metal or carbon plate forming or attached to a dia- 
phragm and a solid fixed block of carbon at the 
back. If piessure is applied to the diaphiagm it 
compresses the granules and in ci eases their con- 
ductivity, so that a greater cunent passes fiom a 
battery through the microphone and the icceiveis 
and reproduces the sound through the piessme 
variations. In the " button " type of microphone, 
on the other hand, the carbon gianules aic en- 
closed in a light metallic box or capsule coveicd 
by a small diaphragm, and the whole aiiangemcrit 
is mounted on a largci diaphragm, so that ifs 
vibrations move the capsule as a whole and shake 
Fig i Non-dnec. U P tne granules, with only such changes of pies- 
tiond Hydrophone sure as resu it f 10 m the inertia of the capsule 
In this case it is the motion 01 displacement of the 
diaphragm which produces the vanations of icsistance in the 
microphone. 

The commonest type of simple hydrophone is diagrammati- 
cally shown in fig. i and illustrated m fig 18 It consists simply 
of a heavy circular metal case of disc form with a hollow space 
covered by a metal diaphragm to the centre of which a button 
microphone is attached. It is fairly sensitive but has no dnectional 
properties. 

Directional Transmission and Reception 

The problems of directional transmission and reception aie 
among the most important as regards acoustic transmission. As 
in the case of wireless telegraphy or telephony, acoustic transmis- 
sion suffers greatly from the difficulty that sound, like wireless waves, 
tends to ladiate more or less uniformly in all directions, with the 



SUBMARINE SIGNALLING 



307 



SOURCE 



isult that its intensity rapidly diminishes according to the inverse 

]uaie law, and there is great difficulty as regards interference and 

r ant of secrecy. Again, as icgards reception, it is of comparatively 

ttle value to have a sensitive receiver which will detect the existence 

F a source of sound at a great distance if it gives no indication of 

le direction or position of the source. On this account the ques- 

on of directional transmission and reception is of at least equal 

npoitance to that of power- 

il transmitters and sensitive 

iceivers. This question of 

irectional transmission and 

sception has received a large 

nount of attention. 
The Binaural Method 

f Directional Listening. 

- Our own ears form a 

51 y efficient directional ic- 

iiving system When a 

idden noise occms we 

istinctively tuin to wauls 

le source, and if we are 

iindfolded we can gene- 

lly tell with consideiable 

cuiacy the dnection liom 

Inch a sound comes. This 
due to the fact that, as 

ir two eais aic on op- 

)site sides of the head 

id about 6 in. apait, the 
und leaches one ear a 

tie soonei than the other, 

iless it anses fiom a point in a plane perpendicular to the 

ic joining the ears, i.e. clnectly in front of, behind, or above our 

k ad. Our eais are exceedingly sensitive to this minute differ- 
ce of time, and as this interval depends upon the direction, 
tting larger the more the source is on either side, we learn to 

timate the direction fairly closely, provided our two ears are nearly 
ually sensitive. This is known as the binaural (two-ear) method 
estimating direction, and it has been developed both for air and 
bmarinc listening For example, if we take two trumpets fixed 
a horizontal bar (fig. a), each of which is provided with a definite 




I- O 



,'/*-* 



Fig a Dmauial Listening with Tuimpcts 



3 o8 THE MECHANICAL PROPERTIES OF FLUIDS 

length of rubber tubing to an ear piece, we can detect and locate 
an aeroplane with considerable accuracy from the noise of its 
engines, as the trumpets magnify the sound, and the sensitiveness 
to direction may be increased by increasing the distance between 
the trumpets. When a sound is heard, the obseiver swings the bar 
carrying the trumpets round in the diiection indicated, and as he 
does so the sound appears to cross over from one ear to the other 
behind his head. The position at which this occurs is called that 
of binauial balance, and when this balance is obtained the bai is 
at right angles to the diiection of the source. 

The same principle can obviously be applied to undei water 
listening with two receivers, but in this case it should be noted 
that, as the velocity of sound in water is about foui and a half times 
that in air, the distance between the receivers must be increased in 
that propoition to obtain the same difference of time, and theiefoie 
equal binaural discrimination As this involves the use of a some- 
what long bar, which is troublesome to turn under water, iccouise 
is generally had to what is called a binaural compensatoi for detei- 
mining the direction. 

Retuining to our pair of trumpets in fig. 2, suppose that the 
source of sound is to the right of the median plane, and that the 
tubes from' the trumpets, instead of being of equal length, are of 
different lengths, so that the additional length of tube to the light- 
hand trumpet is equal to the extra distance fiom the source to the 
left-hand tiumpet. In this case it is evident that the delay of the 
sound in reaching the left-hand tiumpet is balanced by the cxtia 
delay between the right-hand tiumpet and the ear, and that binaural 
balance will be obtained although the source is on one side ot the 
median plane. It is therefore possible to obtain the dnection of 
a source with a fixed bar carrying the leceivers, piovided that airange- 
ments can be made for varying the length of the stethoscope or ear 
tubes, and such an arrangement is called a binauial compensatoi, 
the most simple form of which is shown in fig. 3. Here the two 
equal tubes fiom the trumpets are brought to the two ends of a 
long straight tube, which is, however, made of three sections, the 
middle one sliding in the two end portions. The middle tube is 
blocked at its centie, and is provided with two apertures from 
which two equal rubber tubes are taken to the ear pieces. When 
the centre section of tube is in its middle position the two lengths 
of air path from the trumpets to the ear pieces are the same, and 
binaural balance will therefore only be obtained when the source is 




\ Ml UK \\ ( '<)iMI'l Ns \ I ( )|v i ( Ik I )IM ( I l( >S \l I Is I I MM 



HLOl K L-AMHIf D 
BYUPI'FR F'LATh 




TRUMPET \^ IU U> Xv ^ TRUMPET 

EARPIECES 

FIG. 4($). PRINCIPLE ot AMIKICVN Gt)Mi'i NSVIOK 



SUBMARINE SIGNALLING 



39 




bsmd 



he median plane; but if the source is to the right of this plane, 
that the sound reaches the right-hand trumpet first, sliding the 
tre tube to the left increases the path from the right-hand 
upet and diminishes that from the left-hand one, so that balance 
be restored, and an index on the sliding tube will read off the 
action on a suitably engraved scale which can be divided from 

zd 

relation 2 d = b sin0, or sin0 = , where b is the distance between 

b 

trumpets, d the displacement of the central tube from its mid 
ition, and 6 the angle 
obliquity of the direc- 
i of the source from 

median plane. 
In order to carry out 
jctional listening on 
se lines with the 
atest convenience, a 
ular form of compen- 
)i has been designed 
he United States and 
de by the Automatic 
cphone Company 
4 shows the ex- 
lal appeal ancc oi this 
apensator, and fig. 46 

essential feature oi its constiuction Two concentnc 
;ular giooves aie cut m a fixed plate, and are covered by a 
te which conveits them practically into circular tubes This 
te can be rotated above the fixed plate, and is piovidcd with 
) projections which close the grooves but connect the inner and 
er ones togelhei by two cross channels. The sound from the 
} trumpets, entering the two ends of the outer groove, travels 
ind this groove to the stop and then passes thiough the 
mnels to the inner grooves, returning to its two ends, to which 
; ear pieces are connected. It is evident that as the upper 
te is turned the difference of path between the two systems 
altered by" four times the distance through which the stop 
vels, and a pointer on the top plate indicates the direction on 
dial. 

This binaural principle is of such importance that it has been 
scribed at length, and many applications of it will be seen later, 




Fig 1 binaural Method with Rectilineal Compensator 



3io THE MECHANICAL PROPERTIES OF FLUIDS 

but there are other methods of directional reception which may 
first be referred to. 

Sum -and -difference Method. In the case of electrical 
receivers the binaural method may be replaced by what is called 
the sum- and- difference method. Suppose, m fig. 5, that our two 
trumpets on the bar are replaced by two similar ordinary microphone 
receivers M. I and M a , and that these receiveis are connected to two 
telephone transformers Tj and T 2 , the secondaries of which can be 




M, 




SUM-AND-DIFFERE.NCE METHOD 

Fig. 5 

connected in series as shown. It is evident that if the source is in 
the median plane, so that the sound reaches both receivers simul- 
taneously, they should be similaily affected and produce equal 
electromotive forces in the transformer secondaries. If these 
secondaries are connected so as to assist one another, a loud sound 
should be heard, but if one of them is reversed by the switch s 
the two electromotive forces should be equal and opposite, arid 
silence should result. But if this is the case, and the source moves 
to one or other side of the median plane, the sound will reach the 
two receivers at different times, and cancellation should no longer 
take place, so that the source should appear louder the greater the 
angle of reception from the median plane. By swinging the bar 



SUBMARINE SIGNALLING 311 

til the sound vanishes, or at least becomes a minimum, the direc- 
n of the source is given just as in the binaural method, and in 
s position the sound will be a maximum when the transformers 
ust one another. 

This sum-and- difference method has the advantage over the 
laural method that it does not depend on the binaural sensitive- 
3s of the observer, which may be very poor, especially in the case 
pel sons with partial deafness in one ear; and on this account 
ne observeis prefer it On the other hand, it reintroduces the 
jectionable featuie of swinging the bar unless a cornpensatoi is 
reduced between two sets of icceiveis which intioduces undesir- 
le complication. But in any case this sum-and-difleience method 
of great value in connection with electrical receivers, as it brings 
t a difficulty which has to be overcome before such receivers can 

used ior binaural listening It will be noticed that for silence 
be obtained with the difteience connection the sound must affect 
th receivers equally, but this is very larely the case with oidmary 
crophones, owing to difleicnccs in the propeities of their clia- 
ragms. In fact, if two such icceiveis aic placed close together 
as to receive the same sound, it is not uncommon to find very 
le difference between the sound heaid with the sum-and-difleience 
mections, and in this case such leceivers arc quite useless for 
lauial listening, which depends upon perfect similarity of icsponse. 

replacing the oidmaiy metal 01 caibon diaphragms by i ubber 
mbranes, howevei, much greater equality can be secured, and the 
n-and-difference method can be used in the test loom to test 
s equality and to select peifectly paued receivers either for bmauial 
for sum-and-diffeience diiection finding 

Directional Receivers. It has alicady been pointed out 
t although the picssure changes m an acoustic beam have no 
ection, the displacements take place in the direction of propaga- 
a, and that a displacement receiver should therefoie have direc- 
lal properties. This principle has not actually been employed 

directional listening to any extent, but Mr. B. S. Smith has 
ised a displacement receiver consisting of a small hollow sphere 
ttaining a magnetophone transmitter, the whole arrangement 
ng of neutral buoyancy. Such a sphere vibrates as if it were 
t of the water, and consequently gives maximum effect on the 
gnetophone when its axis is in the direction of propagation and 
o when it is perpendicular to it. 
The type of directional receiver which has been most employed 



Sis THE MECHANICAL PROPERTIES OF FLUIDS 

in practice, however, is of a balanced type, as shown in figs 6 and 7, 
It is similar to the non-duectional hydrophone 
(fig. i), except that instead of a thick hollow metal 




IJ 

51 



\ 




/ \Q 
/ x 

i. 


i / s 

If \ 


T -TI 

\ 


i, . 

1 



Fig 6 Bi-directional 
Hydrophone 



Fig 7 Polar Curves of Intensity for Bi-directional Hydrophone 




case it has simply a heavy brass ring with a cential diaphragm 
having a hollow boss at its centie in which the 
button microphone is fixed. Obviously if such an 
arrangement is placed so that its plane lies along 
the direction of piopagation, the pressuie falls upon 
both faces of the diaphragm equally and simul- 
taneously and no motion results, so that nothing 
can be heard in this position. When the hydro- 
phone is turned with one of its faces towaids the 
source, however, the back face is screened by the 
ring, and the sound reaches it later and with less 
intensity, so that there is a resultant effect. On 
turning such a hydrophone round, therefore, the 
sound is a minimum when the edge points towards 
the source, and rises to a maximum when turned 
through a right angle, the intensity for various 
angles of turning being shown in the polar diagram 
fig. 7. A similar effect is given by the Morris- 
Sykes directional hydrophone (fig. 8), which has 
two similar diaphragms on its two faces connected 
by a rod at their centres, on which the microphone 

is mounted. As the variations in pressure tend to move the two 





Fig 8 Morns-Sykes 
Hydrophone 



SUBMARINE SIGNALLING 



phragms in opposite directions, no movement of the bar is 
)duced and no sound heaid when the hydrophone is edge on. 

These forms of directional hydrophone are fairly effective, giving 
liily sharp minimum, but they do not entirely fill the requirements 

directionality, as it is evident that minimum is given when either 
*e of the disc points to the source, so that the source may be in 
icr of two diametrically opposite directions. For this reason they 

called bi-directional hydrophones; but it has been found possible 
get over this difficulty and to convert a bi- directional into a uni- 




/ I 




Fig 9 Uni-directional Ilydioj lion and Polar Curv o of Intensity 

'ctional hycliophone, by simply mounting what is called a 
iffle plate " a few inches away fiorn one face, as shown in fig. 9. 
s baffle plate may be made of layers of wood or metal 01 have 
ivity filled with shot m it, so that it tends to shield the sound 
n one face. Such a hydrophone gives the loudest sound when 

unbaffled face is turned towaids the source and the weakest 
ad when it is turned directly away from it, the intensity in 
ous directions being shown by the polar curve, so that there 
ow no ambiguity as to direction, and the device is then called 
m-directional hydrophone. It does not, however, give such 
nite indications of direction as the sharp minima of the bi-direc- 
al form, and it is therefore better to couple a uni-directional 

a bi-directional hydrophone at right angles to one another on 



3 i4 THE MECHANICAL PROPERTIES OF FLUIDS 

the same veitical shaft. When maximum intensity is obseived on 
the former and a minimum on the latter, the diiection of the source 
is definitely given. 

Besides the foregoing methods of directional reception there are 
others, such as those of Professors Mason and Pierce, depending on 
the principle of acoustic integration first enunciated by Piofessor 
A. W. Porter, which leads to the use of large flat surfaces for iccep- 
tion, and the Walser gear in which the sound is brought to a focus 
by a lenticular device, as will be described below. 

As regards directional transmission, it may fiist be mentioned as 
a general principle of all radiation that transmission and icception 
are reciprocal problems, that good receivers make good transmitters, 
and that a directional receiver will make a directional tiansmitter 
with the same distribution of intensity in diflerent directions. For 
example, if, instead of listening by means of two ttumpels coupled 
by equal tubes to the ear, we bring the two tubes to a poweiful 
source of sound so that the sound escapes in an exactly similai 
manner from the two trumpets, an observer in median plane will 
hear this sound very loudly, but as he moves to one 01 othei side 
of this plane the sound will appear lainter. Similarly, by vibrating 
the diaphragm of a uni-dnectional hydrophone sound will be 
emitted chiefly in one direction, and by extension ol this principle 
a beam of sound may be sent in any direction we please 

PRACTICAL UNDERWATER TRANSMITTERS AND 
RECEIVERS 

We can now turn to the actual devices employed loi submarine 
signalling, and they may be described under the headings (a) trans- 
mitters, (b) receivers, and (c) directional devices. 

SUBMARINE TRANSMITTERS OR SOURCFS OF SOUND 

The simplest form of submarine tiansmitter is the submarine 
bell which has been used as an aid to navigation for many years. 
Originally suggested by Mr. Henry Edmunds in 1878, it was not 
until 1898 that it was taken up seriously as a practical navigational 
device by Mr. A. J. Munday and Professor Elisha Gray, who 
formed the Giay Telephone Company in 1899, an d employed a 
bell struck under water with a submerged telephone icceiver. After 
Professor Gray's death in 1901, the work was carried on by Mr 



SUBMARINE SIGNALLING 



iday, who stalled the Submarine Signal Company to take over 
>perations. Various forms of submarine bell were expej imented 
, but the form which was finally adopted is shown in fig. 10, 
consists of a bronze bell, weighing 220 Ib. and having a frequency 
115 ~ in water, which is struck by a hammer generally operated 
ompressed air. A twin hose pipe is used to supply the corn- 
led air and to convey away the exhaust air from the appa- 
, and the strokes are legulated by 
ode valve ", which consists of a 
[ diaphragm actuating the main 
apply to the hammei mechanism 
type of bell is generally used on 
ships, in which case it is simply 
overboard to a depth of 18 to 
., but m the case ol lighthouses 
e electric supply is available an 
ically operated bell ol the same 
is employed, which is hung on 
)od stand about 25 ft. high and 
. spread, standing on the bottom 
ly convenient position up to a 
or so from the lighthouse. In 
:ase the hammer is opeiated by 
Dulai iron armatuic atti acted to 
ectiomagnets on a common yoke, 
ole faces being coveied by coppci 
to prevent sticking by lesidual 
etization A foui-coie cable is 
led, two for supplying the 3! 
of operating cuirent, the othei 
)emg connected to a telephone 

nitter m the mechanism case, which enables the operatoi to 
f the bell is working propeily. The first of these electrically 
ted bells was laid down at Egg Rock, near Boston Haibour, 
d States, and a large number of pneumatically and electrically 
ed bells aie now in service round the British and American 




Fig 10 Submannc Signal Company's 
Licit lileuucally operated type 



316 THE MECHANICAL PROPERTIES OF FLUIDS 
Electromagnetic Transmitters 

On account of the ease of the operation and control, electio- 
magnetic transmitters have been most popular, and they are now 
made up to large sizes transmitting hundreds of watts of acoustic 
power. They may be divided into two classes: (a) continuous, 
and (&) intermittent or impulse transmitters 

(a) Continuous Electromagnetic Transmitters. In all 
these transmitters alternating current is employed, of frequency 
corresponding to the natural vibration frequency of the vibiating 
system, and this current may be used either to energize a laminated 
electromagnet which acts on the diaphragm, or to traverse a coil 
in a powerful steady magnetic field, thus developing an alternating 
force which can be applied to the diaphragm. These two types 
of transmitter may be called the " soft-iron " and the " rnovmg- 
coil " types respectively. In the former type the frequency of the 
note is double that of the alternating current as the diaphragm is 
attracted equally when the current flows in either direction, but in 
the latter type the note frequency is the same as that of the current. 

The soft-iron type of continuous transmitter has been greatly 
developed by the Germans, and fig. n shows one of the most gener- 
ally used types constructed by the Signalgesellschaft of Kiel. The 
diaphragm D is provided with a boss at its centre, to which is fixed 
a casting carrying a laminated E-shaped iron core C nearly in con- 
tact with a similar block of stampings C' above The exciting coil 
encircles the inner pole of these stampings, as in the familiar core 
type of transformer, and produces a powerful attractive force at 
each passage of the current m either direction, so that the ficqucncy 
of variation of the force is double that of the current The upper 
block of stampings is not rigidly fixed, but is coupled to the lower 
block through the agency of four vertical steel tubes T with steel 
rods inside them, the lengths of these rods and tubes being such 
that the natural frequency of their longitudinal vibiations is equal 
to that of the diaphragm. The diaphragm is bolted to a conical 
housing with glands for the introduction of the supply cables. A 
transmitter of this type, having a total weight of about 5 cwt. and 
a diaphragm about 18 in. diameter, gives an acoustic radiation of 
300 to 400 watts, the mechanical efficiency being about 50 per cent. 

A great objection to these moving iron transmitters is their 
inherently low power factor owing to their great inductance, which 
involves a large wattless exciting current. This can, of course, be 




I' K II ( ON I INI 01 S I I I < 1 KOM \(,M n< 
1 K \NSMI I I I R \S ( I )\s I M ( III) h\ I 111 
Sl( N \l ( I si I I S( II \l I Hi Kill 



IMC. 20 SIM.I i DiAi'iimc.M HI-DIKII- 

IIONVl I I\ DROl'llONI (ONMKIID 
INK) I'M DIKIt IIONVI iNSIUl'MINl 
BY AUDI I ION Ol HA! MI Pi \ II , 




Fat i 



D C, EXCITING COIL 



MAGNET 



SUBMARINE SIGNALLING 3x7 

tilled by using a large condenser in parallel or series with the 

iting coils, but this is not a very satisfactory expedient, 

On this account the moving-coil type of transmitter has been 

cured, especially by the 

lericans, and its funda- 

ntal principle is diagram- 

tically shown in fig. 12. 

ie coil of wiie travel sed 

the alternating current is 
ached directly to the dia- 
ragm, and moves in the 
nular field of a powerful 
Dot magnet " excited by 
-ect current. This type 
s relatively little induc- 
ice, and therefore a high 




A MOVING COlt 



DIAPHRAGM 
-I'nnuple of Moving-coil Tinmmittcr 

wer-factor, but its con- 
ruction is mechanically difficult, as the coils of wire do not 
rm a rigid mass and are theieloie liable to cause great damping 
id loss of efficiency. 

This difficulty was very neatly got over by Fessenden in the 
nited States, and the Fessenden 
ansmitter is probably the most, 
ficient and poweiful of all electio- 
iagnetic tiansmitters The pnn- 
ple is exactly the same as above, 
tit, instead of mounting the coil 
irectly on the diaphragm so as to 
love with it, Fessenden employs a 
xcd coil which induces currents in 
copper cylinder by transformer 
stlon, and this copper cylinder is 
ttached to the diaphragm. Fig. 13 
tiows a diagrammatic section of a 
essenden transmitter in which the 
irect- current electromagnet is bi 
iolar and encircles the copper cylin- 
der which is attached to the dia- 

hragm. The alternating current traverses a fixed coil wound on an 
aner iron core, the coil being wound in. two halves in opposite direc- 
ions to correspond with the two poles of the magnet, and this coil 




-Diagram of Feaaenclon 
Transmitter 



3 i8 THE MECHANICAL PROPERTIES OF FLUIDS 

induces powerful currents in the copper cylinder which traverse the 
strong field of the magnet and impart longitudinal forces to it of the 
same frequency as that of the alternating current. The arrangemenl 
is therefore very rigid mechanically, and a high power-factor and 
efficiency are obtained at resonance, which is usually for a frequency 
of 500 ~. Transmitters of this type giving an acoustic radiation 
of 500 watts or more have been constructed, and are capable oj 
signalling under water to a distance of 300 miles or theieabout 

Moise signals can be senl 
by either of the above 
types of transmitter b) 
the aid of a suitable sig- 
nalling key. 

(b) Intermittent 01 
Impulse Transmit- 
ters. Reference has al- 
ready been made to the 

" submarine bell, which 

\-: j was the fiist type oi 

intermittent submaimc 
transmitter and which 
'' can be opeiated electro- 

J '/ magnetically A more 

'/^//////////l simple type of impulse 

/ / A// // // / //jr ,, , 

| transmitter is the cha- 

Fig 14 Diaphragm Sounder phl'agm SOimdcr of Ml 

B. S Smith, which has 

the advantage over the bell in that the staking mechanism is 
totally enclosed and therefore does not work m water. Fig. 14 
shows a section of a sounder of this type, which is provided with 
an ordinary steel diaphragm with centre boss against which a cylin- 
drical hammer strikes. This hammer is withdiawn on passing 
direct current through the exciting coil, against the foice of a spnal 
spring, and upon the sudden interruption ol the current the spiin 
causes the hammer to strike the diaphragm with a single sharp blow 
thereupon rebounding and leaving the diaphragm free to vibrate 
A very powerful impulse, though of brief duration, owing to the 
heavy damping of the water, is produced in this way. 

Similar powerful impulse transmitters have been constructed ir 
which the hammer is operated pneumatically by compressed air ai 
a frequency of about a hundred blows per second, and this type oJ 




SUBMARINE SIGNALLING 319 

ismittei can be used for signalling in the Morse code, by means 
a suitable pneumatic key. 

A simple tiansmitter has been specially designed by the 
hor for acoustic depth sounding, the object being to give a 
ies of single impulses to the water without vibration. Here the 
?tic diaphragm is entirely done away with and its place taken 
a square laminated plate, which is attracted to an E-formed 
dnated magnet on passing direct current round an exciting coil 

B 2 

ircling the centre pole. The atti active force is dynes per 

877 

tare centimetre, where B is the magnetic field in gausses, so that 
B = 15,000, the loice is about 14 Kgm. pei square centimetie, 
1 a pole area of 140 sq. cm. gives a total foice of about 2 tons, 
order to impart this foice to the water, the pole faces and plate 
grooved, and india-rubber slnps mseited which are compressed 
the attraction of the magnet On switching on this tiansmitter 
a loo-volt cncuit the current uses compaiatively slowly, owing 
its great inductance, and the plate is giadually drawn up, but on 
Idenly bi caking the cunent the reaction ot the iubbci strips 
>ots the plate suddenly forward with an initial foice of about 
ons, and imparts a single sudden shock like an explosion to the 
ter The use of this transmittci will be explained in connection 
h acoustic depth sounding 

Submarine Sirens 

A number of forms oi submaime siren, in which plates 01 cylin- 
s provided with holes thiough which jets oi water pass when the 
tes 01 cylmdcis aie rotated, have been devised both in this country 
i in Germany, and are extremely powciful By suitably bevelling 
: holes, the watei piessure can, of course, be made to lotate the 
tes, but this is objectionable iiorn the signalling point of view, 
it involves a gradual miming up to speed and a consequent 
iation m the frequency of the note On this account the plate 
cylinder is usually rotated independently at a constant speed by 
electric motor, and signalling is effected by switching on and 
the higlb-pressure water supply. These sirens have not, how- 
:r, come greatly into use, as the electromagnetic transmitters are 
much more convenient, and they will therefore not be described 
detail. 
There are many other forms of acoustic transmitters, but the 



320 THE MECHANICAL PROPERTIES OF FLUIDS 

above are most geneially useful for acoustic signalling or impuls 
transmission. For sound-ranging purposes small explosive charge 
are sometimes employed. 




RECEIVERS OR HYDROPHONES 

The C Tube 

The eaihest and most simple of all subaqueous acoustic icceivers 
as already mentioned was the Broca tube, consisting of a length of meta 

tube with a diaphragm sti etched ovej 
its lower end The Americans have 
improved this form of tube, by le 
placing the diaphragm by a thick- 
walled rubber bulb 01 teat, and have 
called it the C tube (fig 15) from 
Dr. Coohdge, its inventor It is 
fairly sensitive, but the amount oi 
energy communicated to the an 
within the bulb is very small by the 
principle of transmission given above, 
and it sutlers from the inconvenience 
of requiring the obsciver to listen 
at the end of a somewhat shoit 
tube. 

The advantages, as icgaids sen- 
sitiveness and convenience, of em- 
ploying miciophones were also 
appreciated by the Americans who 
enclosed microphones in hollow 
rubber bodies, and a combination 
of three such bodies was often floated 
on a triangular frame and employed 
for binauial listening Fig. 16 
shows a double C tube anangement 
for binaural listening. As has al- 
ready been explained, binaural lis- 
tening on two receivers permits the direction of the source 
to be ascertained, but it is evident that the direction suffers 
from the same ambiguity as in the bi-directional hydrophone, as 
a source symmetrically situated on the other side of the line joining 



i! S 




Fig 15. C Tube 



SUBMARINE SIGNALLING 



321 




r 



two hydiophones would give the same difference of time of 

ral. By using tfcuee hydrophones arranged at the corners of 

equilateral triangle, and 

urallmg on each pair in 

, this ambiguity disap- 

s. The necessity for 

ectly pairing the micro- 

les by the sum-and-dif- 

tice method has been 

idy referred to. 

Magnetophones 

\lthough greatly inferior 
sensitiveness to micro- 
ties, magnetophones have 
e advantages for under- 
T listening, as they are 

from the vagaries of ^_ ^ 

ular microphones and ' 

be more easily paired 
binaurallmg As their 
itiveness can be enhanced 
most any extent by the Q, "J 

Srn Valve amplifiers, Fig 16 Double C Tube Bmaurnl Arrangement 

h cannot be employed 

microphones owing to the grating or " frying " noise pro- 
d by the granules, they can be made equally effective, 
lie Fessenden transmitter described on p. 305 can be used as a 
;rful magnetophone receiver by exciting its magnet and listening 
he coils, which are supplied with alternating current when 
nutting, and it is commonly used as a receiver in signalling, as 
of course, in tune with the note of all such transmitteis This 
> tuning, however, renders it unsuitable for general listening 
oses. 

>ne of the most effective magnetophone devices for inboard 
ing is the " air-drive " magnetophone of Mr, B. S. Smith 
[7). It consists of a massive lead casing (4) fixed to the side of 
lip, carrying a thick mdia-rubber diaphragm (2) in contact with 
/ater. Close behind this diaphragm an ordinary Brown reed- 
telephone receiver (3) is mounted, so that the sound transmitted 

(DS12) , 



322 THE MECHANICAL PROPERTIES OF FLUIDS 

from the water to the air behind it causes the diaphragm and iced 
of the receiver to vibiate and induces cunents m the receiver wind- 
ings ^This type of leceiver connected to a three- valve amplifiei 
and high-resistance telephones gives a fairly faithful reproduction of 
ordmaiy sounds; and if four of these receivers are mounted on the 
hull m positions fore and aft and port and starboard, the screening 
effect of the hull enables the direction of the source to be estimated 
from the relative intensities on the four receiveis a foui-way 
change-over switch being interposed between the receivers and the 
amplifier. Ship noises aie greatly diminished by fixing the lead 



^^^*'^^/^ "'.;%,, 

0| = fxx fvJ U-x^^H^ ~~ 




Fig 17 Air-drive Magnetophone 

ring to the plates with a rubber seating, as the gicat ineitia 
of the lead (4) prevents it from taking up the hull vibiations 
readily. 

Theie are many other forms of receivers, but the above are 
the principal ones which have been used for undei water acoustic 
reception. 

PRACTICAL CONSTRUCTION OF HYDROPHONES 

A few illustrations may now be given of the actual foims of 
some of the most generally used hydrophones. Fig. 18 shows the 
simplest form of non- directional hydrophone, of which a diagiam 
was given in fig. i , in which a heavy hollow bronze casting is pro- 
vided with a diaphragm on one side, to the centre of which a small 
" solid back " microphone is attached. 

Fig. 19 is an illustration of the double- diaphiagm bi-directional 
hydrophone, diagramrnatically shown in Rg. 8, and fig. 20 (see 
plate facing p. 316) shows a single- diaphragm bi-directional hydro- 






Ftitniff pagt )-- 



SUBMARINE SIGNALLING 



323 



ihone converted into a urn-directional instrument by the addition 
if a baffle plate, as m fig. 9. 

In order to be able to listen from a ship in motion and to reduce 
hip and water noises as much as possible, hydrophones, either of 
he rubber-block form or of one of the foregoing types, have been 
nclosed in fish-shaped bodies and towed through the water some 
iistance astern, and combinations of such bodies have been used 
or directional listening by bmaurallmg. The modern tendency, 




Microphone 




Fig 21 Reception by Hydrophone in Tanks 

owever, has been m the direction ol inboard listening, by securing 
Hcient acoustic insulation from the hull. 

The method of listening in tanks inside the hull, fust intio- 
iced by the Submarine Signal Company, has been greatly 
lopted by the Germans. Fig. 21 shows the disposition of a pair 

these tanks with the hydrophones inside. This device avoids 
ie great loss by reflection on passing from water to air, as has 
sen referred to above. 

A remarkably interesting and effective form of directional m- 
>ard listening device, however, is that known as the Walser gear, 
jvised by Lieutenant Walser of the French navy, in which the 



3 2 4 THE MECHANICAL PROPERTIES OF FLUIDS 

sound is brought to a focus, as in a camera obscura, and the direc- 
tion determined by the position of this focus For this purpose a 
"blister", consisting of a steel dome A of spherical curvature and 
about 3 ft. 6 m. diameter, part of which is seen m fig 22, is fitted 
to the hull, and this steel dome is provided with a large number of 
apertures B into which thin steel diaphragms C are inserted. These 




Fig 22 Walser Apparatus 

diaphragms being on the spherical dome collect the sound and 
direct it to a focus at a distance of 5 or 6 ft A trumpet D, to which 
a stethoscope tube is attached, is mounted on an arm E turning on 
a vertical axis, so as to be able to follow the focus and point in the 
direction of the sound from whatever direction it comes. Two of 
these blisters are generally mounted somewhat forward on the two 
sides of the hull, and an observer seated between them applies the 
tubes from the two trumpets to his ears, so that he can follow the 
position of the source on either side, the direction being given on 
a scale when the maximum intensity is obtained. 



SUBMARINE SIGNALLING 325 

DIRECTIONAL DEVICES 

Sound Ranging 

One of the most important acoustic applications in the War 
ras that of sound ranging for the detection of the position both of 
uns and of submarine explosions, the importance of which is 
bvious. There are two chief methods of location, which may be 
escribed as "multiple-station" and "wireless-acoustic" sound 
mgmg respectively, but the former, although less convenient, was 
le only one employed in the War, as it needs no co-opeiation on 
ie part of the sending station. 

Multiple -station Ranging. The multiple-station method 



Fig 23 Sound-ranging Diagram 




/ 
/ 



sound ranging depends on the principle that sound waves are 

tit out as spheres with centre at the source of sound. If three 

more receivers are therefore set up on a circle with centre at 

5 source, the sound will arrive at all of them simultaneously, so 

it if the signals are all coincident the source must be at the centre 

the circle passing through the receivers. If, however, the source 

in any other position the signals will be received at different tunes, 

d if the differences of the times of reception are measured the 

sition of the source can be located by calculation, or graphically. 

A simple diagram (fig. 23) will make this method clear. Let 

5 CD be four receivers in any accurately known positions and 

DC the position of an explosion to be located. If we draw a circle 

h P as centie through the receiver A, it is evident that when the 

md arrives at A it still has the distances bE to travel before arriving 



326 THE MECHANICAL PROPERTIES OF FLUIDS 

at B, and cC and dD before arriving at C and D respectively, so that the 

times of arrival at B, C, and D are t, = t , and 2 3 = 

v v v 

behind that at A. Consequently if we can measure the time inter- 
vals /tj, a , and 3 , and multiply them by the velocity, we get the 
perpendicular distances of the station B, C, and D from the circle 
passing through the source, and if we draw circles round B, C, and 
D with radii to scale representing these distances, the centre of a 
circle tangential to these circles will be the position of the source P. 

The method of determining these time differences almost entirely 
employed during the War was by means of a multiple-stringed 
Einthoven galvanometer, four of these strings being connected to 
four microphones or hydrophones, while a fifth was connected to 
an electric clock or tuning fork, so as to give an accurate time scale. 
The image of the strings was focused on a continuous band of 
bromide paper, which was drawn through the camera and a deve- 
loping and fixing bath by means of a motor, so that it emeigecl fiom 
the apparatus ready for washing and drying, though the times could 
be read off instantly it appeared. To facilitate the reading off of 
the time intervals, a wheel with thick and thin spokes was kept 
revolving in front of the source of light by means of a " phonic 
motor " in sychromsm with a tuning fork, so that a number of lines 
were marked across the paper at intervals of hundiedths and tenths 
of a second. 

Fig. 24 is a reproduction of a sound-ranging recoid so obtained, 
on which the times of reception at four receivers are marked, and 
fig 25 a view of the Einthoven camera outfit employed The re- 
ceivers used in this case were simple microphones, mounted on 
diaphragms bolted on watertight cases mounted on tripods lowered 
on the sea bottom and accurately suiveyed, the microphones being 
connected by cables to the obseivmg station 

On account of the importance of sound ranging as a means of 
locating the position of a ship m a fog, efforts have been made to 
improve it still further, and to eliminate the photographic apparatus. 
The greatest achievements in this direction have been made by 
Dr. A. B. Wood and Mr. J. M. Ford at the Admiralty Experimental 
Station, who have devised what they call a < phonic chronometer " 
for indicating the time intervals directly on dials to an accuracy 
within one-thousandth of a second. The principle of the instru- 
ment is very simple, and can readily be understood by reference 
to fig. 26. A phonic motor with vertical spindle revolves with a 





1'K, .25 - KlN I IIOV1 N C\MIK\ I OK SoiNI) R\N(.INK. 




FlG 26-TlIRM< DIAL PHONIC ClIRONOMGIliR 



Facing page 336 



3 28 THE MECHANICAL PROPERTIES OF FLUIDS 

sist in this case of diaphragms with single-point contacts which aj 
thrown off on arrival of the shock, and remain broken until th< 
are restored by electromagnets. Each of these contacts is connecte 
to the electromagnet windings of the dials as shown, and it will I 
seen that as each contact is broken it breaks one of the circuits i 
either one or two of the dial mechanisms, and starts the pomte 
revolving until the breaking of another contact breaks the secorj 
winding and allows the small wheel to fly away from the revolvir 
wheel and against a brake which immediately stops it. After tl 
shock is received at all four hydrophones, therefore, the three dia 
indicate the time intervals between the ai rival at the first hydi( 
phone and that at the other three directly in thousandths of 
second, each thousandth representing a distance of about 5 ft , fro] 
which the graphical diagram shown in fig 23 can be constiucte 
and the position of the source indicated on a chart 

In order to obtain this position as readily as possible the wnl< 
has devised what he calls a sound-ranging locator (fig 28, sc 
plate facing p. 326). It consists of a long steel bar pivote 
at one end on a ball-bearing, the centie of which can be fixe 
on the chart exactly over the position of one of the hydiophonc 
Three thin steel bands are attached to the other end of this bar b 
means of keys, like the strings of a violin, and pass thiough 
slot in a sliding piece to graduated rods sliding through simiL 
ball-bearing swivels, which are fixed on the chait in positior 
corresponding to those of the other three hydrophones. Tl 
graduations on the sliding bars are maikcd in times to the sea, 
of the chart, so that by sliding them to the readings eoirespondin 
to the time differences indicated on the chronometer, each stn 
is lengthened by the amounts bE, cC, and dD in the diagiam fig. 2; 
and when the slotted slider on the main bar is pushed down an 
the bar turned until all the strips aie tight, the point from wluc 
they radiate indicates the position of the source on the chait withoi 
any calculation, and a marking point just under the edge of th 
slider can be depressed to prick the position. In order to secui 
accuracy, each of the strips is provided with a small tension indi 
cator which shows when the strip is strained to a definite tensioi 

Two strips only are shown in fig. 28, but any number ma 
be employed according to the number of receivers. 

A device on a similar principle has been put forward by Mi 
H. Dadounan in the United States.* 

* Physical Review, August, 1919. 




Pacing 



[n using the multiple-station method of sound ranging for 
iting navigation in foggy weather, a ship desirous of being 
rmed of its position calls up the nearest' sound-ranging station, 
:h instructs it to drop a depth chaige As soon as the record 
sceived on the Emthoven camera or phonic chronometer, the 
tion of the ship is worked out or marked by the locator and 
lessed to the ship. 

iYireless- Acoustic Sound Ranging. A method of sound 
;ing which promises to be of much gieatei value foi navigation, 
which has not yet been fully developed as it was of little value in 
time, is the wireless-acoustic method proposed by Professor Joly. 
he original experiment of Collodon and Sturm in 1826, the 
city of sound in watei was determined by striking an under- 
>i bell and igniting a charge oi gunpowder simultaneously 
mowing the distance from the souice and observing the inteival 
ime between the flash and the sound of the bell the velocity 
determined, as light travels piactically instantaneously over any 
naiy distance Convcisely, if the time inteival and the velocity 
known, the distance oi the souice can be at once dctei mined, as 
be familiar method oi ascei taming the distance of a lightning 
i by noting the time between the flash and the tlumdei clap The 
intage oi employing an nuclei watei method is that sound is tians- 
cd moie clTectively thioiigh watei, and that theie aie no watei 
cnts compaiablc with winds to aflect the velocity appieciably 
Jnf 01 tunately a flash of light is of no value in a fog, but wireless 
2s are little affected by it, and travel with the same speed as 
, so that if a wireless flash and an underwater explosion are 
lated simultaneously at a lighthouse or other known position, 
the ship is provided with a wireless equipment and a directional 
ophone, the distance of the station can be at once determined 
he ship by noting the interval between the two impulses. As 
velocity of sound in sea water is neaily a mile a second, the 
ince can be determined within a quarter of a mile by a simple 
-watch, and the direction of the source found by either the 
;tional hydrophone or directional wireless, without any com- 
ication with the station. If the lighthouse or lightship simply 
s out wireless impulses simultaneously with the strokes of the 
narine bell at convenient inteivals, all ships in the vicinity can 
e their positions fiom time to time without delay or mutual 
ference, and if they aie within the range of two such stations 
can do so without any directional apparatus. 

(D312) 12, 



330 THE MECHANICAL PROPERTIES OF FLUIDS 

The lecent developments m duectional wireless have lendered 
the application of sound ranging to navigation of less impoitance, 
but even now wireless direction finding is not always reliable, 
especially at sunrise and sunset; and there is also liability to error 
on steel ships owing to their distorting effect on the wireless waves. 
As hydrophones become increasingly employed on ships for listen- 
ing to submarine bells, &c , the ability to obtain accurate ranges by 
wireless acoustic signals will doubtless prove of great value 

Leader Gear 

Although not stiictly speaking an acoustic device, some mention 
should be made of the leader gear or pilot cables as an aid to navi- 
gation of harbours and channels in foggy weather For this pur- 
pose it is necessary to be able to follow some well-defined tiack with 
a latitude of only a few yards, so that sound langing is inadequate 
But if a submarine cable carrying alternating current of sonic 
frequency, say 500 ~, is laid along the desired track, and the ship 
is provided with search coils with amplifier and telephones, the 
alternating magnetic field pioduced by the cable induces alternating 
electromotive forces in the coils, and thus gives a sound in the tele- 
phones when the ship is sufficiently near the cable By using two 
inclined coils on the two sides of an non or steel ship it is lound 
that the sound is loudest when the telephones aie connected to 
the coil which is neaiei to the cable, so that the ship can be steeied 
along it, and keep a fairly definite distance to one side of it, so that 
vessels passing in opposite diiections will not collide This device, 
which was first put forward by Mr C A. Stephenson of Edinburgh 
in 1893, was icvived duiing the war by Captain J Manson, and 
is now coming into use both in this country and in the United 
States. An i8-mile cable has been laid by the Admnalty from 
Portsmouth Harbour down Spithead and out to sea. 

Acoustic Depth Sounding 

Another purely acoustic device which promises to be of con- 
siderable value to navigation is that of depth sounding by acoustic 
echoes fiom the bottom. If a ship produces an explosion neai 
the surface, the sound travels down to the bottom and is reflected 
back as an echo, and for each second of interval between the ex- 
plosion and the echo the depth will be half the velocity of sound 
or 2500 ft., say 400 fathoms. Various experimenters, notably 



SUBMARINE SIGNALLING 



t[. Marti in Fiance, Herr Behm in Geimany, and Officeis of the 
anerican Navy, have devised apparatus wheieby the time between 
ring a detonator or other small charge under the ship and the 
sception of its echo fiorn the bottom can be recorded on a high- 
peed chronograph, and veiy accurate results have been obtained. 



T 




Uh- 



DETONATOR' 



Fig 29 Behm's Acoustic Depth-sounding Method 

The method of Behm, called the " Echolot " or echo-sounding 
evice, now being developed by the Behm Echolot Co , Kiel, has 
ttained a high degiee of perfection, and is claimed to give inch- 
ations in a ship at full speed, and even in rough weather, to an 
ccuracy of within a foot. The transmitter consists of a tube 
irough which a cartridge is impelled by air picssurc into a holder 
xed on the hull a little above the water line. The caitndgc is 
red out of the holder on pressing the firing-key, and is shot towards 



332 THE MECHANICAL PROPERTIES OF FLUIDS 

the Impulse Receiver, while a time fuse in the cartridge is airanged 
to explode a detonator just before the cartudge reaches the micio- 
phone. Both the Impulse and Echo Receivers are microphones, 
but the latter is scieened from the direct eflect of the detonator 
by being fitted on the opposite side of the ship. 

The explosion of the detonator causes a sudden drop in the 
current through the impulse receiver and weakens the cuiicnt 
passing round an electromagnet, and causes it to release an " im- 
pulse spring " which suddenly starts a pivoted disc in lotation 
with a uniform velocity until the weakening of the cunent thiough 
the brake magnet, due to the echo i caching the echo microphone, 
stops the disc. The angular motion of the disc is theieforc pio- 
portional to the interval between pressing the firing-key arid 
return of the echo, and a light minor on the disc spindle causes a 
spot of light to revolve round a translucent scale divided in depths, 
and to stop at the depth indicated It is claimed that this timing 
device is capable of indicating shoit mteival of time to an accuiacy 
of one-ten-thousandth of a second, corresponding to only 3 in in 
depth Three keys are provided on the indicator, one for restonng 
the indicator to zeio, one for filing the chaige and obtaining the 
depth, and the third for checking the inclicatoi against a standaid 
time interval A number of detonator charges can be stoied in 
the transmitter magazine, and fiied as requiiecl The whole appai- 
atus can be operated by a few dry cells, as the lamp is lit only at 
the moment of restoration, indication, or checking, and the colour 
of the light is varied at each opciation to eliminate nsk oi mistake 
It is stated that a lock with an upper suiface of only 2 sq. meties 
in area is sufficient to give a correct indication 

The Bntish Admnalty have icccntly developed a very simple 
and accurate echo sounding gear. 

Echo Detection of Ships and Obstacles 

By means of leader gear, sound langing, and echo sounding 
navigation in fogs may be made much safei and more regular, but 
there still lemams the gieat danger of collision in the open sea 
between ships, and especially with wrecks, rocks, and icebergs. 
As far as ships are concerned the difficulty is to some extent met 
already by signalling with sirens, but the curious blanketing and 
reflecting or lefi acting effect of fogs is a source of considerable 
confusion and danger. Undei water signalling does away with this 
difficulty almost entirely, and as hydrophone equipments become 



SUBMARINE SIGNALLING 333 

ore common the nsk of collision between moving ships will rapidly 
imimsh. 

With a good directional hydrophone equipment an ordinary 
eamship can easily be detected and its direction determined up 
> a range of some miles merely by the noise of its engines. But 
i the case of wrecks, locks, and icebergs, which emit no sound, 
te danger is still very great, and nothing but an echo method will 
^tect them. Unfortunately this is a difficult matter, as a ship or 
nail rock at a moderate distance is a very small target for an echo, 
that the echo is of very small intensity, and it may quite easily 
j masked by bottom echoes However, Fessenden, by the use of 
s powerful electromagnetic transmitter, succeeded as early as 
)i6 in obtaining echoes from distant obstacles, and by employing 
rectional transmitting and leceivmg devices, which concentrate the 
und in the desired direction, the strength of the echo can be 
creased, disturbances reduced, and the dnection and approximate 
nge of the obstacle determined. As eaily as 1912, just after the 
name disastei, a pioposal to employ echo detection for avoiding 
mlar dangers was put loiwaid by Mr, Lewis Richardson, and it 
ay be hoped that this method will ultimately eliminate the last 

the serious dangers of navigation 

Acousnc TRANSMISSION OF POWLR 

Beloie concluding this article, reference ought to be made to the 
mderful achievements of M Constantmesco, as showing the possi- 
lities of what may be called acoustic engineering. For the pur- 
ges of underwater signalling the power transmitted, although large 
comparison with what we have heretofore contemplated in con- 
ction with sound, raiely exceeds a hundred watts; and it has been 
t for M Constantmesco boldly to envisage the possibility of 
msmittmg large amounts of power by alternating pressures in 
iter of sufficiently high frequency to be described as sound waves. 
>r many yeais it has been customary to illustrate the phenomena 

alternating electric cm rents by hydraulic analogies, and the 
esent writer has even written a book in which such analogies 
ve been used as a means of giving a complete theory of the subject; 
t the obvious possibility of using such alternating piessures in 
ter for practical purposes was entirely missed until M. Con- 
mtmesco conceived it, and immediately the idea occurred it was 
ident that the whole of the theory was ready to hand from the 



334 THE MECHANICAL PROPERTIES OF FLUIDS 

electrical analogies. In a surprisingly short time, therefore, M. 
Constantmesco has been able to devise generators, motois, and 
transformers capable of dealing with large amounts of power trans- 
mitted by hydraulic pipes in the form of acoustic waves of a frequency 
of about 50 '--'. The generator is, of course, simply a high- 
pressure reciprocating valveless pump, and the motor can be of 
similar construction, but by having three pistons with cranks at 
120, three-phase acoustic power can be generated and employed in 
the motors. The first commercial application of M Constantmesco 's 
devices has been to reciprocating rock drills and riveters, for which 
this method is especially suitable, as the reciprocating motion is 
obtained simply from a cylinder and pistol without any valves 
whatever, and the power is transmitted by a special form oi flexible 
hydraulic hose pipe comparable with an electric cable It is not 
too much to say that M. Constantmesco 's ideas have opened up an 
entirely new field of engineering, and their development may have 
far-reaching effects. 

For a discussion of the theory of hydraulic wave transmission 
of power, see Chapter VI. 

Although this article is necessarily very incomplete, it will at 
least have served its purpose of showing the great importance of 
underwater acoustics, and there can be no doubt that a new depait- 
ment of scientific engineering has been opened up which has vast 
possibilities 

Developments in Echo Depth -sounding Gear. 

Since the first appearance of this volume, the chief advance in 
underwater acoustic devices has been in the improvement of echo 
depth-sounding devices which have proved their gieat value for 
navigation and appeal likely in time to become a standard feature 
of ship equipment Three diflcrent types of such gear are now manu- 
factured m this country: the Admiralty type by Messrs H. Hughes 
& Sons; the Langevin piczo-electric type by the Marconi Sounding 
Device Company, and the Fathometer gear, which has been de- 
veloped from the original Fessenden apparatus by the Submarine 
Signal Company. All these devices have now been made to give 
both a visual indication of the depth on a dial and a continuous 
record on a chart. 

The basis of all methods of acoustic depth sounding is the re- 
cording of the time taken for a signal to travel from the ship to the 
bottom of the sea and leturn, but they differ in the type of the signal 



SUBMARINE SIGNALLING 335 

ad method of indication, and may be divided into impulse or 
some " methods and high frequency or " supersonic " methods 
a the formei class to which the Behm " Echolot " (see p. 331), 
le original Admiralty sonic gear, and the Fathometer belong, the 
gnal is in the form of a single powerful impulse provided by an 
splosive cartridge or an electromagnetic or pneumatic hammer 
nkmg a diaphragm; while in the latter a short train of high- 
equency vibrations is emitted from a quartz piezo-electiic oscillator, 
steel rod which vibrates at a high frequency when struck by a 
ammer, or by a magnetostriction oscillator which is the magnetic 
nalogue of the quartz oscillator 

The single impulse or sonic transmitter is practically non- 
irectional, i.e the disturbance tiavcls equally m all directions under 
ic ship. This has the advantage of making the indications practi- 
illy independent of any rolling of the ship, but it has many dis- 
ivantages Firstly, it is liable to give such a severe shock to the 
iceiver at the moment the impulse is sent out that it does not 
;cover in time to icspond to an echo fiom a very shallow bottom, 
icondly, the greater part of the energy is wasted, thndly, the echo 
mst be very strong to be heard above the noises caused by the ship's 
lachmery and motion through the water, and iomthly, il may not 
ive true depths if the bottom is shelving steeply, as the fust echo 
received from the object which is nearest to the ship With the 
igh-frequency method the sound can be concentiated within a 
3ne of any desired angle, so that the receiver can be fanly close to 
xe transmitter without sustaining any severe initial shock, and the 
>ceiver can be sharply tuned to the transmitted fiequency, so that 
is nearly deaf to any othei disturbances If the ship could be kept 
n. a perfectly even keel, the narrower the beam the better, as it 
ould be equivalent to a vertical sounding line, but on account of 
)llmg it is desirable that it should have an angle of something like 
ilf the maximum angle of roll. For a circular transmitter the 

'imangle of the beam 9 is given by the relation sin 9 = i z~, wheic A 

the wave length of the sound and d the diameter of the tians- 
utter, so that we can obtain any beam angle we please by varying 
le diameter and frequency 

As regards receivers, a granular microphone is the most suitable 
ir the single impulse or some system, and it must be mounted at 
>rne distance from the transmitter and preferably on the other side 

the keel, so as to be shielded as much as possible from the initial 



336 THE MECHANICAL PROPERTIES OF FLUIDS 

shock. This separation is however objectionable, as it seriously 
reduces the accuiacy of sounding in very shallow water, wheie it is 
frequently most important. The device, of course, indicates the dis- 
tance from the transmitter to the bottom and back to the receiver, 
and this varies very little when the depth imdei the keel is small 
compared with their separation. With the high-frequency system, 
however, the transmitter and receiver can be close together, so that 
this difficulty does not arise; and as both the quaitz and magneto- 
striction transmitters will also serve as receiveis, it is even possible 
to dispense with a separate receiver, as is done in the Marconi gear 
The essential function of the indicator is, of couise, the measure- 
ment of the time interval between the impulse and echo As the 
average velocity of propagation of sound in sea water is about 4900 
ft. per second, and the sound has to travel the double distance to 
the bottom and back, each second of interval corresponds to a depth 
of 2450 ft. or about 400 fathoms; and if soundings are leqmred within 
an accuracy of one foot, the time must be measured within an ac- 
curacy of four ten thousandths of a second The most simple and 
reliable method of effecting this is by the contact method employed 
in the Admiralty some gear, m which the receiving earphones arc 
shunted by two brushes, which press on a revolving img which has 
a small gap in it, so that the phones are short-cucuited for all but an 
interval of one or two thousandths of a second in each i evolution 
The transmitter is actuated at a certain moment in each i evolution, 
and the two brushes are carried on an arm which can be turned by 
the observer until the short circuit is icmoved simultaneously with 
the arrival of the echo. The depth is then indicated by the position 
of the arm on a scale which can be divided in feet 01 fathoms Ducct 
visual indication is, of course, piefcrablc, and is seemed in the 
Fathometer gear by a revolving disc mounted close behind a giound- 
glass scale The disc has a narrow slot in it, behind which is a small 
neon lamp, and the echo when sufficiently amplified, causes this 
lamp to flash and show a momentary red streak on the scale at each 
revolution. In the Marconi gear the amplified echo is received by an 
oscilloscope or high-frequency galvanometer the beam fiom which 
falls on an oscillating mirror and shows a luminous streak on a giound- 
glass scale. When the echo is received the momentary kick of the 
galvanometer shows as a kink in this luminous streak at the corre- 
sponding depth on the scale. Messrs. Hughes have produced a 
direct-reading pointer indicator for the high ficquency Admiralty 
gear, which operates on the phase indicator principle. A revolving 



SUBMARINE SIGNALLING 337 

lolenoid is fed with direct current and therefore produces a rotating 
nagnetic field, and a soft iron needle is momentarily magnetised by 
he current from the echo receiver, so that it sets itself along the 
ixis of the solenoid at that moment. 

Any of these devices enable the depth to be observed at intervals 
)f every few seconds even when the ship is running at full speed, 
Tvhich is an enormous advantage over the old lead line, which required 
he ship to be running dead slow Merely for ensuring safety in navi- 
gating shallow waters this is sufficient, but a great gain is secured by 
naking the apparatus record the depths continuously on a chart 
vhich gives a profile of the bottom along its course, as this enables 
i ship to locate its position with considerable precision if the con- 




FIR 30 I- Icctrormgnctic Hammer Transmitter 
for Sh illow Water Gear 

our of the bottom is accurately known. During the last few years 
ecorders have come into general use, and have been found very 
atisfactory The motor which actuates the tiansmitter contacts 
md the receiver mechanism is also employed to move a stylus uni- 
ormly across a band of paper which has been previously soaked in a 
ensitive solution (usually starch and potassium iodide, as in the 
arly Bain printing telegraph), and the amplified and rectified echo 
auses it to make a mark on the.paper at the moment it is received 
The paper band is moved slowly forward at a constant rate by the 
ame motor, or it can be driven from an electrical log so as to move 
>roportionately to the distance covered by the ship, and, as the 
tylus makes a mark for each echo, a practically continuous line is 
Irawn on the paper showing the variation of depth either with time 
>r distance. By simple contact devices the stylus can also be made 
o mark the paper at each five or ten feet or fathoms of depth, and 
t regular intervals of time or distance, so that the record is complete, 



33 8 THE MECHANICAL PROPERTIES OF FLUIDS 

and can be reproduced directly m a hydro-graphic atlas. Fig. 31 
shows such a record of a 15 minutes' run, with a shallow water 
magnetostriction set. 

After the above general description, the only features of the 
various gears which require special consideration are the tians- 





Ftg 32 Pneumatic Hamrnu Tiansmittci 

mitters. For the impulse or sonic transmitters the types employed 
in the shallow water Admiralty gear and the Fathometer gear are 
very similar, and the former is shown m fig. 30. A ring of iron stamp- 
ings, with internally projecting poles, is excited by coils on the poles, 
and the hammer consists of a tapered block of stampings, which is 
drawn into the gap between the poles and compresses a spiral spring 
which drives the hammer down against a diaphragm when the 





O> 



Facmg page 33& 



SUBMARINE SIGNALLING 



339 



rent is broken. For the deep water Admiralty gear, which has 
in used for depths of over 2000 fathoms, the hammer is operated 
2umatically with an electromagnetic release (fig. 32). 
The Marconi high-frequency quartz transmitter, which also 
ves as the receiver, shown in figs. 33 (p. 338) and 34, has a thin 
er of quartz crystals H cemented between two steel discs F and G, 




1 IK 34 Marconi Quaitz Ti. inarm tier 

e lower of which is usually m contact with the water while the upper 
highly insulated and connected to a high-voltage oscillator which 
ves a frequency of about 37,500 periods per second, producing 
ives about 4 cm long in the water, and a somewhat sharp beam 
i 01 dei to provide for the removal and replacement of the oscil- 
tor without diy-dockmg the ship, the housing is sometimes pro- 
dcd with a second resonant steel plate shown at the bottom of C 
Inch is clamped by a central flange and transmits the oscillations 
the water.* 

* Cdr. J. A. Slee, C.B.E., R.N., Journal Institution Electrical Engineers, Dec. 1931. 



340 THE MECHANICAL PROPERTIES OF FLUIDS 

Quartz oscillators, although highly efficient, are somewhat 
costly and require special technique in construction, and hence 
efforts have been made to obtain a high-frequency impulse without 
employing crystals. One simple method, which is fairly effective, is 
to employ the ordinary hammer of the single-impulse transmitter, 
but to substitute a steel rod clamped at its centre like the lower 
plate in the Marconi transmitter for the diaphragm. This lod 

i i 



..Toroidal 
winding 



_Thm sheet 
"paper, & cement 




End load 
c erncni eg 
to nickel 

Fig 35 Magnetostriction Scroll-type Oscillator 

vibrates with its resonance frequency and emits a short train of 
damped oscillations each time it is struck 

But within the last few years a great advance has been made by 
employing the principle of magnetostriction, i.e. the defoimation 
which takes place in magnetic materials when they are magnetized. 
This effect is most marked in pure nickel, and m certain nickel and 
cobalt steel alloys, and it enables oscillators of any frequency to be 
constructed very cheaply and by ordmaiy workshop methods It 
lends itself to very various forms of oscillator, but the two which 
have been found most convenient for echo sounding work are the 
" scroll " and " ring " types shown in figs. 35 and 36, In the former, 
a strip of nickel is simply wound up like a scroll of paper, and is 



SUBMARINE SIGNALLING 



34* 



rovided with a simple toroidal winding like a gramme ring armature, 
/hen this winding is supplied with alternating current the scroll 
tpands and contracts axially, so that a disc cemented to one end 
;rves as the emitting surface. The axial length of the scroll is made 
ich that its mechanical lesonance frequency is that required 



, 

Toroidal winding 



\ u 



-# -! d 

Annular nickel 
stampings 




1'ig 36 Magnetobtnction Ring Oscillator 

isually about 15,000 periods per second), and the winding is fed with 
ternatmg current at a low voltage either from a valve oscillator or 
condenser discharging through an inductance, in either case at the 
'sonant frequency In the disc type the oscillator is made up of a 
imber of ring stampings with holes round its inner and outer 
snphery These stampings are cemented together into a solid 
ock with an insulating cement, and a toroidal winding is wound 
irough the holes. When supplied with alternating current the 



342 THE MECHANICAL PROPERTIES OF FLUIDS 

ring expands and contracts radially, and it is operated at its resonance 
frequency. As the sound is emitted m all directions m a plane 
parallel to the ends of the ring, it is surrounded by a sound reflector 
made of two thin metal cones with india-rubber " mousse " between 



v 



\ J 



(a) 



Watertight 
glarm 

Rubber 
mousbc 




Thm metal 




Fig 37 Magnetostriction Oscillatoia and RefkUors 

them, ay shown m fig 37, and the beam angle can be varied by choos 
mg the diameter of this reflector The transmitters will serve equalb 
well as receivers, but it is found preferable to mount two of then 
close together, one acting as transmitter and the other as receiver 
These magnetostriction transmitters and the associated recorder 
were designed by Drs. A. B Wood and F. D. Smith and Mr. J, A 




gg 

5! 



w 

Q 

P 



Facing '$age 342 



SUBMARINE SIGNALLING 



343 



Peachy,* of the Admiralty Research Laboratory, after a research 
magnetostriction by Dr. E P Harrison, and have been mcor- 
ited with great success in the latest forms of Admiralty depth- 



Iransmittmg key Recording stylus 



Depth marker 



Constant -speed 
motor 




Chemical recorder 



HT generator 



_ Rectifier 
Amplifier 



Condenser- 




Transmitting 
+ keyK 




Fig 38 Magnetostriction Geneial Arrangement 

mdmg gear. As they only require very small power for shallow 
)ths without high- voltage oscillators, it has been possible to mstal 
m with recorders in small motor-boats for the hydrographic survey 
shallow rivers and estuaries, which has enormously increased the 
thty and rapidity of such surveys On the other hand, they have 

* Journal Institution of Electrical Engineer*, Vol 76, No 461, May, 1935, p 55 



344 THE MECHANICAL PROPERTIES OF FLUIDS 

proved equally efficient for moderate depths and deep water geai, 
and soundings have been taken successfully in depths of 2000 
fathoms with the transmitters and receivers in water-filled tanks in 
contact with the hull, and transmitting and receiving through the 
ship's plates. Fig 38 is a diagram of the motor-boat outfit with 
chemical recorder 

Figs. 39, 40 and 41 show external views of the Marconi, Fatho- 
meter, and Admiralty indicating and recording sets (Plates facing 
pp. 342, 346). 

Probably over two thousand naval and mercantile vessels have 
by now been equipped with echo depth-sounding gear of one 01 
other of the above types, and the icports on them show their great 
value, accuracy and reliability For moderate depths down to 200 
or 300 fathoms, an accuracy of a foot is obtainable, while the mag- 
netostriction motor-boat gear actually records depths to an accuiacy 
of three inches even when the boat is almost touching bottom The 
deep water set on the Discovery II was so satisfactoiy in Antarctic 
waters that line sounding was discontinued, and such sets have been 
of great service in cable-laying ships by enabling them to lay then 
cables on the least irregular bottoms The importance of continuous 
and recorded soundings for mercantile ships by facilitating their 
eitry to harbours and locating their positions at sea hah aheady been 
referred to, and its value will increase as more and moie iccords 
along the main trade routes become available Lastly, a icmaikable 
application of such gear has been found for fishing, and many ti awki s 
are now being equipped with it, as it has been found that the quantity 
and quality of catches depends greatly on the depth, and echo sound- 
ing gear enables the ship to follow a contour line of the depth de- 
sired. Echo depth-sounding has proved the most useful application 
of underwater acoustics and seems likely to be universally adopted 

The writer is indebted to the three firms above mentioned for 
particulars and illustrations in this section, and to the Journal of the 
Institution of Electrical Engineers for figs. 34, 35, 36, 37, and 38. 




II Hughes & Son, Ltd 

FIG 41 ADMIRALTY SUPERSONIC RECORDER 



Facing page 



CHAPTER X 

The Reaction of the Air to Artillery 
Projectiles 

Introduction 

All calculations of the motion of a projectile through the air are 
cted to one object to determine the position and velocity of the 
|ectile at any given time after projection in any prescribed manner, 
general the reaction of the air to a rotating projectile is very com- 
ated, the complication is considerably reduced, however, if the 
jectile can be made to travel with its axis of symmetry coincident 
i the direction of the motion of its centre of gravity. It is a 
ter of experience that by giving the projectile a suitable spin it 
be made to travel approximately in this manner for considerable 
ances; m such circumstances the reaction of the air is reduced to 
Qgle force, called the drag, which acts along the axis of the pro- 
lie and tends to retard its motion.* When this drag is known for 
roj'ectile of given size and shape the problem enunciated above 
)mes one of particle dynamics, and its solution for that pro- 
lie can be effected, at all events, numerically. The first and major 
of this chapter is devoted to the consideration of this drag. 
When the angle of elevation of the gun is considerable the cur- 
ire ol the trajectory mci eases too rapidly for these simple condi- 
s to hold. The motion then becomes complicated and the prob- 
becomes one of rigid dynamics in three dimensions; the trajectory 
twisted curve instead of a plane one, and the well-known pheno- 
lon of drift appears Similar complications arise when the pro- 
ile is not projected with its axis coincident with the direction of 
ion, or when the spin is insufficient to maintain this coincidence 

A couple of small magnitude due to skin friction also exists, it acts about the 
and tends to reduce the spin; its effect is generally negligible with modern 
ctiles. 

345 



346 THE MECHANICAL PROPERTIES OF FLUIDS 

In the second part of this chapter the component forces and couples 
of the reaction of the air in these circumstances are briefly con- 
sidered. 

THE DRAG 

Early Experiments: the Ballistic Pendulum 

Most early writers on ballistics* assumed that the resistance of the 
air (the drag) to the motion of projectiles was inconsiderable The 
first experimenter to attempt the determination of the air drag on 
projectiles moving at a considerable speed was Robins, who, in 1742, 
carried out experiments with his ballistic pendulum He found that 
the icsistance encountered was abnormally gi eater foi velocities 
greater than about noo ft. per second than for lesser velocities 
Following Robins, many experiments were performed with the 
ballistic pendulum, notably at Woolwich (by Hutton, T775-88) and 
Metz (by Didion, 1839-40), to determine the drag as a function of 
the velocity of the projectile f 

The method employed by Robins was, briefly, as follows: 

A gun was placed at a known distance from a heavy ballistic 
pendulum; the charge was caiefully weighed and the projectile was 
fired horizontally at the pendulum. The latter received the pro- 
jectile in a suitable block of wood, and the angle through which it 
swung was recorded Knowing the weights of the projectile and 
pendulum and the free period of oscillation of the latter, the velocity 
of the projectile at the moment of hitting could be calculated The 
expeiimcnt was repeated with the same charge, the distance between 
the pendulum and gun being varied from round to round. There 
resulted a series ol values of velocity at known distances from the 
gun, the retardation of the piojectile and hence the resistance of the 
air at these distances could be deduced 

By performing similai sets of expeiiments with various weights 
of charge, the drag could be determined as a function of the velocity 
of the projectile. 

The unceitamty of realizing the same muzzle velocity in each 
set of experiments with constant charge vitiated the reliability of the 
results. Hutton overcame this difficulty by hanging the gun hon- 

* The study of the flight of piojectiles 

f For a full account of these expei imenls see Robins, New Principles of Gunnery, 
1761; Hutton, Tracts, 1812, especially Tract XXXIV; Dicuon, Lois de la resistance 
de I'air (Paris, 1857). 



REACTION OF AIR TO ARTILLERY PROJECTILES 347 

itally from a suitable support, so that the gun itself became the 
D of another pendulum From the angle through which this 
tern swung on firing, the muzzle velocity of the projectile could 

calculated. For each round fired he thus obtained two values 
the velocity one at the muzzle, the other at a known distance 
m the muzzle 

Let v and v> 2 be these values, and let x be the distance between 
a and pendulum. Then, if m be the mass of the projectile, the 

Wl / \ 

s of energy m traversing the distance x is f % 2 v^ J. If R 

2 > ' 

the mean value of the drag we therefore have 
r, mf o 2 N \ 

R = <y_ 2 <Q 2 ). 
2X\ l * J 

avided that the distance x is sufficiently small, this value may be 
.en as the actual value of the drag for the velocity v = ^(V L -f- v z ) 
By varying the charge and the distance between the gun and 
ndulum Mutton determined the drag numerically as a function of 
2 velocity 

From the time of Hutton to the present day, experiments con- 
cted on the Continent and in Ameiica to measure the lesistance 

the air have been based on this pimciple, namely, to determine 
? velocity at two points on an approximately horizontal trajectory 
known distance apart A large number of instruments for mea- 
nng the velocity of a projectile at a given point have been invented 
irmg this time; the reader is refeired to Ealistica Experimental 
Aphcada, by Col Negrotto of the Spanish aimy (Madrid, 1920), 
r an up-to-date and exhaustive account of them. It should be 
ited here that few of these chronogiaphs were invented especially 
r the determination of the resistance of the air; there are, of course, 
any important uses for such instruments m gunnery. 

Since 1865 experiments on the resistance of the air conducted 

England have been based on a different principle. The method 
is first proposed by the Rev. F. Bashforth, B.D., sometime Pro- 
ssor of Mathematics at the Artillery College, Woolwich; it consists 

measuring the times at which a projectile passes a number of 
uidistant points along an approximately horizontal trajectory. 
hese times are then smoothed and differenced, and the velocity and 
tardation of the projectile at a number of corresponding points are 
Iculated by the method of finite differences. This method is 



348 THE MECHANICAL PROPERTIES OF FLUIDS 

evidently more economical m expenditure of ammunition than that 
of foreign experimenters. 



The Bashforth Chronograph 

In 1865 Bashforth invented his now-famous electric chronograph,* 
by means of which he succeeded in measuring small intervals of time 
with an accuracy previously unattained in ballistic instiuments. 

The chronograph consists essentially of two electro-magnets, 
to the keepeis ol which two sciibeis aie attached by hnkwoik; these 
scribeis tiace continuous spual lines on paper fixed on a revolving 
cylindei The two spnals aie genet ated by a mechanical movement 
ol the iiameuoik supporting the electro-magnets in a direction 
piiiallel to the axis of the cylmclei The movement of each keeper 
is eonti oiled by a suitable spring, and any small movement of either 
is identified on the iccoid by a kink on the corresponding spiral 
tiace. 

One of the dcetio-magnets is connected with a clock and the 
cm lent is bioken momentarily every second; one of the spiral traces 
thus constitutes a time record. The othei electro-magnet is con- 
nected in seiies with screens placed at equal distances along the 
tiajectory of the projectile. At the moment the latter passes a screen 
the curient is bioken for a shoit inteival of time and a kink is made 
in the co i responding spual tiace. The mechanism by which the 
em tent is bioken is as follows. 

A boaul us suppoitecl in a honzontal position with its length at 
tight angles to the dhection of motion ol the piojcctilc Tiansverse 
grooves aie cut in the board at equal distances, somewhat less than 
the diameter ol the projectile. Haul spimg-wne staples are fixed 
in the board so that each ptong piojects upwards fiom a gioove 

On the near edge of the boaul a number of copper straps are 
fixed; each strap has two oval-shaped holes which are placed at the 
near ends of adjacent grooves, The prongs of the staples are bent 
down into the grooves and project through the oval-shaped holes; 
the arrangement k such that the butts of the staples and the copper 
straps alternate, so that a current may pass continuously through the 
staples and straps. 

The prongs terminate in hooks from which are suspended small 

* For ft full account of Bashforth's experiments consult his Rmsed Account oj 
thu Eptrimmt$ mad$ with the Sashforth Chronograph (Cambridge University Press, 



REACTION OF AIR TO ARTILLERY PROJECTILES 349 





Fig i The Bashforth Chronograph and Screen Reproduced by 
courtesy from "Description of a Chronograph", by F Bashforth 
B D , Proceedings of the Royal Artillery Institution, 5, 1867 



350 THE MECHANICAL PROPERTIES OF FLUIDS 

equal weights by means of fine cotton; the weights rest against a 
second horizontal board supported some distance below the first, 
and aie sufficiently heavy to maintain the prongs in contact with the 
bottom edges of the holes in the stiaps. 

When the projectile passes it will break at least one of the cottons; 
the corresponding prong will spring from the bottom to the top of 
its hole in the copper strap and so break the current momentarily. 
This mechanism constitutes a " screen ". 

The record, when removed from the cylinder and laid flat, con- 
sists of two paiallel straight lines with kinks in them; m the upper 
line the kinks correspond to the passage of the piojectile thiough 
successive screens; in the lower the kinks indicate seconds of time. 
With a suitable measuring appaiatus it is possible to read off the time 
intervals between the screens to four decimal places of a second. 

Bashforth continued his experiments until 1880, and pioduced 
a table giving values of the air drag for velocities up to 2780 ft per 
second. Contempoiary experiments were also conducted m Europe 
by Mayewski (Russia), Krupp (Germany), and Hojel (Holland), giving 
results in substantial agreement with those of Bashforth 

Later Experiments 

In the early years of the piesent century a Luge amount of work 
was done in England, Fiance, and Germany to obtain acrmate infor- 
mation concerning the air diag In 1906 the Oidnance Board used 
a method similai to that of Bashfoith, but having a moic accuuitc 
timing and recording device; the drag tor velocities up to 4000 it 
per second was determined. In 1912 O von Eberhaid, at Kiupp's,* 
made a large number of experiments with projectiles of various shapes 
and sizes. It is thought that these foim the most exhaustive set of 
expeiiments yet undertaken; the results, which aic ficqucntly used 
in this chapter, aie certainly the most complete yet published openly, 

In this method the velocity at two points on the trajectory was 
measured by means of a spark chionograph; the distance between 
the points varied from 50 m. for small projectiles to 3 Km. for those 
of large calibre The method of deducing the resistance was similar 
to that used by Hutton, and results for velocities up to 1300 m, 
per second were obtained. 

* Cf O von Eberhard, Artillenstische Manatshe/te, 69 (Beihn, 1912). 



REACTION OF AIR TO ARTILLERY PROJECTILES 351 



Krupp's 1912 Experiments 

The velocity at a given point is measured, in this method, by 
irmg the projectile through two screens; one screen is placed a 
cieasured distance (a few metres) m front of the point, and the other 
he same distance behind it The spark chronograph measures the 
ime taken by the projectile to traverse the distance between the 
creens; the average velocity between them is deduced and is taken 
3 be the actual velocity at the point. The distance being short 
o appreciable error arises from this assumption. 

Each screen consists of a square wooden frame across which fine 
oppci wire is stretched backwards and forwards continuously in 



r 



To 1st 
Screen- 



I To 2nd 
-j Screen* 



L, 
L, 



To 3rd 
} Screen, 



To 4tk 



Fig 3 The Spark Chronograph used in Krupp's 1913 Experiments 

ich a way that a projectile passing through is ceitam to break the 
ire. In the experiments four screens are used; one pair serves 
> measure the velocity at the beginning of the measuied range, the 
ther, the velocity at the end of it. 

The spark chronograph is shown diagrammatically m fig 2 A 
tetal drum A is rotated at high speed by means oi a suitable motor, 
ie speed being recorded by means of a Frahm tachymeter; readings 
within one revolution per second can be taken with this mstru- 
icnt. It is essential that the speed of the drum be constant during 
ie flight of the projectile over the range, and this instrument serves 
ie additional purpose of indicating the most suitable moment to 
e the gun. The surface of the drum is silvered and is coated with 
>ot except at one edge where a circumferential scale is fixed. 

There are four induction coils, I; their primary circuits, which 
>ntam batteiies, are connected respectively to the four screens; 
ie terminal of each secondary circuit is connected to the spindle of 
LC drum, the other terminals being connected to sharp platinum 



352 THE MECHANICAL PROPERTIES OF FLUIDS 

points, P. A break in one of the primary circuits will cause a spa 
to pass across the small gap between the corresponding point and tl 
drum; the spark is enhanced by a condenser in parallel with tl 
secondary circuit. The mark on the drum made by the spark 
like a bright pin-point and is surrounded by a sort of halo; it is thi 
easily identified. 

The positions of the marks are read by means of a micioscoj 
mounted on the frame supporting the drum; this microscoj 
can be tiaversed parallel to the axis of the drum. To take a leadir 
the drum is rotated by hand until the mark made by a spark is i 
the field of the micioscope; it is then clamped. The mark is the 
brought to the zero line of the eyepiece by means of a fine adjustmen 
The microscope is then traversed to the edge of the dium and tl 
reading is taken from the circumferential scale The positions < 
the marks made by the other sparks are similarly measured Tl 
time intervals between the pairs of spaiks are then deduced wit 
the aid of the tachymeter leading 

The chronograph is cahbiated by bi caking all the pnmai 
circuits simultaneously and recoiding the iclative positions of tl 
marks made by the spaiks. 

With this instrument such a small time interval as 0-0017 se< 
can be measured with a probable erroi of 7 5 X io~ 6 sec., or 04 
per cent. 

Expeiiments prior to the war thus fell into two types the Hutto 
type, in which the velocity of a projectile was mcasuicd at two point 
a known distance apait, and the Bashfoith type, in which the times c 
passing a number of equidistant points along the tiajectoiy wci 
recorded. 

With regard to the fiist type, unavoidable errois in the measuic 
ment of the velocity would vitiate the icsutts if the distance bet wee 
the points were too short; on the other hand the appioximale mctho 
of deducing the resistance as a function of the velocity cannot giv 
satisfactory results unless the distance is short. It would thus appes 
to be a difficult matter to choose suitable distances, and laws of re 
sistance based on methods of this type must be somewhat uncertair 

With regard to the second type, it is evident that, provided tib 
time readings are sufficiently smooth to ensure that differences c 
finite order vanish, the resistance and velocity at correspondin 
points can be deduced to a known degree of accuracy. Wher 
however, the observed times require appreciable alteration to mat 



EACTION OF AIR TO ARTILLERY PROJECTILES 353 

smooth, considerable uncertainty attaches to the results deduced, 
'o both types there is the objection that no account is taken of 
ble yaw * of the projectile. It is well known that all shells have 
yaw on leaving the muzzle of the gun, and it cannot be hoped 
it is always damped out sufficiently before reaching the points 
ilch observations are made. At high velocities very considerable 
nay develop, and, in particular, such obstacles as screens may 
to increase it. In any case the yaw does not remain constant 
g the flight of the projectile, hence, unless it is at all times 
jible, the resulting law of resistance cannot be consistent, 
i any method which depends on observations of a projectile in 
it is therefore necessary to make some provision for observing 
iw as well as the velocity (or time) at points on the trajectory, 
i yaw is small throughout the flight reliable results would be 
led; in cases of considerable yaw a method would have to be 
2d of correcting for it in the analysis of the records before any 
ce could be placed on the deduced values of the resistance 

Cranz's Ballistic Kinematograph 

method in which provision is made for observing the yaw, at 
qualitatively, was devised by C Cianz, j- who carried out ex- 
ents which wcic contemporary with those of Ebertuird, it was, 
/er, applicable only to rifle bullets and to similar projectiles of 
mall calibre 

series of shadow photographs of the bullet was taken by means 
Ballistic Kmematograph at each end of a 20-m range. The 
ty at each end could be deduced from the positions of the 
;sive images on the kinematograph film and the observed speed 
ich the film moved through the camera The occurrence of 
viable yaw could be detected at once from the photographs, 
le reliability of the records for the purpose of the experiments 
thus be estimated. 

The Solenoid Method 

ice the War technique has developed considerably in the 

rement of high velocity. A very successful method, developed 

Frank Smith, of measuring time intervals in experiments of 

he yaw is the angle between the axis of the shell and the duection of motion 
jntre of gravity 

3i tt full account of Cianz's experiments see Artillenstische Monatshefte, 69 
1912). 
12) 13 



354 THE MECHANICAL PROPERTIES OF FLUIDS 

the Bashforth type, consists in firing an axially-magnetized piojectile 
through the centres of a series of equidistant solenoids which are 
connected in a series with a sensitive galvanometer. The current 
induced in each solenoid reaches a maximum as the projectile 
approaches, falls rapidly to a minimum as the projectile passes 
through, and finally returns to its original value as the projectile 
emerges. The " signature " of the galvanometer is recorded photo- 
graphically on a rapidly moving film on which is also recorded the 
oscillations of a tuning-fork of known frequency. 

Experiments with High -velocity Air Stream 

During the war experiments were undertaken in a new direction. 
Instead of making observations on a projectile m flight, the thrust 
on a stationary projectile in a current of air moving at high velocity 
was directly measured.* The method has subsequently undergone 
considerable development m France and Amenca,f and at the 
National Physical Laboratory "[ 

The projectile (or a scale model) is supported by means of a 
thin steel spindle fixed to the centre of the base m prolongation of 
its axis; this spindle is attached at its olhei end to a mechanism 
designed to measure the thrust on the projectile. Compressed air 
issues from a reservoir through a suitable orifice, thus generating a 
high- velocity stream; the projectile is placed m the centie of the 
stream. The determination of the most suitable size and shape of 
the orifice was a matter of considerable difficulty, aftot a large number 
of trials an orifice was obtained which ensured a steady stream m the 
vicinity of the projectile. 

The temperature and velocity of the air in the stream are com- 
puted on the classical theory from the state of the air in the icseivoir 
before the orifice is opened. When the velocity is greater than that 
of sound m air a check on the computed value can be obtained by 
photographing the head wave caused by the projectile (see p. 357) 
and measuring its slope. 

The possibilities of such an experimental method are innumerable. 
Apart from the direct determination of resistance of air at all angles 
of yaw, its sphere of usefulness extends to the elucidation of many 
problems connected with the general reaction of the air on projectiles. 

* Experiments of various kinds on piojectiles had previously been can led out 
in wind channels, but the velocity of the air stream was at most 30 m, per second. 

t See " The Experimental Deteimmation of the Forces on a Piojectile ", by 
G. F Hull, Army Ordnance, Washington, May-June, 1921. 

f See the Annual Reports of the Dnectoi, N.P.L. for igzz and succeeding years. 



REACTION OF AIR TO ARTILLERY PROJECTILES 355 

Considerable progress has been made in experimental ballistics 
nee the war. The time, the yaw, and the orientation of the axis of 
ie projectile at a series of points along a horizontal trajectory can 
e observed with considerable accuracy; results from such experi- 
tents co-ordinated with those of experiments with the high- velocity 
r stream, have placed our knowledge of the reaction of the air on 
projectile upon a very sound foundation. 

The Drag at Zero Yaw 

The resistance of the air to a projectile of given shape, moving 
ith its axis of figure coincident with its direction of motion, depends 
i the following arguments: 

the velocity, v, of the projectile; 
the calibre (i e. diameter), d, 

the characteristic properties of the air, chiefly the density, the 
the elasticity, and the viscosity 

Dimensional considerations lead us to the form 

R = P d z v 2 f(vja, vd/v) . . (i) 

r the resistance R of the air to projectiles of given shape, where p 
the density of the air, a is the velocity of sound in air, an index 
the elasticity, and v is the kinematic viscosity The function 
;/, vd/v) in this expression is called the drag coefficient Since 
2 a 2 has the dimensions of a toice it follows from equation (i) that 
3 drag coefficient has no physical dimensions, its arguments must 
irefore be so chosen that they shall have no dimensions; vja and 
/v both satisfy this condition, and are the simplest arguments in 
ms of which the function can be expressed. 

The Drag at Low Velocities 

For velocities below the critical velocity it is well known that 
/(#/, vd/v) = A.v/vd 

ere A is a constant, the terms in v/a being negligible. This leads 
the expression 

R = Apdvv 

viscous drag. 

For velocities higher than the critical velocity we have a change 

physical conditions, the air behind the projectile breaks up into 



35 6 THE MECHANICAL PROPERTIES OF FLUIDS 

eddies and the linear law of viscous drag no longer holds.* In su 
circumstances the resistance is found experimentally to be appro: 
mately proportional to the square of the velocity. 

For incompressible fluids an expression of the form 

Av/vd + B 

for the drag coefficient will usually fit experimental data for bod 
completely immersed, A and B being constants depending on t 
shape of the body When v is very small the fust tei m is large co] 
pared with the second, and the linear law for viscous drag reappea 
when, on the other hand, v is large the first term becomes srn 
compared with the second, and we have an approximate quadra 
law. 

An expression of the same form will also hold for the resistar 
of air to a projectile, provided the velocity is not sufficiently high 
cause compression of appreciable amplitude. No uppei limit 
velocity can be fixed for this law, since the amplitude of the compn 
sion will depend on the shape of the head of the projectile as well 
on the velocity; thus, for Krupp noimal shells, which have a mo 
or-less pointed head, the diag coefficient changes extremely slov 
with v even at a velocity of 215 m per second, showing that the i 
proximate quadratic law holds for these piojectiles at tins veloci 
whereas with cylindrical projectiles (fiat heads) the diag cocrfici< 
changes rapidly at this velocity (see fig. 5) 

There is little experimental evidence of the behaviour of 1 
drag with variations of d for projectiles at these velocities r l 
results of wind-channel expenments confirm the loim given .ibc 
for the drag coefficient for velocities up to 30 m. pci second a 
it has generally been found that the drag coefficient is greater 
projectiles of small calibre than for those of laigc cahbic of the sa 
shape. 

The Drag at High Velocities 

For velocities greater than the velocity of sound the phys: 
conditions are again changed. 

At the nose of the projectile the air undergoes condcnsati 
The air being an elastic fluid, a condensation formed at any pr 
in it is transmitted in all directions with a velocity which is, in gene 
the velocity of sound. If, then, the projectile is travelling wit] 
velocity less than that of sound, the condensation of the air at 
* See Chapter III, p, 103. 




Facing page 



REACTION OF AIR TO ARTILLERY PROJECTILES 357 

ose will be transmitted, as soon as it is formed, away from the nose 
i all directions If, on the other hand, the projectile is moving faster 
lan sound is propagated, the condensation of the air at the nose cannot 
e transmitted away from the nose in all directions; it can be trans- 
litted away laterally, but not forwards. The result is that the nose 

always in contact with a cushion of compressed air. Greatly 
(.creased pressure is thus experienced by the projectile when travelling 
ith a velocity greater than that of sound. 

Photographs of bullets moving with such velocities reveal the 
dstence of two wave fronts, somewhat conical in shape, one at the 
2ad and the other at the base. In fig. 3 a photograph taken by Cranz 

his ballistic laboratory is reproduced. 

The wave front at the head can be accounted for by Huyghens' 
inciple; it is in fact the envelope of spherical waves which originate 

the head of the projectile at successive instants of time. If the 
nplitude of the condensation, when first formed, were small the 
ave fiont would be a cone of semi-angle Q, such that 

sioQ = ajv, . . ,(2) 

r hen v is less than a the spherical waves have no envelope, and, 

couise, no wave fiont is formed. 

In the actual state of affairs the amplitude of the condensation 

the nose is not small, but finite. The velocity at which it is 
opagated is theiefore greater than the normal velocity of sound, 
t points on the wave front near the nose we should therefore expect 
e angle Q to be greater than at more distant points where the 
aphtude has become considerably reduced The form of the 
tual wave fiont at the nose is therefore a blunted cone, and the 
liter the head of the projectile the more is the wave front 
unted. 

The foimation of the waves behind the projectile cannot be 
counted for in such a satisfactory manner. Loid Rayleigh * has 
own that the only kind of wave of finite amplitude which can be 
aintamed is one of condensation; his argument refers to motion in 
LC dimension only, but we see no reason for modifying the result 
len applied to motion in three dimensions. We therefore conclude 
at the wave at the base of the projectile is, like the head wave, one 

condensation. An examination of the photographs of bullets in 
^ht veiifies this conclusion. 

* " Aerial Plane Waves of Finite Amplitude ", Scientific Papers, Vol. V. 



35 8 THE MECHANICAL PROPERTIES OF FLUIDS 

The source of disturbance causing this wave might be identifi 
with the relatively high state of condensation of the air flowing u 
the rarefied region at the base 

The angle Q of the straight part of the wave appears genera 
to be less than that of the head wave; the difference 
angle is probably the geometrical consequence of placing t 
source of light close to the bullet; we have, in fact, a p< 
spective view of the waves. When the source of light 
very close to the bullet the consequent difference in angle m 
be consideiable. 

The tendency of the angle to diminish towards the apex of t 
wave is probably due to two effects. In the first place theie m 
be some variation in temperature of the air in the immediate neig 
bourhood. Close to the base the air may be cooler than at pou 
more distant; the wave may theiefore be propagated with less veloc 
in the vicinity of the base than at more distant points. In the seco 
place it seems certain that the air behind the projectile will hav< 
velocity gradient from the axis outwards Near the axis the air v 
be moving faster than at more distant points 

Of these two effects the fiist will tend to dimmish a, wh 
the second will cause an mcicasc m v m equation (2); 1 
values of Q will therefore be less near the apex of the wave 

The change of sign of Q immediately behind the base is piobal 
due to change in direction of the air's motion in the immediate neig 
bourhood Lord Rayleigh has proved,* furthei , that such waves 
condensation cannot be maintained m the absence ot dussipati 
forces. It is therefoie evident that some tcim involving the viscosi 
such as vdjv, must be included m the diag coefficient. 

The Scale Effect 

There is some expciimental evidence of the dependence of the di 
coefficient on d, and hence on some such teim as vdjv. For cxamr- 
Cranzf quotes the following figures for the icsislancc pci squaie cer 
metre of cross section deduced from Krupp's 1912 experiments: 

(a) For cylindrical shell (flat heads). 

Calibie (cm.) foi v = 400 500 600 700 800 m./acc, 
6-5 1-40 2-58 3-80 5-15 6-60 Kgm./cj 

lo-o 1-29 2-20 3-30 4-70 6-30 ,, 

* Loc. at. 'I Lehrbuch der Ballistik, Vol. I (Berlin, 



REACTION OF AIR TO ARTILLERY PROJECTILES 359 
(6) For ogival * shell, 3 cahbres radius. 



Calibre (cm ) 


for v = 550 


650 


750 


850 : 


6 


I -00 


130 


1-58 


194 


10 


098 


1-25 


1-52 




28 


o 62 


081 


I 01 


125 


3 








0-90 


i 06 



In the absence of a term involving d, such as vd/v, from the The 
coefficient the numbeis in each vertical column would be equal, drag 
discrepancies are, no doubt, partly due to differences in the yaw of 
the projectiles, but they cannot be wholly accounted for in this way. 

These results indicate that the drag per unit cross section (i.e 
4R/7rd r2 ) for projectiles of small calibre is largei than that for those of 
large calibre. Didion noticed this so-called scale effect as early 
as 1856! and deduced a relation between R/d 2 and d, but later he 
abandoned it as it would not hold for all velocities encountered m 
gunnery This effect has not been confirmed in recent experiments 
and further evidence is needed before definite conclusions can be 
drawn. 

Dependence of the Drag on Density 

The assumption that R varies as />, other factois being constant, 
has considerable theoretical support, but up to the present the range 
of vanation of p m experiments has been extiemely small; we cannot 
therefore claim piactical veuft cation for this assumption When 
more work has been done with the high-velocity an stream more light 
may be thrown on this question, as considerable variations of air 
density are easily obtained m this method. 



The Function f(v/a, vdfv) 

At the present time no satisfactory mathematical expression for 
the drag coefficient has been derived from theoretical considerations 
We are therefore forced to accept values of the function deiivcd from 
experiment alone In ballistic calculations it has generally been 
assumed that the term vdfv could be neglected, that is to say, that 
the drag coefficient is independent of the calibre; the function 
f(v/a, vdfv) has, in consequence, been determined as a function of 
v/a only. 

* Sec p. 360, f Lots d"e la resistance de Vavr (Pans, 1857). 




360 THE MECHANICAL PROPERTIES OF FLUIDS 

Shape of Projectile 

In our discussion we have so far considered the air resistance 
to projectiles of the same shape. Our next step is to consider the 
changes that occur in the resistance when the shape is altered 

All modern projectiles have a cylindrical body and a more-or-less 
pointed head. The head is usually ogival, that is to say, it is generated 
by the rotation of an arc of a circle about the axis of the projectile. 
The shape of the head is identified by the length of the radius (m 
calibres) of this arc. Thus in fig. 4 a head of 3 -calibres radius is 

depicted 

When the point is 
rounded the radius of 

-o 26 d the rounding is also 

stated m calibres Thu 
dotted head m fig 4 

-- 3d would be described as a 

""" ~-- ----^ 3~cahbres radius head 

"T-L^-^ with a o 26-caIibre 

loimdcd point 

In fig 5 the diag co- 

_ _ efficients of a i^-cm. 

| piojeclile with flat-head 

Fig 4 Shape ofl lead and pointed heads of 

vaiious lengths are 

plotted against the latio, velocity of shell to velocity ol sound. The 
curves are deduced from British and ioieign experimental data 

The values of the drag co-efficient given m these curves may be 
used with any self-consistent system of units, foi example, if the 
fundamental units used aie the metic, kilogiamme and second, these 
values of the drag coefficient when used in equation (i) will give 
the drag in metre-kilogramme-second units of foice (i unit = 100,000 
dynes) Again, if the ft.-lb. and second be used, these values of the 
drag coefficient used in equation (i) will give the drag in poundals. 
This property arises, of course, from the fact that f(vja) has no 
physical dimensions. 

The shape of the head, provided it is more-or-less pointed, 
does not appear greatly to influence the resistance at lower velocities, 
At velocities greater than about 350 m. per second, however, the 
effect of the length of the head is appreciable. At velocities greater 
than about 750 m. per second it appears that the shape of the point 



REACTION OF AIR TO ARTILLERY PROJECTILES 361 

is more important than that of the rest of the head Thus the resis- 
tance is less, at these velocities, for a sharp-pointed 3 -calibres radius 
head than for a 5-5 calibres radius head with a blunted point. For 
k he same shape of point the resistance is less, at velocities greater 
han about 350 m per second, for a long head (e.g. 5-5 calibres 
adius) than for a short one (e g. 2-cahbres radius). 

07 



06 



05 



04 




Fai.itedShett 

02 ( ^ 

g- , >. -^ 1 calibre 

* ^v ^*^*^ 

-2 ca fibres 

01 / 3 calibres 



v/a. 



2.0 so to 



Fig 5 The Drag Coefficients for is-cm Projectile with Various Shapes of Head, 
plotted as a Function oi Velocity 

Experiments and trials lead to the conclusion that at high velocities 
long head with a sharp point encounters considerably less resistance 
lan a short head. For example an 8-calibres radius head experiences 
ily about half the resistance of one of a-cahbres i adius. Little 
Ivantage appears to be gained, however, by lengthening the head 
iyond 8-cahbres ogival radius; thus 10- and iz-calibres radius 
2 ads are only slightly more effective than those of 8-calibres radius 

reducing the air resistance. 

(D312) 13 t 




362 THE MECHANICAL PROPERTIES OF FLUIDS 

The Base 

The importance of the shape of the rear pait of a body moving 
in a resisting medium has been realized for many years; the torpedo, 
the racing automobile, and the fusilage of an aeroplane are examples 
of " stream-lining " familiar to all. The suggestion has frequently 
been made that artillery projectiles should have a tapeied (so-called 
" stream-line ") base with a view to reducing the air resistance. 

Experiments with rifle bullets have shown, however, that the 
stability is so seriously affected that any possible advantage gained 
by a pointed base is entirely eclipsed by the effects of a lapidly 
developing yaw. 

In recent times a compromise has been effected in a shape known 
as the " boat-tail ", which is illustiated m fig. 6. The base is tapered 

for a short distance and is then tut off 
squaie. The stability of the piojectile 
is not appreciably affected by this modi- 
Fig 6 Boat-tail Projectile fication of the base. Bullets ot this 
shape were tried in Fiance as fai back 

as 1898, but they were found to have no gieat advantage over the 
flat base. It has, however, been shown recently that, although such 
a base has no particular advantage at high velocities, it has appreci- 
able superiority over the flat base at velocities below about 450 rn 
per second. In the trials mentioned above, the French cxpeu- 
mented at high velocity over short ranges and so failed to discovci 
the merits of this shape. 

Some extremely interesting and suggestive results aic iccorclcd 
by G. P. Wilhelm * of comparative experiments with bullets having 
the boat-tail and the flat base. The following table gives A summary 
of these icsults: 

MUZZLE VELOCITY, 1500 FELT PER SFCOND 

Angle of Range (Yards), Range (Yards), 

Departure. Flat Base Boat-tail 

20' 200 220 

o 40' 360 410 

1 o' 500 580 

i 20' 630 730 

1 40' 740 840 

* In " Long Range Small Arms Firing ", Part VII, Army Ordnance, Washington, 
March-April, 1923. 



REACTION OF AIR TO ARTILLERY PROJECTILES 363 

MUZZLE VELOCITY, 2600 FEET PER SECOND 

Angle of Range (Yards), Range (Yards), ' 

Departure Flat Base Boat-tail. 

20' 570 600 

o 40' 930 990 

1 o' 1050 1200 

5 o' 2250 2700 

10 o' 2900 3600 

15 o' 3250 4200 

20 o' 3500 4650 

It is clearly seen that for velo- ^ / 

cities lower than 1500 ft. per IT 
second the boat-tail bullet con- ^, 
siderably out-ranges the flat-base ** 
Diillet. On the high-angle trajec- 
ories with a muzzle velocity of 
z6oo ft per second the velocity 
)f the bullet is, during the greater 
)art of its flight, less than 1500 ft. 
)er second, in fact it is generally 
mly a few hundied feet per 

, . 06 

lecond; on the low-angle trajec- 
ones, on the other hand, the 
r elocity is less than 1500 ft. per 

11 1 r- i r 04 

econd only in the rmal stages or 
he flight. The appreciable gam 

n range by the boat-tail bullet Q 

ired with high muzzle velocity 05 o 6 os 10 v/a 

t high angles, and the Small Or Fig 7 The Dms Coefficient deduced from 
111 .1 i .1 High-speed Air-struim I 1 xpt-nments 

egligible gain at low angles, thus 

n^Grmc- fVi 1r<rr*ntKpeic tKdt thf Curve A Foi s-cal -rad head and flat base 
OnlirmS tile nypOtneSlS mat me CutveB ^. Same hend as A, boat-tail base, taper, 

hor>A f\f tV>P KIOP 1 of OTPfltfM* '5 Curve C Same head as A, boat-tail biae, 
nape Or me DaSC IS Oi grcdici tape , )7 o CwveD Same head as A, boat-tail 
-nnm-fonr-p. if Inw tVinn at fiitrh base, taper, 9 Dotted Curve The diag co- 

Bportance at low man at nign c[Ilclcnt glven m flg s y\ at base). 
elocities 

The results of experiments with 

ic high-speed air stream are interesting in this connection. In 
g, 7 some of the results of experiments conducted by the Ord- 
ance Department of America are reproduced.* 

* From " Experimental Determination of Forces on a Projectile ", by G. F 
ull, Army Ordnance, Washington, May-June, 1921. 




364 THE MECHANICAL PROPERTIES OF FLUIDS 

These curves tend to show that foi velocities below 350 m. pet 
second the drag is greatly influenced by the shape of the base; on 
the other hand, as we have already seen, provided it is more or less 
pointed, the actual shape of the head has very little influence on 
the drag at these lower velocities 

From these results we may fairly conclude that the greater part 
of the air resistance to pointed projectiles at these velocities is due 
to the drag (suction) at the base, and that this drag is appreciably 
reduced by boat-tailing 

The divergence of curve A in fig. 7 from the dotted curve is not 
clearly understood It is possible that the assumptions made in 
deducing the velocity of the air stream aie not altogethei sound and 
lead to values which are too high, it is also possible that the tod sup- 
porting the model (p. 354) may materially affect the air flow at the 
baseband so modify the drag 

We have seen that at high velocities the drag is gieatly affected 
by the shape of the head, whereas no appieciablc effect is produced 
by modifying the base. The probable explanation of this is not fai 
to seek. At velocities greater than 750 m per second (the so-called 
" cavitation " velocity of air) the vacuum at the base must be of high 
order, and, as the velocity of the projectile increases, it must tend 
asymptotically to a perfect vacuum. We should therefore expect 
that the component of the air resistance due to the base is tolerably 
constant at these high velocities, wheieas the total icsistance is 
rapidly increasing with velocity The component due to the base, 
with increasing velocity, soon becomes a small part ot the total 
resistance, and theiefore any possible modification of it, clue to shape 
of base, can have little influence on the whole 

Our observations on the effect of shape of pointed piojectiles 
may now be conveniently summarized At velocities less than about 
350 m. per second the drag at the base contributes the gi eater part 
of the air resistance, so that the shape of the base is of greater im- 
portance. At velocities between about 350 m. and 750 in. per second 
we have an intermediate stage in which the shape of the head gradu- 
ally gams ascendancy. At velocities greater than about 750 m. per 
second the greater part of the resistance is due to the head, and the 
shape of the latter is of greater importance than the shape of the base. 

Before leaving the subject of shape we must refer to some extremely 
interesting experiments designed to determine the pressure distribu- 
tion on the head of a piojectile moving with high velocity. 



REACTION OF AlR TO ARTILLERY PROJECTILES 365 

The Pressure Distribution on the Head of a Projectile 

The pressure at any point of a body moving through a fluid 
:onsists of two components the static pressure, which is the pressure 
>f the fluid when the body is at rest, and the dynamic pressure, which 
s due to the motion. The sum of these two at any point is the total 
>ressure at that point and is essentially positive; the dynamic pressure 
nay be either positive or negative. 

A series of experiments was carried out by Bairstow, Fowler and 
laitree to determine the dynamic pressure at various points on the 
lead of a shell moving with high velocity.* The fundamental idea 
>f the experiments is the use of a set vice time-fuze j- as a manometer 
o determine the pressure under which the powder is burning 

Projectiles were fitted with hollow caps which entnely enclosed 
he fuze, each cap had a number of holes drilled m it at the same 
hstance from the nose, the pressure on the fuze was thus practically 
qual to thai at the holes. 

The projectiles were fired along the same trajectory at various 
uze settings and times to buist weie observed, a relation between 
aze setting and time was thus obtained, whence was deduced a 
elation between late of burning and time 

Since rate of burning is a function of the pressure on the fuze, by 
ompanson with laboiatory experiments it is possible to convert 
iis relation to one of pressuie and time Knowing the velocity of 
le piojectile at various times of flight on the trajectory, it is thus 
ossible to deduce the pressure in terms of the velocity 

By repeating the expenment with other caps of the same size 
ad shape with holes at other distances from the nose, the pressure 
istnbution over the head is obtained at a number of velocities. 

The results of the experiments are leproduced m fig 8 The 
rdmates are values of p/pv 2 , as this quantity has no physical dimen- 
ons the values given may be used with any self-consistent system of 
nits. The abscissae are distances from the nose of the projectile 
bservations were made at four positions on the head, indicated by 

* For a full account of these experiments see " The pressure distribution on the 
'ad of a shell moving at high velocities ", Proc Roy Sac. A, 97, 1920 

f " A service tirne-fuze contains a train of gunpowder, which is ignited, by a 
'tonatoi pellet on the shock of discharge flora the gun The ' setting ' of the 
ze can be varied so that a length of powdei train depending on the setting is burnt 
'fore the magazine of the fuze is ignited and the shell exploded The setting is 
>ecified by a number which defines the length of composition burnt on an arbitral y 
ale The time of burning is taken as equal to the time interval between the firing 

the gun and the bursting of the shell," Loc. cit. 



366 THE MECHANICAL PROPERTIES OF FLUIDS 



Contin lous curves ( - ) are drawn through points marked 
according to the values of via to which they reftr, obtained 
from firing trials, and, for the nosu of the shell, (torn cal- 
culation 

Dotte 1 curves (----) are extrapolations of the continuous 
cuives, which are needed for mtegiatmg up the nessurc on 
the head 

Dash and dot curve ( ) is drawn throunh points 
marked X, determined by wind channel experiments, at a 
velocity of 40 ft -sec , 01 value of (via) of o o f 



v/a=/2 
v/a=JO^A 
#0=0*4 
6' 



(Reproduced from Proc Roy. Soc A, 97, 1930 ) 




. 

V/cli/Q 




of shell 



Beginning of 
Cylindrical fartian 



INCHES 
Fi. 8. Distribution of Pressure on tha G-cal.-rad Head of a i 3-In Shell 

the points A, B, C, and D. The value op/pv* at the nose was calcu- 
lated from Rayleigh's formula * in eaclf case. 



pv* 7 i_7 $ (7 i)a*lv*-> 7Z) ^( 

deduced from the foimulte given m hia Scientific Papers* Vol. V p. 610, 



REACTION OF AIR TO ARTILLERY PROJECTILES 367 

These curves reveal most emphatically the necessity of a sharp 
Doint at the nose of the projectile. Compared with the pressure 
encountered at the nose the pressures at other points of the head are 
juite small. 

The authors integiated numerically the observed pressures over 
he head in each case, and derived values of the drag coefficient for 



125 



SO 




05 10 15 2.0 i S 30 35 V'u. 

Fig 9 Drag in Kilogrammes Weight on a 3 3-in 6 C R H Projectile 

the dynamic icsistance on the head. From these results we have 
computed the actual dynamic lesistance on the head; it is plotted 
igamst velocity in fig 9. 

The total drag on projectiles of this shape is also shown (ap- 
proximately) in the figure. By subtracting the head resistance from 
the total resistance we have derived an approximate curve of the 
drag at the base The horizontal dotted line indicates the drag due 
to a complete vacuum at the base in this case, and the dotted exten- 
sions of the curves represent a tentative extrapolation of the results 
of the experiments. 



368 THE MECHANICAL PROPERTIES OF FLUIDS 

The Effect of Yaw on the Drag 

Before approaching the complicated reaction of the air to a yawing 
projectile it will be convenient to consider, briefly, the effect of yaw 
on the drag; we now define the latter as the force exerted on the 
projectile in the opposite direction to the relative motion of the an 
and the centre of gravity of the projectile. 

If 8 be the angle of yaw the drag coefficient will now take the foim 

f(v/a, vd/v, B), 

and, in this notation, the drag previously considered takes the form * 

f(vja, vd/v, o) 

The manner m which the drag coefficient vanes with yaw at veiy 
low velocities has been determined experimentally m wind channels 
The results of such experiments on a 3-m piojcctilc with a 
2-calibres- radius head and o i5-calibie rounded point arc given in 
fig io;f the ordmates aie values of the ratio 

f(vfa, vd/v, $)/f(vfa, vd/v, o) 

The velocity at which the experiments were conducted was 40 it 
pel second (vja o 04). 

We have seen that the drag on a body moving in air is appioxi- 
mately piopoitional to the square of the velocity,'] piovided that the 
shape is such that the condensation m Iront is ol small amplitude, 
for example, this quadiatic law holds loi pointed projectiles lor values 
of v\a not greater than 0*65 when the yaw is zeio 

We might icasonably expect the quadratic law to hold ioi yawing 
pointed projectiles within the same limit ot velocity, provided that 
the yaw is such that the air is encounteicd point fust, for the con- 
densation would be of the same order as when the yaw is zci o Within 
this limit for yaw, with values of v/a less than 0-65, we should theicfoic 
expect that the ratio f(v fa, vd/v, ^)f(vja, vd/v, o) is independent of v 



* Approximately. Except in the case of lesulls quoted from tur-stienm experi- 
ments we cannot be certain that the values of the drag coefficient hhheito used are 
for zero yaw All we can affum is that they aie the values foi very small 01 /eio 
yaw. 

f This cuive is derived fiom one given in " The Aeiodynamics of a Spinning 
Shell ", by Fowlei, Gallop, Lock, and Richmond, Phil. Trans, A, 591, 1920. 

J Except, of course, for such low velocities that the diag is due to viscosity 
alone. 



REACTION OF AIR TO ARTILLERY PROJECTILES 369 

and therefore a function of 8 only. The limit of 8 for the projectile 
under consideration appears to be about 45. 

Experiments with the high-velocity air stream tend to verify the 
independence of velocity of this ratio within the limits mentioned, 



12,5 




75" 



50 




30 



45" 



G0 



75 



90 



Yaw 



Fig 10 The Ratio f(vla, vdjv, fi)//(w/a, vdjv, o) plotted against the 
yaw & for via => o 04 



but at present the number of lesults available is insufficient to justify 
our drawing definite conclusions. 

The dotted curve in fig. 10 gives, approximately, the ratio between 
the total plane areas encountered by the projectile with yaws 8 and 
zero. The former is, of course, equal to the area of the shadow cast 



370 THE MECHANICAL PROPERTIES OF FLUIDS 

by the projectile, in a paiallel beam of light inclined at angle 8 with 
the axis, upon a plane normal to the beam; the latter is simply the 
cross-sectional area of the projectile. The two curves in the figure 
are coincident for small angles of yaw, but rapidly diverge as the yaw 
increases; both curves appear to have a maximum at about the same 
value of the yaw. 

We have at present no knowledge of the effect of yaw on the drag 
at higher velocities; when more work has been done with the high- 
velocity air stream it is hoped that our knowledge in this direction 
will have been considerably extended 



The Drag Coefficient; Concluding Remarks 

In our consideration of the drag coefficient we have limited the 
number of arguments of the function to three only, namely v/a, vd/v, 
and 8. There appear to be two other possible arguments, namely 
y, the ratio of the specific heats, and ajd, where a is the effective 
diameter of the molecules of the air. 

Variations of y are so small in practice that no evidence of its 
effect is available. Expressions deduced from thermo dynamic 
theory, by various authors, for the drag in one-dimensional motion 
may give an indication of the manner in which it occurs * 

There is at present no evidence of the necessity of the aigument 
aid. If further experimental results show that the argument vd/v 
does not adequately account for the " scale " effect (e g. with vaiymg 
v), it will of course be necessary to include some other argument 
involving d, such as cr/d 

Finally, in the case of a projectile moving In air, as distinct from 
one which is stationary in an air stream of constant velocity, there is 
the question of retardation. It is just possible that some such argu- 
ment as rd{v*, r being the retardation of the projectile, may be re- 
quired to co-ordinate the results of air-stream experiments with 
those of experiments on a projectile moving in air, but it is difficult 
to see how the retardation can ever have an appreciable effect on the 
drag, except, perhaps, when the velocity is in the neighbourhood of 
the velocity of sound in air. 

* See for example the footnote on p. 346; also Vieille, Comptes Rendus, 130 
(1900), and Okmghaus, Monatsh.fiir Mathem. . Phys. t 15 (1904). 



REACTION OF AIR TO ARTILLERY PROJECTILES 371 

REACTION TO A YAWING, SPINNING 
PROJECTILE 

The most complete specification in existence of the system of 
>rces acting on a projectile is that given by Fowler, Gallop, Lock, 
id Richmond in " The Aerodynamics of a Spinning Shell ", Phil, 
"rans A, 591 (1920). In this paper the authors describe expeiiments 
inducted to deteimme numerically the principal reactions, other 
ian the drag, to which a spinning shell is subjected 

The experiments are confined to the study of the angular oscil- 
itions of the axis of the shell relative to the direction of motion of 
le centre of gravity. The projectile is fired horizontally through a 
sries of cardboard targets fixed, veitically, along the range at 30 ft. 
ad 60 ft. intervals Initial disturbances at the muzzle give rise to 
scillations of the projectile of sufficient amplitude for measurement; 
le details of the oscillations aie obtained by measuring the shape of 
le holes made in the caids If, on passing thiough a caid, the shell 
; yawing, the resulting hole will be elongated, the length of the 
>nger axis of the hole determines the yaw; the orientation of this 
xis determines the azimuth of the plane containing the axis of the 
hell and the direction of motion; these two angles determine the 
irection of the shell's axis completely 

The range containing the cards is so short, and the velocity so 
igh, that the effect of giavity is negligible If, then, we ignore 
amping forces, the angular motions of the axis of a top and the 
xis of a shell' are identical, provided that (i) the top and shell have the 
ame axial spin and axial moment of inertia; (2) the tians verse moment 
f inertia of the top about its point of support is equal to the trans- 
erse moment of inertia of the shell about its centre of gravity; (3) 
be moment of gravity about the point of the top is equal to the 
cioment of the force system on the shell about its centre of gravity, 
lie foimal solutions of the two problems are then identical. 

From the periods of the oscillations of the axis of the top we can 
[educe the moment of the disturbing couple, and -vice versa', similarly 
he moment of the force system on the shell can be deduced. The 
lamping forces can then be determined from the nature of the decay 
>f the oscillations. 

The reactions are described by the authors as follows: 



372 THE MECHANICAL PROPERTIES OF FLUIDS 

The Principal Reactions 

When the shell, regarded for the moment as without axial spin, 
has a yaw 8, and the axis of the shell OA and the direction of motion 
OP remain in the same relative positions, the force system can by 
symmetry be represented, as shown in fig. ir, by the following com- 
ponents, specified according to aerodynamical usage: 

(1) The drag R acting through the centre of gravity O, in the 
direction PO. 

(2) A component L, at right angles to R, called the cross-wind 
force, which acts through O m the plane of yaw POA, and is positive 
when it tends to move in the direction from P to A 

(3) A moment M about O, which acts 
A m the plane of yaw, and is positive when it 

tends to increase the yaw. 

The following foims aie assumed for 
L and M: 

L = pv z d z sinS/ L (?;/<2, 8). 
M = pv*d 3 sin8/ M (;/fl, 8). 

>L 

These equations are of the most natural 
forms to make / L and / M of no physical 
Fig u. dimensions. The form chosen is suggested 

by the aerodynamical treatment of the force 

system on an aeroplane Since L and M, by symmetry, vanish 
with 8, the factor smS is explicitly included in these expiessions 
in order that / L and / M may have non-zero limits as 8 o. 



The Damping Reactions 

The yawing moment due to yawing In practice the dncction of 
the axis of the shell, relative to the direction of motion, changes fahly 
rapidly. By analogy with the treatment of the motion of an aero- 
plane, we assume, tentatively, that the components of the force 
system R, L, and M are unaltered by the angulai velocity of the axis, 
but that the effect of the angular motion of the latter can be repre- 
sented by the insertion of an additional component, namely, a couple 
H, called the yawing moment due to yawing, which satisfies the equation 

H 




REACTION OF AIR TO ARTILLERY PROJECTILES 373 

here w is the resultant angular velocity of the axis of the shell, 
'he form is chosen to make / H of no physical dimensions and is the 
nly one suitable for the purpose. 

The couple is assumed to act in such a way as directly to diminish 
i (see fig. 12). It is suggested by, and is analogous to, the more 
nportant of the " rotary derivatives " in the theory of the motion 
f an aeroplane. 

The coefficient / H may be expected to vary considerably \vith 
'/a, and it may depend appreciably on other arguments such as 
id\v and 8 

The effect of the axial spin, The spin N gives rise to certain 
dditional components of the complete force system. 




Fig 12 

There will be a couple I which tends to destioy N, and, when 
the shell is yawing, a sideways force, which need not act thiough the 
centre of gravity, analogous to that producing swerve on a golf or 
tennis ball. This force must, by symmetry, vanish with the yaw; 
it is assumed to act normal to the plane of yaw (any component it 
may have in the plane of yaw is inevitably included m either R or L). 
The complete effects of the spin N can therefore be represented by 
the addition to the force system of the couples I and J and the force 
K, acting as shown in fig. 12. 

To procure the correct dimensions we may assume that these 
reactions have the forms 

I = pvNd% 

J = pvNd* sm8/ r 

K = pvNd* sin8/ 8 . 




10 M 

o fc 



REACTION OP AIR TO ARTILLERY PROJECTILES 375 

The coefficients /j,/ p / K may depend effectively on a number of 
lables which we can make no attempt to specify in the present 
te of our knowledge. 

It will be seen that this specification is equivalent to a complete 
stem of three forces and three couples referred to three axes at 
,ht angles. Owing to the complex nature of the reactions the 
thors considered it essential to construct the specified system 




Vertical 

Vertical 




Radial Seals 
afdeqrees 



Fig 14 Examples of Path of Nose of Shell relative to the Centre of Gravity 
(FromP/H/ Trans A, 591, 1930) 

stead of attempting to analyse a complete system of three forces 
id three couples and to assign each component to its proper causes. 

The experiments were designed to determine L * and M and to 
ve an indication of the magnitude of the chief damping couple 
. It was not possible to determine I, J, and K; all three are 
resumably small compared with H and very small compared with 
> and M; no certain evidence that they exist was given by the 
qpenments. 

The results of the experiments are exhibited in fig 13, which is 
^produced from the paper. The units in which the coefficients are 
^pressed are suitable for use with any self-consistent system. The 

* Values of L were deduced from values of M for shell of the same external 
lape, with centres of gravity at different positions along the axis of figure. 



376 THE MECHANICAL PROPERTIES OF FLUIDS 

curve of the drag coefficient is also given; in comparing this with 
/ L and jf M it must be remembered that the latter should be multiplied 
by sinS. The shape of the curve for/ L is rather unexpected; the 
curves for / M appear to exhibit the same tendencies as that for the 
drag coefficient. 

The values of/ H deduced fiom the experiments were rough; they 
varied from about 1-4 to 5-0. It was impossible to deduce any 
details concerning the variations of / H with velocity, but the right 
order of magnitude is represented by these limits In comparing 
/ H with / M it must be remembered that the former is multiplied by 
zvd, whereas the latter is multiplied by the much larger quantity v. 

The general features of the motion of the axis of the shell and of 
the damping are shown in the examples, reproduced from the paper 
cited, in fig. 14. 



SUBJECT INDEX 

(See also Name Index) 



coustic properties of matenals, 304 
coustics under water, 298-344 
ir, clouds in, 40 

Displacement, 9 

Resistance coefficients, 207. 

Viscosity, 118 
vogadro's hypothesis, 43. 

alhstic pendulum, 348-51. 
arometei, the first, 2-3 
aum hydiometer, n 
eanngs, cylindrical, 142-6. 

Flexible, 155-6 

Lubrication, 133-58 

Pivoted, 148-55 

Self-adjusting, 139-42 

Width, 146-9 

ernoulh's equation, 57-9, 96 
maural method of sound icception, 

306-10 

oiler tubes, gases in, 183 
oyle, barometer experiment, 2-3 

Law, derivation of, 42 
rass, acoustic propeities, 304 
ndgman, high-pressure expenments, 

1 6-8 

ronze, acoustic properties, 304. 
ubble, collapse of, 75 

allendar's equation, 49 
apillary tubes, action in, 25. 

Rayleigh equation, 27 

Viscometer tubes, 112-6 
'arbon bisulphide, viscosity, 118. 
'ast iron, acoustic propeities, 304. 
'astor oil, compressibility, 219 

Viscosity, 1 1 8 
hannels, flow in, 179-80 
harles's law, derivation of, 42-3. 
lausius equation, 47-8 
louds, suspension of, in air, 40. 
eofficient of viscosity, 32, 103-4, 117-9 
ollision, prevention of, by echoes, 
332-3 



Compiessibility of liquids, 15-9, 23, 

212, 218-9 
Conduction of heat and viscosity, 99- 

100 

Contact angle, meaning of, 25. 
Continuity, equation of, 60- 1. 
Coohdge tube, 320 
Critical point, 44 
Critical temperatuie, 44 
Critical velocity, 165-72 
Crystals, elasticity of, 4-5 
Cube, piessure on, 4-5 
Cup-and-ball viscometer, 125-8 
Cylinders and wires, icsistance of, 

214-7 

Deformation and rigidity, 3-4. 
Density, 7-13 
Diaphragm soundei ,318 
Dietenci equation, 47-8 
Dimensional homogeneity, 187-203 
Diiectional acoustics, 306-12 
Directional sound receivers, 311-3 
Displaced air, correction factoi, 9 
Drops, deteimination of weight of, 28-9 
Dynamical similarity, 193-203 

Echo depth-sounding geai, 335-44 

Echo, detection of ships by, 332-3 

Eddies and turbulence, 166 

Elasticity of crystals, 4-5 

Elasticity, phenomena due to, 218-36 

Electrical measurement of pressure, 17 

Energy at surface of liquids, 19-29 

EotvSs equation, 23 

Equations of motion, 36 

Equations of state, 4149 

Errors in soap film experiments, 244-7. 

Explosions under water, 76 

Fessenden sound transmitter, 317, 333. 
Flow, measurement, 172-8 

Stream-line, 160-90 

Sudden stoppages, 219-21, 



377 



378 



THE MECHANICAL PROPERTIES OF FLUIDS 



Flow-meters, 173-8 

Diaphragm, 175-6 

Pitot tube, 176-9 

Venturi, 172-5 
Fluids, definition, 1-2, 6. 

Motion, 56-101. 

Pressure, 17 

Solids irm^ersed in, 33. 
Friction in pipes, 221-3. 

Gas, perfect, definition of, 6. 

Gas equation, 3. 

Gases, liquids, and solids, classification, 

1-2 
Glycerine, viscosity, 118 

Heat transmission, flow effects, 180-90. 
Hydrodynamics. See Stream-line Mo- 
tion 

Hydrodynamical resistance, 191-217 
Hydrometers, calibration defects, 10-2 
Hydrophones, 305-6, 320-4 

Bi-directional, 312. 

C-tube, 320-1. 

Constructional details, 322-3. 

Magnetophones, 321-2. 

Morns-Sykes, 312 

Uni-directional, 313 

Incompressible fluid, equations of, 36 
Irrotational stream-line motion, 62, 
65-71- 

Kinetic theory of gases, 41-4. 

Laminar motion, 103 

Lead, acoustic properties, 304. 

Liquids, compressibility, 13-9 

Definition, 6. 

Molecular viscosity, 39 

Surface energy, 19-29 

Surface tension, 19-29 
Liquids, gases, and solids, classification, 

1-2 
Lubrication, 128-59 

Bearings, 133-58 

Bibliography, 158-9 

Inclined planes, 131-3. 

Viscous, 129-31 
Lubricating oils, compressibility, 219 

Viscosity, 118-9. 

Matter, states of, 1-2 

Mercury, viscosity, 118 

Mineral oils, viscosity, 118 

Mohr's balance, 12 

Molecular viscosity of liquids, 39. 

Motion, equations for incompressible 

fluids, 36 

Motion and heat transmission, 180-90 
Motion of fluids, mathematics of, 56- 

101. See also Stream-line Motion, 



Laminar Motion, Vortex Motion, 

Wave Motion 
Naval architecture, model experiments, 

209-1 i 
Nitrogen, Amagat's experiment with, 

44 

Oils, compressibility, 219. 

Viscosity, 118-9 
Olive oil, compressibility, 219 
Osmosis and osmotic pressure, 50-4, 
Ostwald's viscometei, 39 

Parallel planes, viscous flow between, 

119-25 

Permeability, 50 
Petroleum, compressibility, 219. 
Pipes, critical velocities, 168-72. 

Elasticity, 223-6 

Flow resistance, 198-203. 

Flow stoppage, 226-7 

Friction, 221-3 

Opening valves, 228-9 
Pitot tube, 59 
Poiseuille's equation, 37-8 
Power transmission, acoustic, 333-4 
Prandtl's analogy, 239-42 
Pressure, 56 

Electrical measuiement, 17. 

Viscosity, 117-9 
Projectiles, air density, 359 

Ballistic pendulum, 346-8 

Base shape, 362-4 

Bashforth chronograph, 348 

Boat-tail bullet, 362-4 

Cranz's ballistic kmematogi aph, 353 

Drag coefficient, 359, 370 

Kiupp's 1912 experiments, 351-3 

Piessure distribution on, 365-7. 

Reaction of an on, 345-76 

Scale effect, 358-9 

Shape, 360-4 

Spark chronograph, 350-3 

Spinning, 371-6 

Yawing, 371-6 
Pyknometei, descuption of, 9-10 

Rape oil, viscosity, 119 

Rayleigh's equation for capillary tubes, 

27 

Redwood's viscometer, 116 
Resistance and compressibility, 212 
Resistance of square plates, 211-2. 
Resistance of submeiged bodies, 206- 

9 
Resistance of wires and cylinders, 

214-7 

Rigidity and deformation, 3-4 
Rubber, acoustic properties, 304. 

Scale effects, 211-3 
Shearing of a cube, 4-5. 



SUBJECT INDEX 



379 



hips, resistance of, 209-11. 

irens, undei water, 319. 

km fuction, 203-6 

oap film stress deteimmations, 237-52 

Contour mapping, 244-7. 

PrandtPs analogy, 240 

Sheai stress in twisted bar, 240. 

Toique on twisted bar, 241. 

Twisting of bais, 237-9. 

Warping of sections, 238 
olid, scientific definition of, 5 
olids, liquids, and gases, classification, 

1-2 

olids and fluids, mtei action of, 195 
olutions and solvents, 50-4 
ound, nature of, 299. 
ound ranging, 325-33 
Depth sounding, 330-2 
Leader geai, 330 
Multiple station system, 325-9 
Wixeless acoustic method, 328-30 
ound icceivers, 304-6 
ound transmission, 301-4, 316-8 
ound velocity, 300 
Wave-length, 301 
ources and sinks, 75-81 
pecific gravities, air displacement, 9. 
perm oil, compressibility, 219 
Viscosity, 1 1 8 
pheie in viscous fluid, 33 
quaie plates, resistance of, 211-2 
tate, equations of, 41-9 
teel, acoustic properties, 304 
tethoscope, action, 306-9 
Cokes' foimula, 41 
tone's viscometer, 114-6 
tream function, definition of, So 
tream-line forms, typical, 216-7 
tream-lme motion, 57-85, 160-90 
Application to naval aichitecture, 91 
Axial symmetry, 75-81 
Bernoulli's equation, 57-9. 
Circulation, 61 
Continuity equation, 60-1 . 
Critical velocity, 165-72 
Equations, 83-5 

Hele Shaw's expenments, 162-5 
Iirotational, 62, 65-71. 
Stability, 1 6 1-2 
Steadiness, 63-5 
Tracing stream -lines, 81-3. 
Tubes, 162 

Turbulent flow, 161-90 
Two-dimensional, 60-2 
Velocity potential, 71-4 
Vortex rings, 75 
Vorticity, 61-2 
ream-tube, definition of, 57 
ress determinations from soap films 

See Soap Film 
ruts, strearn-lme, 216-7 
ibmanne signalling, 298-334 



Submeiged bodies, resistance of, 206- 

9- 

Sum-and-difference method of sound 

reception, 310-1. 
Surface tension, 19-29. 

Tempeiature effects, 3, 20-2, 39, 44 
Thermodynamics of compression, 15-8. 
Timber, acoustic propeities, 304 
Torsion, examination by soap films, 

237-54 

Trotter oil, viscosity, 119 
Tubes, converging, 171-2 

Stream-line motion in, 162 

Viscosity in, 109-12 
Turbulent and stream-line flow, 160-90 
Two-dimensional motion, 60-2 

Valves, sudden closing of, 219-21, 226 

Sudden opening, 228-9 
Van der Waals' equation, 46, 48 
Velocity, critical, 16572 

Measurement, 172-80 
Ventun meter, 172-5 
Viscometeis, 35-9 

Capillary tubes, 112-6. 

Cup-and-ball, 125-8 

Redwood, 116 

Secondary, 116-7 

Stone's, 114-6 
Viscosity, 31-41, 102-28 

Bibhogiaphy, 158-9 

Bounding surfaces, 107-8 

Coefficients of, 32, 103-4, 117-9 

Equations, 97-9 

Laminar motion, 103 

Lubrication, 102-59 

Measuiement, 34-6 

Paiallel planes, 119-25 

Piessure vanation, 117-9 

Relative motion, 123-5 

Relative velocities, 105-7 

Solids, effect of, 33 

Temperature effects, 39 

Tubes, flow-in, 109-12 

Two-dimensional cases, 99-101 

Velocity giadient, 32 
Voitex motion, 85-9 

Isolated vortices, 87-9 

Persistence, 85-7 

Rings, 75, 85-9 

Water, acoustic properties, 304 

Compiessibihty, 218-9. 

Density, 9 

Viscosity, 1 1 8 
Water-hammer, 219-23. 
Wave motion, 89-97. 

Canal waves, 89-91 . 

Deep-water waves, 91-4. 

Group velocities, 94-7. 



380 THE MECHANICAL PROPERTIES OF FLUIDS 



Superposed liquids, 93-4 
Transmission of energy, 229-36. 
Wind structure 

Altitude and velocity, 276-81, 28, 
Anemometer records, 263-4. 
Anti-cyclone, 260-1. 
Clouds, cumulo nimbus, 282 
Cyclone, 260-1, 286-94 
Eddy theory, 272-7. 
Egnell's law, 284 
Geostrophic component, 263. 



Giadient wind, 260-1. 

Rain, cause of, 282 

Stiatospheie, 281 
[-9. Strophic balance, 262 

Surface winds, 266-71. 

Troposphere, 281. 

Wind variation, 263-6. 
Wires, heat dissipation fiom, 183 
Wires and cylinders, resistance of, 
214-7. 



ii f 



NAME INDEX 



Akerblom, 274. 
Allen, 191. 
Archbutt, 158 
Archimedes, 8-n. 
Avogadro, 43 

Bairstow, 203, 213, 365 

Barnes, 168-9 

Bashforth, 347-52 

Bassett, 36. 

Baume', n 

Beauchamp Tower, 158 

Behm, 331-2 

Bennett, 23 

Bernoulli, 57-8, 63, 65, 

90-1, 96, 165 
Bessel, 148 
Booth, 203-13. 
Borda, 59 
Boinstem, 63 
Boswall, 159 
Boyle, 2-3, 7, 42, 44 
Bndgman, 16-9 
Bailie", 302-5 
Bnlloum, 158 
Broca, 305, 320 
Bryant, 186 
Buckingham, 213. 
Bunsen, 115 

Callendai, 49 
Canovetti, 212 
Carey Foster, 17. 
Carothers, 158 
Cave, 281-2. 
Charles, 43 
Clausius, 48-9 
Clement, 169 
Cleik Maxwell, 19. 
Coker, 168-9 
Collodon, 339 
Constantmesco, 13, 230, 

23 6 . 2 9S> 333~4 
Cook, 218. 
Coohdge, 320 
Cranz, 353, 357 
Crombie, 264. 



Dadounan, 328. 


Hershall, 172. 


Daicy, 1 68 


Hun, 158 


Datta, 230 


Hodgson, 175 


Davis, 1 8 


Hojel, 350 


Deeley, 158. 


Hoskmg, 158 


"De F", 159 


Hull, 354, 363. 


Didion, 346, 359. 


Hunsaker, 213. 


Dietenci, 48 


Hutton, 346-7, 352 


Dines, 212, 268, 271 


Huyghens, 357 


Dobson, 274, 279-81 


Hyde, 119, 159, 218-9 


Eberhard, 350, 353 


Jacobs, 35 


Eden, 170, 216 


Joly, 329 


Edmunds, 302 


Jordan, 183. 


Egnell, 284 




Eiffel, 212 
Emthoven, 326, 329 
Ekman, 169 
Eotv&s, 23 


Kmgsbuiy, 159 
Kobayashi Toras, 159 
Krupp, 350-60 


Ewmg, 49 


Ladenburg, 35, 




Lamb, 36, 164 


Faust, 159 


Lanch ester, 159 


Fenanti, 155 


Landholt, 218 


Fessenden, 317, 321, 333 


Lasche, 158 


Ford, 326 


Lees, 203 


Fowlei, 365, 368, 371 
Fiahrn, 351 
Froude, 203, 205, 209-10 


Lempfert, 293 
Lock, 368, 371. 
Lorentz, 186. 




Love, 238. 


Gallop, 368, 371 




Gibson, 35, 171, 177, 180 


Macleod, 22 


200-1, 223, 225, 250 
Goodman, 158 


Manson, 330, 
Marks, 18 


Giassi, 218 


Marshall, 186. 


Gray, 314 


Marti, 330 


Griffith, 242 


Martin, 159. 


Gnffiths, 252-3. 
Grmdley, 171 
Giimbel, 159, 


Martini, 219. 
Mason, 314 
Matthews, 23. 




Maxwell, 108, 119, 133 


HSlstiom, 7 


Mayewski, 350 


Hartree, 365. 


Michell, 146-50 


Hele Shaw, 162-4 


Mitchell, 23. 


Hellmann, 271 


Morns, 312 


Helmholtz, 100 


Morse, 318-9, 330. 


Hersey, 159 


Munday, 314-5. 


381 





THE MECHANICAL PROPERTIES OF FLUIDS 



potto, 347- 
nst, 23. 
vbigm, 159. 
vton, 33, 301. 
holson, 183. 
iselt, 183. 

n, 302. 
nghaus, 370. 

nell, 170, 184, 200-3 
ions, 218. 
dn, 9. 
fer, 51 
ee, 314 

t, S9> *72 

emlle, 37-9, 104-5, 

C2, 1x6, 158, 200, 234 

er, 52, 3H- 

itmg, 43-4, 230 

vdtl, 239-40 

icke, 219, 
kine, 81. 



Rayleigh, 27, 102, 108, 

146, 159, 187, 357-8. 

366. 

Redwood, 116 
Reynolds, 128-9, 139, 142, 

146, 158, 161, 165-9, 

181-3, 186, 200 
Richards, 23. 
Richardson, 333. 
Richmond, 368, 371, 
Ritchie, 250. 
Robms, 346 

St. Venant, 238 

Shaw, 262, 288-96. 

Shore, 159 

Smith, 311, 318, 321. 

Soenneker, 186 

Sommerfeld, 142-5, 158 

Sprengel, 9 

Stanford, 10 

Stanton, 170, 181-7, I9 1 . 

200-3, 212-3 
Stephenson, 330, 
Stokes, 33, 158, 164 
Stone, 114, 159. 



Stoney, 159 
Stuim, 329. 
Sykes, 312. 

Tait, 5, 218 

Tayloi, 185-6, 242, 252-3, 

264, 272-3, 275 
Thomson, 218, 230. 
Torncelh, 58 
Twaddell, 10 

Van der Waals, 46, 48. 
Van't Hoff, 51 
Ventun, 172. 
Vieille, 370. 

Walser, 314-23. 
Watson, 230 
Webster, 36, 
Weston, 227 
Wilhelm, 362 
Wood, 300, 326. 

Young, 5, 48- 
Zahm, 205,