Applied Physics Series
THE MECHANICAL PROPERTIES
OF FLUIDS
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Warwick Huuw, Iurt Httu't,
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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 ViceAdmiral
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 VIOEADMIEAL 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.
reamline 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 Cupandball 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 Selfadjustment
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
Streamline and Turbulent Flow
By PROFESSOR A H GIBSON, D Sc.
Streamline Motion Stability of Streamline 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 nonUniform 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 VICEADMIRAL
SIR GEORGE GOODWIN, K.C.B., LL.D.
(Late EngineerinChief 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 flowmeters, and should prove
eful to engineers associated with highpowei 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 flowmeters 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, wavelength; 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 pigeonhole 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 sealingwax 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 twentyone
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 elementplane 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 elementplane 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 elementplane at that point. Thus if #, y, z
aie the coordinates 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 timerates,
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 0000001690
b = 0000000105
/ = 0000058815
m = 00000062168
n = 000000001443
a 0000052939
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 4108 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 099987 42 099147
2 o 99997 44 o 99066
4 i ooooo 46 o 98982
6 o 99997 48 098896
8 o 99988 50 o 98807
10 099973 52 098715
12 099953 54 098621
14 099927 56 098525
1 6 o 99897 58 098425
18 099862 60 098324
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 099440 85 o 96865
36 099371 90 096534
38 099299 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 specificgravity bottle may be
jed, but for precision measurements some form of pyknometer is
icessary The pyknometer is usually a Utube 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 specificgravity work is carried
out by means ol vaiiableimmersion 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 1035 the standard of
reference being very often doubtful and the Twaddell hydrometer
is so constructed that the specific gravity s is given by
s = iooo ~~ 0005 r,
where T is the reading in Twaddell degrees. Clearly on the common
LIQUIDS AND GASES 11
lydrometer the " waterpoint " is 1000, on the Twaddell hydrometer
,ero, and, unless the hydrometer carries some reference to the tem
>erature at which its waterpoint 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 waterpoint being at 10. Now it is usual ^
to mark the point to which the hydrometei sinks in sul
phunc acid of density 1842 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 ciosssectional 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 = 1842 when n = 66, we find that k = 1443, and
theiefore for any other liquid giving a reading x,
s = I44 ' 3
1443  *'
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 ooi
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 1374 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 12. 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 supercooled 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 03 A logarithmic plot of Y
gainst X shows, in general, very good straight lines, whose slope
leviates very little from the value 03. 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 091
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 boilingpoint under noimal pressure is very approximately
wothirds of the cntical temperature Putting, therefore, m = f ,
fc have
Pfc = 2/40 91(1 !) 03 h(io 5 x*)],
geneial relation between the density at the normal boilmgpomt
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 piceminent 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 chiomevanadiurn
steel cylinder, and the pressure is produced by the advance ^of a
piston of known crosssection, 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 highpressure 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 pressurevolume 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
10195
i 0224
10255
I 0287
500
99 6 5
9992
I 'OO2O
I 0049
10075
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 lighthand 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 settlmgout 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 surfaceaiea
of the oilwatei 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 indiarubber, 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 12. 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
0003553
I
210
0004695
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
2140
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 nonassociation
We now tuin to the discussion of a problem of fundamental
mportance that of the i elation between the pressureexcess (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 airbubble 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 pressureexcess equation, combined with a know
ledge of the fact that a liquid meets a solid at a definite angle called
the contactangle, will suffice to solve many important surfacetension
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 wellknown 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 semiaxes 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
2nry 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
Piessuieexcess = gph = ,
or za 2 = Rh.
Now R, the radius of curvature at the end of the semiaxis 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 = ^/! _)_ ^ _ 01288^ +
\ 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
mzenewater 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 liquidau 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 physicochemical works is to
juate the weight of the detached diop to zirry, a procedure which,
Lit for the fact that the di opweight 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
?"
2rrry
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
258 413
116 397
o 708 3 80
o44i 3 73
0277 378
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 38. 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 12, 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 12
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 righthand 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 = fofv
\ 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 015 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 onethird of the total depth
of the liquid.
LIQUIDS AND GASES 35
Two important corrections are necessary one for " walleffect ",
e for " endeffect "; 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 walleffect 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 endeffect
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 0002 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 coordinates,*
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 lighthand 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 00013. The paradox is cleared up by an application oi
Stokes' formula,
V =  (^ZJ ^ 3 .
9 P
Taking the viscosity of air as 000017 m C.G S units, the leader is
recommended to calculate the teiminal velocities of spheres of water
say oi, ooi, . . . 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 grammemolecular 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
mmemolecular volumes of all gases, measured under the same
iditions of tempeiature and pressure, are the same.
The giammemoleculai volume, measured at o C and 760 mm.
Hg, is 2238 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
Utube, the open lirnb being installed on the side of a shaft 327 m.
deep. The pv 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 " equalroot " 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 pvp 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 375 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 267 and 3 oo respectively, and the corresponding equations
of Dieterici give 375 and 369, 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 semipermeable 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 semipermeable 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 semipermeable, is called the
osmotic pressuie of the solution.
It will thus be seen that osmotic pressure is defined in teims of
a semipermeable 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 semipermeable 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 sopoiificpowerofopium 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 = ^rUh + + )
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 COORDINATES
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 coordinates 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 singlevalued function of the
coordinates 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 " pressuieintensity " 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 $x is parallel to
MATHEMATICAL THEORY OF FLUID MOTION 57
he axis of #, and suppose that the dimensions of the crosssection
w) are small compared with Bx. The pressureintensities at the
wo ends may then be
lenoted by pa} _ + Q ;)+( P <.$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 streamline 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 streamlines drawn through the contoui of any small area
ill mark out a tube, which we may call a streamtube. Since the
ime volume of fluid must ti averse each ciosssection in the same
me, we have
heie co!, o> 2 are the areas of any two crosssections 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 streamline
P + W + /v = c, ............. (2)
where C is a constant for that particular line, but may vaiy from
one streamline 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
pressuregauge 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 streamline. 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. Twodimensional Motion, Streamfunction
There are two types oi streamline motion which ate specially
simple and important. We take first the twodimensional 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
crosssection, 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 streamline, 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
coordinate 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 coordinates
', 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 lineintegral 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 lineelement 8s In lectangulai cooidmates, 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 twodimensional case)
ic vorticity] we denote it by . Its value, in terms of rectangular
62 THE MECHANICAL PROPERTIES OF FLUIDS
coordinates, is found by calculating the circulation round an
elementary rectangle PQRS whose sides aie $x and Sy. The por
tions of the lineintegral (n) due to PQ and RS
S * R aie together equal to the difference in the cor
icspondmg values of uSx, i.e. to ~SySx.
\ I The portions due to QR and SP are in like
manner equal to SxBy. Equating the sum to
p > Q x
Fig. 7 &*%> we have
c. dv __ du , ^
dx 9y'
or, from (6),
f\e* i r\n t
o O^/r O^ift
The value of at any point P is related to the aveiage lotation
relative to P of the particles in the immediate neighbourhood To
examine this, we calculate the circulation round a circle of small
radius r having P as centre The velocity at any point Q on the circum
ference may be regarded as made up of a general velocity equal to
that of P, and the velocity relative to P. The former of these contri
butes nothing to the required circulation The latter gives a tan
gential component cur, where cu is the angular velocity ol QP The
circulation is therefore
fZtr rZt
I corrdO = r 2 /
./ o J o
where 6 is the angular coordinate of Q. Since the same thing is
expressed by nr 2 , we have
TTJ
which is twice the average value of co on the circumference. For this
reason a type of motion in which is everywhere zero, i.e in which
the ratio of the circulation in every infinitesimal circuit to the area
within the circuit vanishes, is called irrotational.
MATHEMATICAL THEORY OF FLUID MOTION 63
3. Condition for Steady Motion
We can now ascertain the dynamical conditions which must
e satisfied in order that a given state of motion may be steady,
'or this purpose we consider the forces acting on an element,
'QQT', of fluid included be
tfeen two adjacent streamlines
tid two adjacent normals. The
itter meet at the centre of cur
ature C. We write PQ = &,
P' = Bn, PC = R. The mass
f the element is therefore p$s8n.
'he forces acting on it may be
ssolved in the direction of the
ngent and normal, respectively,
1 the streamline 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 streamline, but will in general vary fiom one stieamlme
to another. If we fix our attention on two consecutive stieamlmes,
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 streamline. 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 vortexmotion 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 coordinates,
?! 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 streamlines 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 linesource
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 streamline 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
streamfunction then has the form
Fig n
so that the relative sti camlines 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 hnevortex 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. Velocitypotential
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 coordinates, 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 coordinates
; 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 coordinates, 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 " velocitypotential ". 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 streamlines at light angles If on the other hand Sn
be the perpendicular distance between two adjacent equipotential
lines, we have 80 == q$n If, therefoie, we imagine a whole
system of such lines to be drawn for equal small increments S(f>,
the perpendicular distance between consecutive lines will be evciy
where inversely propoitional to the velocity. If, further, we suppose
the streamlines to be drawn for intervals Si/f each equal to S</>, we
have Si/r = #3$', where Ss' is the interval between consecutive
streamlines 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 streamfunction 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 linesouice 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 linevortex 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 cooidmates 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 streamline 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 streamlines 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 crosssection must remain constant.
us is a particular case of a general theorem that the strength of a
rtexfilament (in this case a vortexring) lemams unaltered as it
If = o, the aigument for the existence of a velocitypotential
11 hold as befoie One or two simple cases may be noticed If
: imagine a pointsource 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 / WJ.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 footpound
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
3 627
4 1176
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 = 316 X io 4 cm /sec. We find that the radius is doubled
rrhj sec., and multiplied fivefold 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 doublesource. 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 " endon ",
and that under " k z " to motion " bioadsideon "
c\a &! k z
i (sphere)  
15 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 0029 "945
90 o 024 o 954
10 o 0021 0960
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 277 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 streamlines, and /r is called
the streamfunction
To find for the case of a
pointsource, 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 streamline 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 Streamlines
Theie aie vanous methods by which diawmgs of systems of
tream lines can be constructed. For example, suppose that the
trcamfunction 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, 02, 03, ... a system of
streamlines is leadily drawn The same numerical work comes in
useful in the case of (39) A similar process can be applied to tracing
the stieamlmes 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 righthand 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 streamline 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 threedimensional 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 lefthand 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 ;r8#)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  = fWV + 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 twodimensional 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 vortexline according to a general definition to be given
presently, and the vortexlines 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
crosssection, 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 velocityfunctions 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 vortexpair 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 coaxial circles having A, B as limiting points If we super
)ose a uniform flow K/zna in the direction of y negative, the case
s reduced to one of steady motion, and the streamfunction is now
277
, .
(99)
The streamline 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
ortexpair in the original form of the problem.
If a flat blade, e g a paperknife, held vertically, be dipped into
rater, and moved at light angles to its breadth for a short distance,
nd then rapidly withdiawn, a vortexpair 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 vortexpair 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 cooidinate 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 lighthanded as
regards the positive directions of the coordinate axes. Now, if
Fig 18
/, m, n be the diiectioncosines 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 vortexline. The vortexlines 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 vortexfilament, 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 crosssections of the filament,
we learn that the product of the resultant
vorticity into the crosssection 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 vortexring 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 voitexnng 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 vortexpairs.
Wave Motion
I. Canal Waves
Waterwaves are by no means the simplest type of wavemotion
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 wavelength 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
surfaceelevation is 77 will be
ch f ^
*=*+} .......... (103)
The pressure along the waveprofile, which is now a streamline,
is given by Bernoulli's equation
P
/ \ _ t>
I TI \
constant f# 2 g*rj = constant ik 2 ( i f ~) gq
\ 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 wavevelocity 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   = cL ........... (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 halfpotential
nd halfkinetic.
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 waveform 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. Deepwater 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 streamfunction 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 waveprofile may be a streamline 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 wavelength, i.e. the distance
between successive crests or hollows is A = 27rjk, so that
//#A\ , N
Vw ^ II4 ^
This gives the wavevelocity 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 wavelength, y = A,
= e~ zv = 000187. 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 semiaxes
coshk(y + h) smhk(y ~r h)
smhM ' smhkh
e horizontal and vertical, icspectively
The eneigy, per unit aiea of the surface, of deepwater 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, halfpotential
id halfkinetic The total energy per wavelength (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 waveresistance due to this cause is sometimes
experienced
The preceding theory of surface waves is restricted to the case
of a simpleharmonic profile It is true that any othei form can be
resolved into simpleharmonic constituents of different wavelengths,
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 wavelengths which are present with sensible amplitude
are all large compared with the depth, in which case there is a
common wavevelocity \/(gh) as found above.
3. Group Velocity
One consequence of the dependence of wavevelocity on wave
length is that a gioup of waves of approximately simpleharmonic
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 simplehaimonic trains of equal amplitude but
slightly different wavelengths, 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 groupvelocity If c is constant, as
when the wavelength 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 onehalf the wavevelocity The geneial
foimula, obtained fiom (119), is
U , . kh
This expression diminishes from i to  as kh increases from
o to oo.
The groupvelocity U determines the rate of propagation of
energy across a vertical plane. To take the case of deepwater 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 wavelength, by which the tiansition to
smooth water is effected. We infer from the piecedmg aigument
that the approximately simpleharmonic portion of the tiain is
lengthened only by half a wavelength 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 wavemotion 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 downstream side. In the former case the encigy spent in
producing the train measures the waveresistance expencnced by
the disturbing agency. If E be the mean energy per unit length of
the wavetrain, the space in front of the disturbance gains in unit
time the energy cE, whilst the energy transmitted is UE, where
U is the groupvelocity. The waveresistance R is therefore given
by
Rc= (cU)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 wavevelocity 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 cooidmates 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 ir 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 fullscale 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. Twodimensional 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 lefthand 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
2jrrdr = 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 onehalf its original value when
v&t  loga, or t = ........ ( IS o)
Putting v = 0134 (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 " yieldpoint " 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 wellknown
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 coordinates,
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 11 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 squareended tube opening into large vessels
at each end, the loss of pressure is approximately 112 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 squareended 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 oooi to 0014 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 enddisturbances 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 waterjacket, 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
constanthead apparatus, and the piessuiegaugc.
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 wideboie 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 tensionrods 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 constanthead 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 pressuiegauge K is of the oidmary U pattern, with meicury
as the woiking fluid A thieebianch pipe P connects the burette,
pressuregauge, and constanthead 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
lirpressure in B (fig 8) and thus draw the liquid under test into it,
lowering the suiface in A below the lowei platinum gaugepoint.
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 gaugepoints in A is taken by means of a stopwatch 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 17 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 shoittube 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 bestknown 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 cupandball 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
1575 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 cooidinates, 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
SyU = $y f ud% ~ I (^ %h)d%
J o 2ft OXJ o
_ ... A 8 3?
or U = *!.* (I7 )
Similarly the flow pei unit width in the y direction
^ 7 h z "dp , ON
is V =  ~. . . ...... (18)
I2[JiOy
Thus the total flow in any diiection across a unit width perpendiculai
to that diiection is equal to the rate of decrease of piessure in that
h 3
direction multiplied by the constant 
I2/A
The same relation evidently holds for the flow of a viscous liquid
in the space between two concentric, fixed cylinders, in either the
axial or the circumfeiential direction, provided that the ladn of the
cylinders are so nearly equal that their difference can be neglected
compared with either of them.
In both of these cases, as well as in all othei cases of flow between
parallel surfaces plane or curved, it is evident, considering any small
rectangular element S#, Sy, which extends in the z direction from
one surface to the other, that since the same amount of fluid must
flow out of, as flows into, the element in unit time,
= *
d s t> 2
or from (17) and (18)
VISCOSITY AND LUBRICATION 123
it being lemembeied that the surfaces # = o and % = h, are
assumed to be fixed.
Flow between Parallel Planes having
Relative Motion
If the plane % = h is moving parallel to the plane % = o, with
components of velocity % and z^ in the X and Y directions, uniform
rates of shear ~ and ~ in these two directions will be superimposed
h h
on the fluid velocities u and v of (15) and (i$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)
Gupandball Viscometer
The type of viscous motion which has just been discussed is
that on which is based the action of the cupandball 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 Cupandball Viscometer
the cup to fill the groove and interspace, and pressing the ball home,
to suspend the whole instrument and note the time by stopwatch
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 cupandbali 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 bearingbrass,
tirustbearing shoe, or engine piston). In fig. 14, let XY be axes
f coordinates (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
cooi 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 filmthickness, 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 tsjtM . I I */7'V* I 1 ^ 1
i r(, ^ _ i . w^ *, . , I S { J
117 f yini/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 ^ f7^
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 coordinates, 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 coordinates 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 ingbrass of the axle.
The lesults as to the fluid pressure which have been calculated
above evidently remain true if, instead of
the bearingbrass 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 stuffingbox, 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 bearingbrass 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 onethird of the
3
gth of the sleeve or bearingbrass 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 0431 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
12
02775
0333
04818
3285 X 103
14
03465
o 4851
o 4664
2428
16
03793
o 6079
04532
2 065
18
03955
o 7119
04416
1858
2
04026
o 8051
04313
I 722
2 2
04043
08895
04221
1025
24
o 4027
o 9665
04137
1553
26
03991
10375
o 4061
1497
28
o 3942
1 1037
03991
1451
30
03884
I 1652
o 3926
1414
40
03559
14237
03662
i 298
50
03247
I 6237
03465
i 239
60
02982
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 _ . , ,.,.,, ^TITMHTI
.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 = 22 . . . 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, = 22
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 = 22 is 278 X io~ 4 c. c. per second
<h.
per centimetre of the transverse dimension of the bearing.
VISCOSITY AND LUBRICATION 139
Selfadjustment 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. 1517, 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 selfadjusting, 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 = 04221 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 selfadjustment 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
Selfadjustment 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 semicylmdiical 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 lighthand 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 lefthand 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 lefthand 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 lefthand
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
ghthand 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 pressureproducing 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 selfadjustment 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 selfadjustment in
journal bearings with rigid (i.e. nonpivoted) 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 oilfilms, 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 semicylindrical 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
coordinate measured from the direction OO', the coordinates for the
ends of the bearingbiass being
~, and iA f  as indicated in
2 2
the figure. Thus the linear coordinate, 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,
rH
.
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,
Vrr/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 ifci . 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
II3
129
 098
X 004
X 091
1'5
135
 093
X 008
X o 92
24
133
088
X 014
X i oo
63
128
o 72
X 029
X 134
33'9
120
o 50
X 062
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
125 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. 131133.
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)HB' 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 0421 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 042% from the rear edge, as compared with
o 431%, in the infinite beaiing. (See Table III, p 136.)
The coefficient of friction is 103^. 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 onesixth
passes out at each of the sides and the remaining twothirds at the
rear edge.
Similarly, in the case of a bearing whose width transverse to the
motion is only onethud of its length, so that
a z ~" a i a \ ~ 3^
the total pressure is
ind the centre of piessure is 039%, 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 bearingshell 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 bearingshell, 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 twodimensional 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: 103.
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 twodimensional 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 coordinate measured from the middle circum
irence, and h the varying thickness, determined by
h = 8 j cos#.
Experimental Results
The curves given in the righthand 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 635
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 righthand
art of fig 24, make Fy'(P//xM 1 ) approximately constant. The
ifthand 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
lanesurfaced 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 coordinates 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 volumeflow in viscous liquids, and potential
and current in an electrical conductor The conductor used is a
conducting liquid contained within solid, nonconducting 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 wellloaded
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 onehalf of its value at entry,
ie conditions being otherwise the same as those for the fullline
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 lefthand 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 nonpivoted 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 lastnamed fluid, doubtless on account of the peculiarity
its viscositytemperature 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 (GautliieiVillars, 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
quaitzcrystal 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 Fluidfilm Lubrication " Trans
of Amer Soc of Mech Engineers, M S P 51/21, Sep Dec , 1929
) KINGSBURY, A " On Pioblems in the Theory of Fluidfilm 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
Streamline 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 streamline 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 sharpedged 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
STREAMLINE 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 streamline 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 streamline 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 wellknown 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 sharpedged 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 streamline 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 streamline 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 streamline 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.
STREAMLINE 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 twodimensional
v of a perfect nonviscous 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. 1434.
STREAMLINE ANt> TURBULENT FLOW 165
hich is Bernoulli's equation of energy for a perfect nonviscous
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 streamline flow of a nonviscous 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
ellmouthed 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 streamline 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 colourband method, and Reynolds took advantage of the
STREAMLINE 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 coordinates. 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 . . 20
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 smoothwalled 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 nonsinuous, 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 streamline to tuibulent motion
is marked by a sudden increase in the temperature of the discharge,
* Proc. Roy, Soc. A, 74,
STREAMLINE 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 258 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 footpou
second units, so that
>v k = _ zL ft.sec , wheie d is in feet.
While for air at o C.,
v = 1415 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 lefthand curve, coi responding to speeds belo
the critical value, is calculated from the foimula
,
d
theoietically conesponding to stieamhnc flow f It will be seen tin
the points for both air and water flow he closely on this cuive, an
that the breakdown of the streamline 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 , 191011, p 48.
AM
MLIf
VD
V
46
STREAMLINE 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 nonsinuous. 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 im 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 hotwire 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.
STREAMLIKE 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
Jtube 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 1259 times the difference of
;vel of the tops of the mercury counter
olumns By using an inverted
Jtube 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 096 to 0995,* 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 (1408 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, 19145, 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., 19167,
Part II, 204, p. 108.
STREAMLINE 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
097
08
096
09
094
I'O
091
Measurement of Flow by Diaphragm in
Pipe Line
The coefficients of discharge of standard sharpedged 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 sharpedged
*Proc. /Twit Mech E , Oct 1919, p. 593.
tProe. IKS/. C.E., 191617, Part II, p. 108.
176 THE MECHANICAL PROPERTIES OF FLUIDS
orifice in a plate of thickness oozd, and for ratios of d/D less than
o 7 is 0608 for water or air when p z /pi is greater than o 985 and is
equal to 0914 0306 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 094 in small nozzles to 099 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 welldesigned 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,
STREAMLINE AND TURBULENT FLOW
177
and for wind tunnel investigations. A tube of this type having
the dimensions shown gave K = ioo within i per cent.*
The tube illustrated in fig. 16 gave a value of C = 0926 when cali
brated by towing through still water, and 0895 when calibrated in
a 2in. 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 0916. 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
STREAMLINE 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 stillwater or stillair 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 086 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 2nrvdr
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
droelectric 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,
VV 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.
STREAMLINE 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 streamline 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 streamline 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 stieamlme 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 stieamlme 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, 19133.
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
STREAMLINE 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 185. 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
508
328
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
472
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
262
149
RffCnT,,,) Fluid
V,W
1235
10 5
108
65
o 0109
00155
00266
00267
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, 19123
STREAMLINE 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 ir
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 i6ju,C ft where C v is the specific heat at constant volume, and
C f . = , so that ~ = 088. In the case of water, however, at
i '4 k
20 C., ft. = ooi, k = 00014, cr io, 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 029, and that
similar experiments by Soennekei f require a mean value of 034.
Taylor, from an examination of data by Lorentz,J concludes that
the ratio is approximately 038. 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 014. 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 = 030 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 = 030, the ratio is 30 for watei at
20 C., and 038 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 /" = 000018
and V m 1850 cm. per second This makes U O'l^Vm
J Chapter V.
STREAMLINE 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 *riTl = *&= ^" a/ {(I + aTs)(l + ^ TJ} ' ' (l8)
here, in C G S units, a = o 004 and ft = ooi 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 00104 With flow in the other
tection k was, however, distinctly less, having a mean value of
proximately 00075.
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 OT 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 aircooled
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 05 and
io, 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
STREAMLINE 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 / 1n (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
_
oT VttT 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 streamline 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 streamline 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 nonturbulent 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 airship 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 175, and with shipshaped 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 stieamlme,
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 nontuibulent.
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 streamline 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 streamline 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 sharpedged orifice under the action
of gravity is an example of the first type of interaction, while the
resistance of an airship, 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 streamlines 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 nonaccelerated
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 , JVg .
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 camline 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 175 and 20 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
ra so
~ C
40
so
t Z 3456789
Values of v" Scale t division  2870 units
Fig i
meters ranging from 0125 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 5318 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 ...
04
A B Smooth p!ps diameter 4 33 cm
cc' Velocity Distribution In hvo rough pipes soscm Siisscm
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 08 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 streamline 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
8
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H
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10
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00
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<+$
" dj <D H M CO
3 H 0) <U "Cl
3 ft
O rr 1 J
H S . O
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 4ft 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 i6ft. board at this speed is
R = o 000457 X (3 <3 5) J 8s = 000402 Ib ,
and R <z; a = o 00000380
'he value of R v z for the 4ft. board in water is 000325, the
* Phil Mag., 8, 1904, pp. 5866.
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 airship. 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 = 1JL
V 20 7 OT '
o that with a i/io scale model, the corresponding speed in the
vmd tunnel would be onehalf 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 shipshaped body at
the speeds usual in piactice is proportional to v", where n is approxi
mately 183, and the nearness of this index to 20 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 wellknown " 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 wavemaking
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 wavemaking 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 + eddyiesistj _ __   p Ib
ance of model . J "~ '" J ^ V "  r 1D *
.'. Wave  eddyiesist^ ,
ance of ship in salt I = DF X ^~ Ib.
62 4
water . ' ^
Total resistance of 6 DF , fAv n lb
_
ship .. ../ 624
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 largescale
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 000126 and
10000007 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, 19101, 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 0002 in. to 125 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
24 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 > 19101, 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 streamline 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 152
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, 19112, 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 432 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 165 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
,11.
V P
Static
V P
V
Pressure
Attnaf
Tressure
Fig i Ideal Case of Waterhammer in Pipe Line
t = lrV 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 pressuretime 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 Waterhammer 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 = 432 X io 6 Ib sq ft.,
,, w 624 Ib c ft ,
g = 32 2 ft.sec. 2 ,
this becomes p = gi6ov Ib.sq ft.
= 6372; 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 castiron pipe line 375 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 Waterhammer 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
MW
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 crosssectional 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 028 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
637 >! Ib. per square inch above normal is transmitted to the June
tion of pipes i and 2. Here the pressure changes suddenly to 6372
above normal. This is maintained in the second pipe during th
passage of the wave, and is followed by a change of pressmc to 6372
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 637^2 an d of velocity ^ v z , to be reflected from the valve
a wave of zero velocity and pressure 637 {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 637^3 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 nonuniform 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 6m pipe
58 ,, 2
99 4
4 J "
ii ft of 6in. pipe
58 2
48 il
3 3
48 ij
4 i
82 ft of 6m pipe
66 , 4
4
I , 2
7
Calc
154
322
Piessures, Pounds per Squaie Inch.
Obs Calc Obs Calc
73
129
ist i jin pipe
ijin pipe
6 9
143
71
127
3in pipe
2jm 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
6m pipe
1 20
49
52
22
90
48
H9
62
645
36
II 2
66
223
82
97
52
16 7
158
466
122
2OI
99
35
368
* 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 2m. globe valve on
a 3fin. 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
oio of a turn the drop was 20 Ib. per squaie inch, while with 005
of a turn it was n Ib. per squaie inch In each case the time oi
opening was less than 013 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
4olhfyj"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 pumpingstation 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 wavelength 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 mstroke 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 springloaded 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
reexpand 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 wavelength,
d which is provided with branches at C, D, E, and B, respec
ely onequarter, onehalf, thieequarteis, and one wavelength
>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 halfwave 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 threephase system is usually employed, as giving
re uniform torque and ease of starting A threecyhndei gener
% having cranks at 120, gives vibrations to the fluid in thiee
es, which are received by the pistons of a threecylinder 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 * sm2rrn(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
jrdu
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 halfcycle.
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 horsepower 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 _ 624
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 alternatingcm rent electr
power transmission; in fact, in the " threepipe system " tl
known facts of threephase 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 coordinate 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 / \ .
 ~ Ijf^ 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 cooidmates 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 castiron 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
3tnon 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 " autocollimatoi " 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 Vneck B and a pinhole 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 autocollimator the soapfilm 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 autocollimator of the inclina
tion of the spherical film at a point on its edge.
Accuracy of the Method
Strictly speaking, the soapfilm 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 soapfilm 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 testplate 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 recj 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 autocolhmator 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 4m. 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 198
per cent. In the case of the average error is only 062 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
jo 7
3133
1456
M75
13
3528
1749
1741
+03
36 oo
0858
0848
+ 12
* On 4m 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
oapfilm 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 Lshaped 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 35 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 LSHAPED 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 1445
050 I 430
060 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 onefifth 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 angleirons.
No. i is the angleiron 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 onefifth of the thickness. This angleiron
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 /" '
angleiron 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
angleiron. Hence the sti esses at these points were deduced,
The results are given in Table IV.
TABLE IV
SHOWING THE RATE OF FALLINGOFF OF THE STRESS IN THE
INIERNAL CORNER OF THE ANGLEIRON
Distance fiom Ratio Stt ess at Point
Boundary * Boundary Sti ess m Arm"
Inches
000 189
O 05 i 36
O 10 I 12
O'2O 077
030 o 49
o 40 024
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
soapfilm methods, while m column 3 is given the conesponding
experimental results taken from Mr. Ritchie's papei .
Section
Angle:
Angle.
1175 X 1175 in.
ioo X i oo in.
Tee 158 X 158 m. _
Ibeam: 501 X 802 in.
Ibeam. 301 X 300 in.
Ibeam: 175 X 478 in
Channel 1 o 97 X 2 oo m.
TABLE V
C (Soap Film)
001234111 4
o 0044 m 4
01451 m 4
1 160 in 4
o 1179 m 4
00702 m. 4
00175 m 4
C ('Dnect Toision
Experiments)
o 01284 m 4
000455 m 4
o 01481 in l
1140 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 soapfilm
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 singlevalued 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 soapfilm apparatus
) not correspond with singlevalued 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
singlevalued 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 singlevalued is therefore
f&Fj , . .
~ds2&, ......... (14)
J
05)
if erring again to the soapfilm 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 singlevalued
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 soapfilm
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 reentrant angles at the bottom of the keyway, and they
frequently failed owing to cracks due to torsion which started at
the reentrant 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 58 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 soapfilm 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 03 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 greatcircle
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 tEaaf
&/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
crosssectional 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
righthand 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 Ihfem 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
selfrecording 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 treetops. 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 18811910
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.
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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 southwest of the station lies the
city, and we find that in this direction the surface wind has the
lowest percentage, while in the northeasterly directions the pei
centages are largest. Towards the northwest 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 snowcoveied and devoid of forests,
this wind often reaches gale force. Such a wind is known as a katabatic wind.
WIND STRUCTURE
269
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MNM
MSM
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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 halfkilometre 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 2Km. level
e coi relation coefficients gradually diminish until at the surface
actically no connection at all is found. Above the 2Km. 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
halfkilometre 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,
76 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 072 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 audessus 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 = 00023 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 0635
10 775 0338
12 585 0192
14 44 8 0116
16 349 o 069
18 273 0042
20 219 0027
22 167 00156
24 12 9 00094
26 99 o 0055
28 74 00031
30 55 o 0017
32 37 000085
34 26 o 00038
36 17 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 = 0338, which means that undei these
three conditions G must have the values o 9, 38, and 54 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 10
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 Eddymotion. (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
rtheast winds the gradient velocity is reached at 915 m., for
itheast below 300 m , for southwest 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
thwest below 300 m Also the winds in the northeast and
itheast quadrants show little or no increase after reaching the
(Strophic value, those in the northeast often showing a decrease,
lie those in southwest and northwest quadrants are marked by
ontmual increase beyond the geostrophic value, the velocity in
northwest being at 2500 m., 145 per cent of this value. The
iation in direction also differs according to the quadrant. In
northeast 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 southeast quadrant the direction of the isobars is
ched at 600 m , and above this level the wind veers still farther,
e southwest 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 northeast to
northwest, 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 westeast component than
in the southnorth, as is to be expected from the exposure of the
station. On the whole, the westeast component shows a tendency
to increase with height, while any eastwest velocity gradually dies
out. The southnorth diagram gives mainly negative values,
*Q.J. R. Met. Soe., 41, p. 133
WIND STRUCTURE 281
e. the winds observed had mainly a northsouth 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 northsouth 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 southwesterly 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 southwesterly
Here we have a warm current passing over a colder, and the
result is generally rain. Very often in summer theie is found
a southwest current passing over a southeast, the two being
associated with shallow thunderstorm depressions, the southwest
current supplying the moisture to form the cumulonimbus 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 northwesterly 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 northsouth 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 = 2869 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.JgJ>,
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 thejyaxis
drawn towards the north, the xaxis 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/iv J
>ng the tfaxis 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 pressuredifference per kilometre
icight may be written as
A P/AT AP\ , .
A* = 34'^r(^r  pJ, ........... (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 southwest 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 3ooom. 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 northeast to southwest, 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 southwest, especially about the 6oooJt. level. The velocities
however, are compaiatively small and therefore a breakup 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 appoximate 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
counterclockwise velocity at the centre of low pressure. When
actual example of a lapidly moving circular storm such as that
lothnth 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 xothiith
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&Afei 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
BII,, _ (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
270
104
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 26,
(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 waveform
is sawtoothed 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 + iSit 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 37 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 wavelength 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 wavelength 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 wavelength 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 141 times this amount, while the density
air p = 000129 8 m< P er cu bi c centimetre, so that the velocity
= A/ 4 = 33,000 cm. per second, or about 1085 ft. per
" 000129
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 22 X io 10 dynes pei squar
centimetre, and p 1028, 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 = LV,, 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 5913.
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 oi 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 0004 P er cent > or 000052 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
016 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.
78
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 fixedend reflector, and a perfect dis
placement receiver a freeend 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 pressuremea
ning device although it takes a small current, and an ammeter a
irrentmeasuring 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 Nondnec. 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 (twoear) 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
lefthand tiumpet. In this case it is evident that the delay of the
sound in reaching the lefthand tiumpet is balanced by the cxtia
delay between the righthand 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 LAMHIf 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 righthand trumpet first, sliding the
tre tube to the left increases the path from the righthand
upet and diminishes that from the lefthand 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,
SUMANDDIFFERE.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 sumand 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 sumanddifleience 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 sumanddifleience
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
nanddifference method can be used in the test loom to test
s equality and to select peifectly paued receivers either for bmauial
for sumanddiffeience 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 nonduectional 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 Bidirectional
Hydrophone
Fig 7 Polar Curves of Intensity for Bidirectional 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 MornsSykes
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 bidirectional hydrophones; but it has been found possible
get over this difficulty and to convert a bi directional into a uni
/ I
Fig 9 Unidirectional 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
mdirectional hydrophone. It does not, however, give such
nite indications of direction as the sharp minima of the bidirec
al form, and it is therefore better to couple a unidirectional
a bidirectional 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 unidnectional 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 fouicoie 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 " softiron " 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 softiron 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 Eshaped 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 HIDIKII
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 movingcoil 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 Movingcoil Tinmmittcr
werfactor, 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 powerfactor 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 Eformed
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 indiarubber slnps mseited which are compressed
the attraction of the magnet On switching on this tiansmitter
a loovolt 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 higlbpressure 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 soundranging 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 bidirectional 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 sumanddif
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 " airdrive " magnetophone of Mr, B. S. Smith
[7). It consists of a massive lead casing (4) fixed to the side of
lip, carrying a thick mdiarubber 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 highresistance 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 fouiway
changeover switch being interposed between the receivers and the
amplifier. Ship noises aie greatly diminished by fixing the lead
^^^*'^^/^ "'.;%,,
0 = fxx fvJ Ux^^H^ ~~
Fig 17 Airdrive 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 bidirectional
hydrophone, diagramrnatically shown in Rg. 8, and fig. 20 (see
plate facing p. 316) shows a single diaphragm bidirectional hydro
Ftitniff pagt )
SUBMARINE SIGNALLING
323
ihone converted into a urndirectional 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 rubberblock form or of one of the foregoing types, have been
nclosed in fishshaped 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 "multiplestation" and "wirelessacoustic" sound
mgmg respectively, but the former, although less convenient, was
le only one employed in the War, as it needs no coopeiation on
ie part of the sending station.
Multiple station Ranging. The multiplestation method
Fig 23 Soundranging 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 multiplestringed
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 soundranging 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 onethousandth 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 26TlIRM< DIAL PHONIC ClIRONOMGIliR
Facing page 336
3 28 THE MECHANICAL PROPERTIES OF FLUIDS
sist in this case of diaphragms with singlepoint 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 soundranging locator (fig 28, sc
plate facing p. 326). It consists of a long steel bar pivote
at one end on a ballbearing, 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
ballbearing 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 multiplestation method of sound ranging for
iting navigation in foggy weather, a ship desirous of being
rmed of its position calls up the nearest' soundranging 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 wirelessacoustic 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 welldefined 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 i8mile 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 Depthsounding Method
The method of Behm, called the " Echolot " or echosounding
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 firingkey, 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 firingkey 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 onetenthousandth 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, threephase 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
farreaching 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
depthsounding 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 piczoelectric 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 piezoelectiic 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
ighfrequency 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 highfrequency 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 shortcucuited 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 highfrequency 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
directreading 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 hydrographic 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 highfrequency 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 highvoltage 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 diydockmg 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 highfrequency impulse without
employing crystals. One simple method, which is fairly effective, is
to employ the ordinary hammer of the singleimpulse 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 Scrolltype 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 indiarubber " 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 motorboats 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 waterfilled tanks in
contact with the hull, and transmitting and receiving through the
ship's plates. Fig 38 is a diagram of the motorboat 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 depthsounding 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 motorboat 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 cablelaying 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 depthsounding 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 wellknown 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, T77588) and
Metz (by Didion, 183940), 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 uptodate 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 nowfamous 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 electromagnets,
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 electromagnets 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 dcetiomagnets 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 electromagnet 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 spimgwne 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 ovalshaped 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 ovalshaped 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 pinpoint 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 00017 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 20m 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 axiallymagnetized 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 tuningfork 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, MayJune, 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 coordinated 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
orless 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 windchannel 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 semiangle 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,
65 140 258 380 515 660 Kgm./cj
loo 129 220 330 470 630 ,,
* 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
158
194
10
098
125
152
28
o 62
081
I 01
125
3
090
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 socalled 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 highvelocity 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 moreorless
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
"TL^^ with a o 26caIibre
loimdcd point
In fig 5 the diag co
_ _ efficients of a i^cm.
 piojeclile with flathead
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 coefficient given m these curves may be
used with any selfconsistent 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 metrekilogrammesecond 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 moreorless 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 sharppointed 3 calibres radius
head than for a 55 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. 55 calibres
adius) than for a short one (e g. 2cahbres 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 iscm 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 8calibres radius head experiences
ily about half the resistance of one of acahbres i adius. Little
Ivantage appears to be gained, however, by lengthening the head
iyond 8cahbres ogival radius; thus 10 and izcalibres radius
2 ads are only slightly more effective than those of 8calibres 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 " streamlining " familiar to all. The suggestion has frequently
been made that artillery projectiles should have a tapeied (socalled
" streamline ") 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 " boattail ", 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 Boattail 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 boattail 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 Boattail
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,
MarchApril, 1923.
REACTION OF AIR TO ARTILLERY PROJECTILES 363
MUZZLE VELOCITY, 2600 FEET PER SECOND
Angle of Range (Yards), Range (Yards), '
Departure Flat Base Boattail.
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 boattail bullet con ^,
siderably outranges the flatbase **
Diillet. On the highangle 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 lowangle 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 boattail 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 Highspeed Airstruim I 1 xptnments
egligible gain at low angles, thus
n^Grmc fVi 1r<rr*ntKpeic tKdt thf Curve A Foi scal rad head and flat base
OnlirmS tile nypOtneSlS mat me CutveB ^. Same hend as A, boattail base, taper,
hor>A f\f tV>P KIOP 1 of OTPfltfM* '5 Curve C Same head as A, boattail biae,
nape Or me DaSC IS Oi grcdici tape , )7 o CwveD Same head as A, boattail
nnmfonrp. 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 highspeed 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, MayJune, 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 boattailing
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 socalled
" 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 timefuze 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 selfconsistent 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 tirnefuze 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 Gcal.rad Head of a i 3In 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 3in 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 3m piojcctilc with a
2calibres radius head and o i5calibie 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 065, 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 turstienm 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 highvelocity 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
crosssectional 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 onedimensional 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 coordinate the results of airstream 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 crosswind
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 nonzero 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 selfconsistent 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 14 to 50. 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, 298344
ir, clouds in, 40
Displacement, 9
Resistance coefficients, 207.
Viscosity, 118
vogadro's hypothesis, 43.
alhstic pendulum, 34851.
arometei, the first, 23
aum hydiometer, n
eanngs, cylindrical, 1426.
Flexible, 1556
Lubrication, 13358
Pivoted, 14855
Selfadjusting, 13942
Width, 1469
ernoulh's equation, 579, 96
maural method of sound icception,
30610
oiler tubes, gases in, 183
oyle, barometer experiment, 23
Law, derivation of, 42
rass, acoustic propeities, 304
ndgman, highpressure expenments,
1 68
ronze, acoustic properties, 304.
ubble, collapse of, 75
allendar's equation, 49
apillary tubes, action in, 25.
Rayleigh equation, 27
Viscometer tubes, 1126
'arbon bisulphide, viscosity, 118.
'ast iron, acoustic propeities, 304.
'astor oil, compressibility, 219
Viscosity, 1 1 8
hannels, flow in, 17980
harles's law, derivation of, 423.
lausius equation, 478
louds, suspension of, in air, 40.
eofficient of viscosity, 32, 1034, 1179
ollision, prevention of, by echoes,
3323
Compiessibility of liquids, 159, 23,
212, 2189
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, 16572
Crystals, elasticity of, 45
Cube, piessure on, 45
Cupandball viscometer, 1258
Cylinders and wires, icsistance of,
2147
Deformation and rigidity, 34.
Density, 713
Diaphragm soundei ,318
Dietenci equation, 478
Dimensional homogeneity, 187203
Diiectional acoustics, 30612
Directional sound receivers, 3113
Displaced air, correction factoi, 9
Drops, deteimination of weight of, 289
Dynamical similarity, 193203
Echo depthsounding geai, 33544
Echo, detection of ships by, 3323
Eddies and turbulence, 166
Elasticity of crystals, 45
Elasticity, phenomena due to, 21836
Electrical measurement of pressure, 17
Energy at surface of liquids, 1929
EotvSs equation, 23
Equations of motion, 36
Equations of state, 4149
Errors in soap film experiments, 2447.
Explosions under water, 76
Fessenden sound transmitter, 317, 333.
Flow, measurement, 1728
Streamline, 16090
Sudden stoppages, 21921,
377
378
THE MECHANICAL PROPERTIES OF FLUIDS
Flowmeters, 1738
Diaphragm, 1756
Pitot tube, 1769
Venturi, 1725
Fluids, definition, 12, 6.
Motion, 56101.
Pressure, 17
Solids irm^ersed in, 33.
Friction in pipes, 2213.
Gas, perfect, definition of, 6.
Gas equation, 3.
Gases, liquids, and solids, classification,
12
Glycerine, viscosity, 118
Heat transmission, flow effects, 18090.
Hydrodynamics. See Streamline Mo
tion
Hydrodynamical resistance, 191217
Hydrometers, calibration defects, 102
Hydrophones, 3056, 3204
Bidirectional, 312.
Ctube, 3201.
Constructional details, 3223.
Magnetophones, 3212.
MornsSykes, 312
Unidirectional, 313
Incompressible fluid, equations of, 36
Irrotational streamline motion, 62,
6571
Kinetic theory of gases, 414.
Laminar motion, 103
Lead, acoustic properties, 304.
Liquids, compressibility, 139
Definition, 6.
Molecular viscosity, 39
Surface energy, 1929
Surface tension, 1929
Liquids, gases, and solids, classification,
12
Lubrication, 12859
Bearings, 13358
Bibliography, 1589
Inclined planes, 1313.
Viscous, 12931
Lubricating oils, compressibility, 219
Viscosity, 1189.
Matter, states of, 12
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, 18090
Motion of fluids, mathematics of, 56
101. See also Streamline Motion,
Laminar Motion, Vortex Motion,
Wave Motion
Naval architecture, model experiments,
2091 i
Nitrogen, Amagat's experiment with,
44
Oils, compressibility, 219.
Viscosity, 1189
Olive oil, compressibility, 219
Osmosis and osmotic pressure, 504,
Ostwald's viscometei, 39
Parallel planes, viscous flow between,
11925
Permeability, 50
Petroleum, compressibility, 219.
Pipes, critical velocities, 16872.
Elasticity, 2236
Flow resistance, 198203.
Flow stoppage, 2267
Friction, 2213
Opening valves, 2289
Pitot tube, 59
Poiseuille's equation, 378
Power transmission, acoustic, 3334
Prandtl's analogy, 23942
Pressure, 56
Electrical measuiement, 17.
Viscosity, 1179
Projectiles, air density, 359
Ballistic pendulum, 3468
Base shape, 3624
Bashforth chronograph, 348
Boattail bullet, 3624
Cranz's ballistic kmematogi aph, 353
Drag coefficient, 359, 370
Kiupp's 1912 experiments, 3513
Piessure distribution on, 3657.
Reaction of an on, 34576
Scale effect, 3589
Shape, 3604
Spark chronograph, 3503
Spinning, 3716
Yawing, 3716
Pyknometei, descuption of, 910
Rape oil, viscosity, 119
Rayleigh's equation for capillary tubes,
27
Redwood's viscometer, 116
Resistance and compressibility, 212
Resistance of square plates, 2112.
Resistance of submeiged bodies, 206
9
Resistance of wires and cylinders,
2147
Rigidity and deformation, 34
Rubber, acoustic properties, 304.
Scale effects, 2113
Shearing of a cube, 45.
SUBJECT INDEX
379
hips, resistance of, 20911.
irens, undei water, 319.
km fuction, 2036
oap film stress deteimmations, 23752
Contour mapping, 2447.
PrandtPs analogy, 240
Sheai stress in twisted bar, 240.
Toique on twisted bar, 241.
Twisting of bais, 2379.
Warping of sections, 238
olid, scientific definition of, 5
olids, liquids, and gases, classification,
12
olids and fluids, mtei action of, 195
olutions and solvents, 504
ound, nature of, 299.
ound ranging, 32533
Depth sounding, 3302
Leader geai, 330
Multiple station system, 3259
Wixeless acoustic method, 32830
ound icceivers, 3046
ound transmission, 3014, 3168
ound velocity, 300
Wavelength, 301
ources and sinks, 7581
pecific gravities, air displacement, 9.
perm oil, compressibility, 219
Viscosity, 1 1 8
pheie in viscous fluid, 33
quaie plates, resistance of, 2112
tate, equations of, 419
teel, acoustic properties, 304
tethoscope, action, 3069
Cokes' foimula, 41
tone's viscometer, 1146
tream function, definition of, So
treamline forms, typical, 2167
treamlme motion, 5785, 16090
Application to naval aichitecture, 91
Axial symmetry, 7581
Bernoulli's equation, 579.
Circulation, 61
Continuity equation, 601 .
Critical velocity, 16572
Equations, 835
Hele Shaw's expenments, 1625
Iirotational, 62, 6571.
Stability, 1 6 12
Steadiness, 635
Tracing stream lines, 813.
Tubes, 162
Turbulent flow, 16190
Twodimensional, 602
Velocity potential, 714
Vortex rings, 75
Vorticity, 612
reamtube, definition of, 57
ress determinations from soap films
See Soap Film
ruts, strearnlme, 2167
ibmanne signalling, 298334
Submeiged bodies, resistance of, 206
9
Sumanddifference method of sound
reception, 3101.
Surface tension, 1929.
Tempeiature effects, 3, 202, 39, 44
Thermodynamics of compression, 158.
Timber, acoustic propeities, 304
Torsion, examination by soap films,
23754
Trotter oil, viscosity, 119
Tubes, converging, 1712
Streamline motion in, 162
Viscosity in, 10912
Turbulent and streamline flow, 16090
Twodimensional motion, 602
Valves, sudden closing of, 21921, 226
Sudden opening, 2289
Van der Waals' equation, 46, 48
Velocity, critical, 16572
Measurement, 17280
Ventun meter, 1725
Viscometeis, 359
Capillary tubes, 1126.
Cupandball, 1258
Redwood, 116
Secondary, 1167
Stone's, 1146
Viscosity, 3141, 10228
Bibhogiaphy, 1589
Bounding surfaces, 1078
Coefficients of, 32, 1034, 1179
Equations, 979
Laminar motion, 103
Lubrication, 10259
Measuiement, 346
Paiallel planes, 11925
Piessure vanation, 1179
Relative motion, 1235
Relative velocities, 1057
Solids, effect of, 33
Temperature effects, 39
Tubes, flowin, 10912
Twodimensional cases, 99101
Velocity giadient, 32
Voitex motion, 859
Isolated vortices, 879
Persistence, 857
Rings, 75, 859
Water, acoustic properties, 304
Compiessibihty, 2189.
Density, 9
Viscosity, 1 1 8
Waterhammer, 21923.
Wave motion, 8997.
Canal waves, 8991 .
Deepwater waves, 914.
Group velocities, 947.
380 THE MECHANICAL PROPERTIES OF FLUIDS
Superposed liquids, 934
Transmission of energy, 22936.
Wind structure
Altitude and velocity, 27681, 28,
Anemometer records, 2634.
Anticyclone, 2601.
Clouds, cumulo nimbus, 282
Cyclone, 2601, 28694
Eddy theory, 2727.
Egnell's law, 284
Geostrophic component, 263.
Giadient wind, 2601.
Rain, cause of, 282
Stiatospheie, 281
[9. Strophic balance, 262
Surface winds, 26671.
Troposphere, 281.
Wind variation, 2636.
Wires, heat dissipation fiom, 183
Wires and cylinders, resistance of,
2147.
ii f
NAME INDEX
Akerblom, 274.
Allen, 191.
Archbutt, 158
Archimedes, 8n.
Avogadro, 43
Bairstow, 203, 213, 365
Barnes, 1689
Bashforth, 34752
Bassett, 36.
Baume', n
Beauchamp Tower, 158
Behm, 3312
Bennett, 23
Bernoulli, 578, 63, 65,
901, 96, 165
Bessel, 148
Booth, 20313.
Borda, 59
Boinstem, 63
Boswall, 159
Boyle, 23, 7, 42, 44
Bndgman, 169
Bailie", 3025
Bnlloum, 158
Broca, 305, 320
Bryant, 186
Buckingham, 213.
Bunsen, 115
Callendai, 49
Canovetti, 212
Carey Foster, 17.
Carothers, 158
Cave, 2812.
Charles, 43
Clausius, 489
Clement, 169
Cleik Maxwell, 19.
Coker, 1689
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, 3467, 352
Dines, 212, 268, 271
Huyghens, 357
Dobson, 274, 27981
Hyde, 119, 159, 2189
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, 35060
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, 20910
Lempfert, 293
Lock, 368, 371.
Lorentz, 186.
Love, 238.
Gallop, 368, 371
Gibson, 35, 171, 177, 180
Macleod, 22
2001, 223, 225, 250
Goodman, 158
Manson, 330,
Marks, 18
Giassi, 218
Marshall, 186.
Gray, 314
Marti, 330
Griffith, 242
Martin, 159.
Gnffiths, 2523.
Grmdley, 171
Giimbel, 159,
Martini, 219.
Mason, 314
Matthews, 23.
Maxwell, 108, 119, 133
HSlstiom, 7
Mayewski, 350
Hartree, 365.
Michell, 14650
Hele Shaw, 1624
Mitchell, 23.
Hellmann, 271
Morns, 312
Helmholtz, 100
Morse, 3189, 330.
Hersey, 159
Munday, 3145.
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, 2003
ions, 218.
dn, 9.
fer, 51
ee, 314
t, S9> *72
emlle, 379, 1045,
C2, 1x6, 158, 200, 234
er, 52, 3H
itmg, 434, 230
vdtl, 23940
icke, 219,
kine, 81.
Rayleigh, 27, 102, 108,
146, 159, 187, 3578.
366.
Redwood, 116
Reynolds, 1289, 139, 142,
146, 158, 161, 1659,
1813, 186, 200
Richards, 23.
Richardson, 333.
Richmond, 368, 371,
Ritchie, 250.
Robms, 346
St. Venant, 238
Shaw, 262, 28896.
Shore, 159
Smith, 311, 318, 321.
Soenneker, 186
Sommerfeld, 1425, 158
Sprengel, 9
Stanford, 10
Stanton, 170, 1817, I9 1 .
2003, 2123
Stephenson, 330,
Stokes, 33, 158, 164
Stone, 114, 159.
Stoney, 159
Stuim, 329.
Sykes, 312.
Tait, 5, 218
Tayloi, 1856, 242, 2523,
264, 2723, 275
Thomson, 218, 230.
Torncelh, 58
Twaddell, 10
Van der Waals, 46, 48.
Van't Hoff, 51
Ventun, 172.
Vieille, 370.
Walser, 31423.
Watson, 230
Webster, 36,
Weston, 227
Wilhelm, 362
Wood, 300, 326.
Young, 5, 48
Zahm, 205,