# Full text of "The Mechanical Properties Of Fluids"

## See other formats

Applied Physics Series THE MECHANICAL PROPERTIES OF FLUIDS > t M | (H/rt^A s , , , ,-- )i<i4 u )( .r ( " , 4 ,'"'J '" . U S BLACKIK & SON LIMITKD 66 Chando* Pface, lONtX)N i/ tatatili(n htitct, Gl.AMjOW ni.ACKlE & SON (IND!A) LIMiTKil Warwick Huuw, I-urt Httu't, BLACKIK & SON (CANADA) UMtH I) TOHUNHI THE MECHANICAL PROPERTIES OF FLUIDS A Collective ffiork by C. V. Drysdale F. R. W. Hunt ,-^^^:==;-.JteJBaB 1. ^ D Sc.(Lond ) M A,, B Sc Allan Ferguson Horace Lamb M A , D St (Lond ) LL D , Sc D , F R S A. E. M. Geddes A. G. M. Michell O B E , M A , D Sc M C E , F R S A. H. Gibson Sir Geoffrey Taylor D Sc , M Inst C E. M A , I R S. Engineer Vice-Admiral Sir George Goodwin K.C B , LL D. SECOND EDITION BLACKIE & SON LIMITED LONDON AND GLASGOW BOOK PRODUCTION WAR ECONOMY STANDARD THE PAPER AND- BINDING OF THIS BOOK CONFORM TO THE AUTHORIZED ECONOMY STANDARDS Fiist ^ssue, 1923 jKejjnnled 1025 Second edition, tevwd and cnlaiged, 1036 Repmtcd, with comclions, 1937 Mepnnted Wd4, 1946 Printed m Great Britain by Blackie & Son, Limited, Glasgow PUBLISHERS' NOTE In recent years a great many researches have been made ito the mechanical properties of fluids by physicists and igineers. These researches are of the utmost practical iportance to engineers and others, but it is not unusual find that the people who are called upon to apply the suits m industry have considerable difficulty in finding mnected accounts of the work It is hoped that this Election of essays, ^many of which are written by men 10 are the actual pioneers, will prove of use in making e recent discoveries in the mechanical properties of fluids are generally known The mathematical notation has en made uniform, and the different chapters have been Hated as far as possible. CONTENTS Introduction By ENGINEER VIOE-ADMIEAL SIE GEOEGE GOODWIN, KO.B., LL.D. itioduction CHAPTER I Liquids and Gases By ALLAN FEEGUSON, MA, DSc(Lond) efinitiona Density Compressibility Surface Tension Viscosity Equations of State Osmotic Pressure 1 CHAPTER II Mathematical Theory of Fluid Motion By PEOFESSOR HORACE LAMB, LL.D., ScD, F.R.S. ream-line Motion Vortex Motion Wave Motion Viscosity - - 56 CHAPTER III Viscosity and Lubrication By A G. M. MIOHELL, M.O.E., F.E.S. VISCOSITY. Laminar Motion Coefficient of Viscosity Relative Velocities Conditions at the Bounding Surfaces of Fluids Motion Parallel to Bounding Surfaces Viscous Flow in Tubes Use of Capillary Tubes as Viscometers Secondary or Commercial Viscometers Coefficients of Viscosity of Various Fluids Variation of Viscosity Vli viii CONTENTS Page with Pressure Viscous Plow between Paiallel Planes Mow between Parallel Planes having Eelative Motion Cup-and-ball Viacometer - 102 B LUBRICATION The Connection between Lubrication and Viscosity Essential Condition of Viscous Lubrication Inclined Planes Un- limited in One Direction Applications to Actual Bearings Self- adjustment of the Positions of Bearing Surfaces Self-adjustment in Journal Bearings Exact Calculation of Cylindrical Journal and Beaimg Approximate Calculation of Cylindrical Bearings Plane Bearings of Finite Width Cylindrical Bearings of Finite Width- Experimental Eesults Types of Pivotal Bearings Flexible Bearings Limitations of the Theory Bibliography 128 CHAPTER IV Stream-line and Turbulent Flow- By PROFESSOR A H GIBSON, D Sc. Stream-line Motion Stability of Stream-line Motion Hele Shaw's Experiments Critical Velocity Critical Velocity in Converging Tubes The Measurement of the Velocity of Mow in Fluids The Ventun Meter Measurement of Flow by Diaphragm in Pipe Line The Pitot Tube The Effect of Fluid Motion on Heat Transmission- Application of the Principle of Dimensional Homogeneity to Problems involving Heat Transmission - ..... 160 CHAPTER V Hydrodynamical Resistance By PROFESSOR A H GIBSON, DSc. Dimensional Homogeneity and Dynamical Similarity Eesistance to the Uniform Flow of a Fluid through a Pipe Skin Friction Besistance of Wholly Submerged Bodies Eesistance of Partially Submerged Bodies Model Experiments on Eesistance of Ships Scale Effects Eesiatance of Plane Surfaces, of Wires and Cylinders, of Strut Sections Resistance of Smooth Wires and Cylinders ... 19] CHAPTER VI Phenomena due to the Elasticity of a Fluid By PROFESSOR A. H GIBSON, D.Sc Compressibility Sudden Stoppage of Motion . Tdeal Case Effect of Friction m the Pipe Line Magnitude of Eise in Pressure, following Sudden Closure Effect of Elasticity of Pipe Line Valve Shut CONTENTS Suddenly but not InstantaneouslySudden Stoppage of Motion la a Pipe Line of non-Uniform Section Sudden Initiation of Motion- Wave Transmission of Energy Theory of Wave Transmission of Energy CHAPTER VII The Determination of Stresses by Means of Soap Films By Sir GEOFFREY TAYLOR, M.A., FJt , Prandtl'a Analogy Experimental Methods Accuracy of tlio Mothod Example of the Uses of the Method Comparison of Soap Film Results with those obtained in Direct Torsion ExperimentsTorsion of Hollow Shafts Example of the Application of the Soap film Method to Hollow Shafts 237 CHAPTER VIII Wind Structure By A E. M. GEDDES, QBE, MA, I) So. Wind Structure 200 CHAPTER IX Submarine Signalling and the Transmission of Sound through Water By V DRYSDALE, D So (Loml ) Fundamental Scientific Principles Velocity of Propagation Wave length Transmission of Sound through Various SubMlances- - Pressure and Displacement Receivers Directional Transmission and Reception PRACTICAL UNDERWATER TRANSMITTERS AND HKOBIVKUS , SUBMARINE TRANSMITTERS OR SOURCES OF SOUND Electromagnetic Transmitters Submarine Sirens RECEIVERS OR HTDROPHONKS - The C Tube Magnetophones PRACTICAL CONSTIUJOTIQN on 1 HYDRO- PHONES DIRECTIONAL DEVICES Sound Banging Leader Q oar- Acoustic Depth Sounding Ecbo Detection of Ships and Obstacles ACOUSTIC TRANSMISSION os 1 POWER Developments in Echo Depth- sounding Gear 208 x CONTENTS CHAPTER X The Reaction of the Air to Artillery Projectiles By F. E. W. HUNT, M.A i'age Introduction THE DRAG Early Experiments. The Ballistic Pendulum The Bashforth Chronograph Later Experiments Krupp's 1912 Experiments Cranz's Ballistic Kmeinatograph Experiments with High- velocity Air Stream The Drag at Zero Yaw The Drag at Low Velocities The Drag at High Velocities The Scale Effect Dependence of the Drag on Density (The Function f(v/a, vdjv Shape of Projectile The Base The Pressure Distribution on the Head of a Projectile The Effect of Yaw on the Drag The Drag Coefficient Concluding Eeniarks EEACTION TO A YAWING, SPIN- NING PROJECTILE The Principal Eeactions The Damping Eeactions 345 INDEX 377 INTRODUCTION BY ENGINEER VICE-ADMIRAL SIR GEORGE GOODWIN, K.C.B., LL.D. (Late Engineer-in-Chief of the Fleet) To those engaged in the practice of engineering, and ble and willing to utilize the information that mathematical ad physical science can offer them, it is of great assistance ) have such information readily available in direct and ilevant connection with the problems with which they e confronted. The collective work of this book, issuing as it does om authors highly qualified and esteemed in their respec- ve fields, whose views and statements will be accepted as ithontative, supplies concisely and consistently valuable formation respecting the mechanical properties of fluids, id elucidates the evolution of many successful practical >plications from first considerations. Much has been done in this direction in regard to solids, id this has been assimilated and usefully applied by any; but much less has been produced on the subject of lids, especially in compact form, and this collective work ill doubtless on this account be very welcome. The necessities of the war brought us face to face with any new problems, a large number of which required >t only prompt application of the knowledge available, xii INTRODUCTION but intensive research and rapid development in order to comply with the constantly Increasing standards of quality that were demanded. Most of the results are well known: the principles by which they were reached, especially in regard to fluids, are perhaps only vaguely understood, except by a few. The results are certainly appreciated, but further application is probably hindered in many direc- tions for want of this knowledge. The several contributions to this work enunciate clearly the principles involved, and indicate that a wide field is open for the application of these principles to those who are engaged in industrial avocations and pursuits, as well as to others whose duties continue to be confined exclusively to preparation for war. Chapters dealing mainly with theoretical considerations form a prelude, clarifying ideas of the physical properties of fluids and providing a sketch of the mathematical theory of fluid motion, with indicators to practical utility. The sections devoted to practical applications are de- veloped from the underlying theoretical and mathematical considerations. The retention of this method of treatment of the several subjects throughout the work is a valuable feature. These sections will interest a variety of readers, some parts being of particular value to specialists such as the gunnery expert, the naval architect, and the aeronautical engineer, but by far the greater portion of the book will be of general interest to the large body of engineers who have to deal with or use fluids for many purposes in theii everyday work. The chapter on viscosity and lubrication should point the way to the better appreciation and further application of the correct principles of lubrication. The best known present application, that of enabling propeller thrust loads of high intensity to be taken on a single collar, has been highly successful, and it is gratifying to observe that other develop- INTRODUCTION xiri nents are already In contemplation, and that some are well dvanced. The description of the determination of stress by means f soap films is fascinating and deeply interesting; and it is heering to know that certain forms of stress in members f irregular form under load, not amenable to calculation, nd hitherto not determinable and therefore provided for by factor of safety, can now be closely approximated to by scperimental means, and it may be hoped that this or some ther experimental process can be extended practically in the ear future to determine other forms of stress. Success in lis direction would be directly attended with economy of taterial and would facilitate design. The chapter on submarine signalling indicates the arch of progress in a new branch of engineering, and e author makes the important and significant remark that e science of acoustics shows signs of developing into the igmeermg stage, a statement worthy of the careful con- ieration of all thoughtful engineers. The section dealing with the wave tiansmission of energy mes opportunely in view of the laige number of practical ^plications of this form of power transmission that are ing developed, and of those that have matured The same ction gives information respecting the principles govern- 2j the various foims of flow-meters, and should prove eful to engineers associated with high-powei installations 10 are, by reason of the magnitude of the individual in- illations, being forced to use flow measurements in lieu the definite bulk measurements hitherto favoured by my, and should give a greater confidence in the use and curacy of flow-meters designed on a sound basis. The preceding cases are merely mentioned as examples; sry chapter contains a great deal of matter of practical )ical engineering interest connected with the mechanical )perties of fluids. I have selected these examples as some xiv INTRODUCTION of those familiar to me in which I have personally felt the want of some preparatory and explanatory information, such as that given in this book; and it is my recollection that such information was more difficult or inconvenient to obtain in regard to fluids than for solids. My own experiences must, I feel, be those of many others. The whole series of articles has been to me most interest- ing, and they show clearly that engineering in the present day requires a great deal of help from pure and experimental science, and is adapting itself to the utilization of branches of science with which it has hitherto not been closely associated. Engineering practice to be worthy of the name must keep itself abreast of and well in touch with those sciences and the developments and discoveries connected with them. This is an onerous task and can only be effected collectively; it is too big for one individual; but works such as this will tend to ease the burden, and convert the task into a pleasing duty. LIST OF SYMBOLS USED p or P, pressure. v or V, volume; velocity. N, modulus of iigidity. E, Young's modulus. t, temperature C. (or F.), time. p, density. s, specific giavity. K, bulk modulus; eddy conductivity T, absolute temperature in C. T, absolute temperature in F. C^, specific heat at constant volume. C p , specific heat at constant pressure 7, surface tension. jS, compressibility. M, molecular weight. m, mass. /*, viscosity. j>, kinematic viscosity z= . P u, v, w, component velocities; displacements N, Avogadro's constant tu, specific volume of water; cross section; angular velocity. S, shearing stress. 5E, twist. T fl , torque A, wave-length; film energy. R, acoustic resistance. , frequency (cycles per second). THE MECHANICAL PROPERTIES OF FLUIDS CHAPTER I liquids and Gases Definitions We propose to discuss in this chapter some of the moie important leial piopeities of fluids Common knowledge enables us to ociate with the teims fluid, liquid, vapour, gas, certain properties ich we regaid as fundamental, and which serve to difteientiate 'se forms of existence from the form which we know as solid A id is, etymologically and physically, that which flows, and the uid or the gaseous state is a special case of the fluid foim of exist- :e A liquid, in geneial, is only slightly compiessible and pos- ses one free bounding surface when contained in an open vessel. gas, on the other hand, is easily compressible under ordmaiy :umstances, and always fills the vessel which contains it * The st elementary observation forces upon our notice distinctions h as these just mentioned, but it still remains to be seen whether se can be made the basis of a satisfactory classification Indeed, it is doubtful whether we can make a classification which I conveniently pigeon-hole the different states of matter, for, as shall see in the sequel, these different states shade over, under cial circumstances, one into the other, without the slightest ach of continuity. Ordinarily the change from solid to liquid as when ice becomes * But compare the quotation on p. 2. 2 THE MECHANICAL PROPERTIES OF FLUIDS water or fiom liquid to vapour as when watei boils is quite shai p, and the propeities of any one substance m the three states aie cleaily marked off. But substances such as pitch or sealing-wax behaving under some circumstances as solids, under others as liquids aie distinctly troublesome to the enthusiast for classification. Thus a bell or tuning fork, cast from pitch, will emit a note perfectly clear and distinct as that given by a bell of metal. Nevertheless a block of pitch, left to itself, will in time flow like any ordinary liquid Steel balls placed on the top of pitch contained in a vessel slowly sink to the bottom, and corks placed at the bottom of the vessel will in time appear at the upper surface of the pitch. Such anomalies serve to emphasize the difficulties attendant on any attempt at a rigorous classification. Indeed it is sometimes held that the diffeience between the solid and liquid states is one of degree, and that all solids in some measuie show the properties of liquids Howevei this may be, it is enough to note now that the diffeiences between the solid, liquid, and gaseous states are sufficiently pronounced to make it convenient to attempt a classification which shall emphasize these differences We shall therefore discuss certain properties of matter which serve to define ideal solids, liquids, and gases. We shall find that no substances in nature confoini to our ideal, which will therefoie be but a first approximation to the truth, and later we shall find that small corrections, applied to the equations of state which are the expression of our fundamental definitions, will serve to make the equations represent with considerable accuracy the behavioui of actual substances. This process, involved though it may appear, is both historically correct and physically convenient Thus the reader may remember that m 1662 the Honourable Robert Boyle took a long glass tube " which by a dexterous hand and the help of a lamp was in such a manner crooked at the bottom that the part turned up was almost parallel to the rest of the tube, and the orifice of this shorter leg . . . being hermetically sealed, the length of it was divided into inches . . . Then putting in as much quick- silver as served to fill the arch, . . we took care, by fiequently inclining the tube, so that the air might freely pass from one leg into the other, . . . (we took, I say, care) that the air at last included m the shorter cylinder should be of the same laxity with the rest of the air about it. This done, we began to pour quicksilver into the longer leg, . . till the air m the shorter leg was by condensation reduced to take up but half the space it possessed (I say possessed not filled} before; we cast our eyes upon the longer leg of the glass, . . LIQUIDS AND GASES 3 and we observed, not without delight and satisfaction, that the quick- silver in that longer part of the tube was 29 in. higher than the other."* The pressure and volume of a gas at constant temperature are therefore in reciprocal propoition; that is, at constant temperature L he equation of state of a gas is given by pv = k. Succeeding experiments emphasized the truth of this result, and it vas not until instrumental methods had advanced considerably that imall deviations from this law were shown to exist under ordinary onditions It was then proved that an equation of the type = k nore closely represented the behaviour of even the more permanent ;ases, later work has shown this equation does not represent with ufficient accuracy the icsults of expenment, and vaiious other quations of state have, from time to time, been proposed To these quations we shall latei have occasion to icier Again, the physical convenience of such a method of appioach may be illustrated by lesults deduced from the principles ol rigid ynamics No body in naluie is peifcctly ngid that is, is such that line joining any two pai tides of the body remains invariable in ingth dm ing the motion of the body but considerable simplification f the equations of motion results if we make this assumption, and le results obtained are in many cases of as high an 01 der of accuracy s is required. We can, if necessary, obtain a closer approximation D the truth by consideimg the actual defoimation suffered by the ody the problem then becoming one in the theory of elasticity, uppose, for example, that it is oui object to deduce the acceleiation ue to giavity fiom observation of the period of a compound pendu- im It would be possible to attack the problem, taking into account b initio such effects as are due to, say, deformation of the pendulum i its swing, yielding of the supports and the like, afterwards ne iecting such effects as expenence has shown to be very small. But ich a method would lender the problem almost unbearably complex, esides tending to distract attention from the essentials, s and it is 3th moie convenient and moie philosophic to focus one's mind on ie more impoitant issues, to solve the problem first for an ideal * Boyle's works (Birch's edition, Vol. I, p. 156, 1743). 4 THE MECHANICAL PROPERTIES OF FLUIDS rigid body, and afterwards to introduce as small corrections effects due to elasticity, viscosity, and so forth. This, then, is the course which we shall follow in discussing the properties of fluids, and we shall seek, using elastic properties as a guide, definitions which will emphasize those differences which undoubtedly exist between the solid, liquid, and gaseous states of existence. If we wish to describe completely the elastic behaviour of a crystalline substance, we find that in the most general case twenty-one coefficients are required. For isotropic substances, fortunately, the problem is much simpler, and the coefficients reduce to two, the bulk modulus (K) and the rigidity modulus (N) These coefficients are B h (b) (c) Fig i easily specified. Thus, if a cube of unit edge be subjected to a uniform hydrostatic pressure P, so that its volume deci eases by an amount Sv, the sides of the cube deci easing by an amount e, then, $v being the change in volume per initial unit volume, the ratio of stress to strain, which is the measure of the bulk modulus, is given P by = K, and, to the first order of small quantities, $v = 32. ov Suppose now that our unit cube is strained in such a way that in one direction the sides are elongated by an amount e, in a per- pendicular dnection are contracted by an amount e (fig i a), the sides perpendicular to the plane of the drawing being unaltered in length Such a strain is called a shearing strain, and may be supposed to be produced by stresses (P) acting as shown. Considering the rect- angular prism BCD, which is in equilibrium under the stresses acting normally over the faces EC and CD, and the forces due to the action on BCD of the portion ABD of the cube, we see that the resultant of the two forces P is a force Pv 2 acting along BD. The LIQUIDS AND GASES 5 >rce due to the action of ABD on BCD must be equal and opposite > this. But the area of the face BD is \/2 units, and there is there- ire a tangential stress of P units, in the sense DB, acting over the agonal area of the cube, due to the action on the prism BCD of ie matter in the prism ABD, and called into being by the elastic splacements Thus the shearing stress, which produces the shearing r atn, may be measured by the stress on the areas of purely normal or purely tangential stress. If we suppose the directions of the pimcipal axes of shear to be mg the diagonals AC and BD, so that these diagonals are conti acted d elongated respectively by an amount e (per unit length), then it n easily be shown that, assuming the strains to be small, the side of e square, the area of the square, and the perpendicular distance tween its sides are, to the fiist order of small quantities, unaltered the strain. Hence (fig. i b) the square ABCD strains into the Dmbus abed, and by rotating the rhombus through the angle cEC iich rotation does not involve the introduction of any elastic forces arrive at the state shown in fig. i c. Hence, rotation neglected, shearing stiam may be icgarded as being due to the sliding of *allel planes of the solid through horizontal distances which are >portional to their vertical distances from a fixed plane DC, the ling being brought about by a tangential stress P applied to the ne AB. The angle Q is taken as a measure of the strain, and the idity modulus (N) is given by the equation It is to be remembered that an elastic modulus such as Young's dulus (E) is not independent of K and N, but is connected with m, as can readily be proved, by the relation E = 9* N * 3/c + N' We are now in a position to define formally the terms " solid " " fluid ". A solid possesses both rigidity and bulk moduli. If subjected to iring stress or to hydrostatic pressure it takes up a new position quilibrium such that the forces called into existence by the elastic dacements form, with the external applied forces, a system in ilibrium. ait, Properties of Matter, p 155, or Morley, Strength of Materials (1931), p. iz. T> THE MECHANICAL PROPERTIES OF FLUIDS A fluid possesses bulk elasticity, but no rigidity. It follows, there! 01 e, that a fluid cannot permanently resist a tangential sh ess, and that, however small the stress may be, the fluid will, in time, sensibly yield to it. In a solid, the stress on an element-plane may have any dheetion with reference to that plane. It may be purely normal, as on the plane BC (fig. i a), or purely tangential, as on the plane BD. In a fluid at test the stress on an element-plane must be normal to that plane And it follows at once from this normality, as is proved m all elementary treatises on hydrostatics, that the pressure (p) at a point in a fluid at rest under the action of any forces is independent of the orientation of the element-plane at that point. Thus if #, y, z aie the co-ordinates of the point in question, In upofett fluid, no tangential stresses exist, whether the fluid be at icst, or whether its different parts be in motion relative to each othci. In all fluids known in nature, tangential stresses tending to damp out this relative motion do exist, persisting as long as the relative motion persists The fluid may be looked on as yielding to these stresses, different fluids yielding at very diffeient time-rates, the late oi yield depends on the propeity known as viscosity A perfect liquid may be defined as an incompiessiblc peifect fluid, No fluid in nature is completely incompiessiblc, and the quantitative study ol the bulk moduli of liquids and their i elation to othci constants oi the liquid substance is a matter ol gieat theoietical and piactical importance Jt must be icmcmbeicd that the magnitude of the bulk modulus depends on the conditions under which the compicssion is earned out Two moduli aic of primary importance that in which the tcnipeialuic of the substance icmains constant, and that m which the compicssion is aduibatic, so that heat ncithci cntcis noi leaves the substance under compiession Remembcnng this, we may define a perfect gas as a substance whose bulk modulus ol isothermal elasticity is numeiically equal to its pressure. this we have at once by definition or pdv -[- vdp = o. LIQUIDS AND GASES 7 Whence, integrating, pv k, nd our perfect gas follows Boyle's Law. Density Having obtained woiking definitions of the substances with which ^e have to deal, we pioceed to discuss in order certain of their more indamental properties and constants One of the most important f these constants is the density of the fluid, defined as the mass of nit volume of the fluid. The density of a liquid is accepted, in a lemical laboratory, as one of the tests for its identification, and the aportance in industry of the " gravity " test needs no emphasis, fe shall therefore detail one or two methods for the measurement the density of a liquid methods for the measurement of gaseous vapoui densities aie peihaps moie appropriately discussed in a eatise on heat. To determine the density of a substance we have to measure ther (a) the mass of a known volume or (b) the volume of a known ass Fluids must be weighed in some soit of containing vessel, d if we know the volume of the containing vessel, the measuiement the mass of fluid which fills it at a given temperature at once /es us the density of the fluid. The most convenient way of librating a containing vessel is by finding the weight o$ some uid of known density which fills the vessel at a known temperature us assumes, of course, that the density of the standard liquid lally water or meicury has been determined by some inde- ident method, and much labonous research has been done on the asuiement of the vanation of density with tempeiature of these } fluids. Thus, Halstrom * measured carefully the linear expansion of lass rod, the relation between length and temperature being ex- ssed by the formula L = L (i + at + bt z ). A piece of this rod of volume V was taken and weighed in water iifferent temperatures, the loss in weight on water at f being ;n by W = W (i + lt + mt* -f nt*). *Ann. Clum. Phys , 28, p. 56. 8 THE MECHANICAL PROPERTIES OF FLUIDS The quantities a, Z>, /, m, n are determined by experiment, and it is clear that the volume at t of the portion used is V = V (i + at -|- fo 2 ) 3 Now since the loss of weight in water is, by Archimedes' principle, equal to the weight of water displaced, and since the volume of this displaced water is equal to the volume of the glass sinker, we have for the density of water at t p =.__ = ,_. ^ + at + or where a, /?, y are known in terms of /, m, n, a, b. The figures obtained in an actual experiment are quoted below: a 0-000001690 b = 0-000000105 / = 0-000058815 m = 0-0000062168 n = 0-00000001443 a 0-000052939 whence -J/8 = 00000065322 \y = o 00000001445 It may be noted in passing that the temperature of maximum density of water may be determined from these results with consideiable accuracy. For when p is stationary we have J- o, and hence 3y2 2 -|- 2/fe + a = O This equation is a quadiatic in t, one of the roots is outside the lange ot the expeiimental figures, the other is 4-108 C From expeiiments of which this may be quoted as a type, Table I (p. 9) has been drawn up It will be seen that, if water be used as the calibrating liquid, the determination of the density of a liquid becomes identical with the opeiation of determining its specific gravity that is, we find by experiment the ratio of the weight of a certain volume of liquid to that of an equal volume of water at the same temperature. The magnitude of this ratio is conveniently denoted by the symbol s* t , and may be reduced to density mass per cubic centimetre by means of Table I on p. 9. It is more usual to compare the LIQUIDS AND GASES 9 TABLE I DENSITY OF WATER IN GM /c.c AT VARIOUS CENTIGRADE TEMPERATURES Temperature. Density. Temperature. Density. Degrees Degrees. O 0-99987 42 0-99147 2 o 99997 44 o 99066 4 i ooooo 46 o 98982 6 o 99997 48 0-98896 8 o 99988 50 o 98807 10 099973 52 0-98715 12 099953 54 098621 14 099927 56 098525 1 6 o 99897 58 0-98425 18 0-99862 60 0-98324 20 o 99823 62 o 98220 22 099780 64 098113 24 o 99732 66 o 98005 26 o 99681 68 o 97894 28 o 99626 70 o 97781 30 099567 75 097489 32 099505 80 097183 34 0-99440 85 o 96865 36 099371 90 096534 38 0-99299 95 096192 40 o 99224 100 o 95838 eight of the liquid with that of an equal volume of water at 4 C., id this value, s* 4 , may also be deduced from the experimental figures y means of the table. It should be noted that the specific gravities and s* 4 are often doubtful in meaning, for they refer sometimes the ratio of true weights, sometimes to the ratio of apparent eights, no correction being made for displaced air. In experimental oik of high accuracy, it is well both to make this correction, and indicate that it has been made. For ordinary work, the common specific-gravity bottle may be jed, but for precision measurements some form of pyknometer is icessary The pyknometer is usually a U-tube of small cubic content , 5 ends terminating in capillary tubes. Three forms are outlined in r.2 . (i) is the original Sprengel type, (ii), a modification introduced 1 Perkin, possesses several advantages. The instrument, filled by iction, is placed in an inclined position in a thermostat, and excess (D312) io THE MECHANICAL PROPERTIES OF FLUIDS liquid is withdrawn from a by means of filter paper, until the level in the othei limb falls to b. The tube is now removed and restored to the veitical position, when the liquid recedes from a. If now expansion takes place before weighing, the bulb above b acts as a safety space, and all danger of loss by overflow is obviated. The form shown in (id) was introduced by Stanford, and reduces to a minimum those paits of the vessel not contained in the thermostat, whilst its shape does away with the necessity for suspending wires, as the bottle can be weighed standing upon the balance pan. In technological practice much specific-gravity work is carried out by means ol vaiiable-immersion hydrometers Hydrometer piac- tice and methods can hardly be said to be in a satisfactory state. il/c FIB Not only has one to plough through a jungle of aibitiaiy scales, but the reduction of these scale leadings to specific giavities, defined accurately as we have defined them alteady, is no easy mattei All hydrometers should cany, maiked permanently on their surfaces, some indication of the pimciple ol their graduation, so that then- readings may be reduced to s|, s*, or some othei definitely known standaid For rough woik, of course, the aibituuy giaduations suffice, and a workman soon learns to associate a reading of, say, * degrees Twaddell with some definite pioperty ol the liquid with which he is working. But with moic delicate hydromcteis an absence of exact reference to some definite standard is distinctly unsatisfactory. Thus the common hydrometer is graduated so that a reading of 1035 corresponds to a specific gravity of 1-035 the standard of reference being very often doubtful and the Twaddell hydrometer is so constructed that the specific gravity s is given by s = i-ooo ~|~ 0-005 r, where T is the reading in Twaddell degrees. Clearly on the common LIQUIDS AND GASES 11 lydrometer the " water-point " is 1000, on the Twaddell hydrometer ,ero, and, unless the hydrometer carries some reference to the tem- >erature at which its water-point is determined, it becomes impossible atisfactorily to compare the performances of two different hydro- neters. Confusion is woise confounded when we introduce Baume" eadmgs. In the original Baume* hydrometer water gave the zero 3omt, and a 15 per cent solution of sodium chloride gave the 15 nark. This for liquids heavier than water. For liquids lighter han water a 15 per cent solution of salt marked the zero point, the water-point being at 10. Now it is usual ^ to mark the point to which the hydrometei sinks in sul- phunc acid of density 1-842 as 66 B We can easily work out a foimula of i eduction giving the specific gravity in terms of these fixed points. Thus, let __o" V be the total volume of hydiometer up to o, p l5 the density of water, v, the volume of hydrometer between o and w, - n p, the density of liquid m which hydiometei floats at mark n, and a, the cioss-sectional aiea of neck By Aichimedes' pimciple the mass of the hydiometer is given by the two expiessions VPJL and (V v)p Hence Fig 3 VftL = (V - v) P = (V - an} P - and if 5 is the specific gravity of the liquid, Pi Hence we have ns __ V __ , s i a If we put 5 = 1-842 when n = 66, we find that k = 144-3, and theiefore for any other liquid giving a reading x, s = I44 ' 3 144-3 - *' It is obvious that we do not know where we are unless the densities u 'TUH MKCHANICAL PROPERTIES OF FLUIDS used in eahbiatum aie sharply defined, and the clouds are not appiet uhly lightened by the piactices of Dutch and Ameiican Iiydiometei inakeis, who take the constant k as 144 and 145 respec- tivcl} pitsumably ioi the convenience of dealing in integial numbers. IMohr's balance is exceedingly convenient for use in those techno- logical laboratories m winch a laige number of determinations of density are made. As fig. 4 shows, it is a balance of special form, one arm being divided into ten equal parts and canying, suspended horn A hook by a silk fibre, a glass thermometer which also serves as a sinker The weights piovided are m the form of riders, the two FIB i I digest being equal, the other two being o i and o-oi respectively ol the largest weights The hook is so E^Ir^ uljiisted that a body suspended from it is in position EI~^:~z it the tenth maik ~ ~7 ~ The othci arm of the balance canies a counter- - - _- ioit,e \Mth a pointed end, which point, when the H~~ ulanee is in ec[uihbinnn, is exactly opposite to a iducial nuuk on the fixed suppoit Suppose now hat the balance is levelled and is in equilibrium, the sinker )eing in an Place the tmiket in water at 15. // will be found that we o/ the ]i<'(r<<i("<f wciqhh, \mf>em1ed from tJic hook, will restore cqmh- ntuin A little thought, based on a knowledge oi the law ol moments, ihould convince the stmlcnt that the specific gravity of a liquid, sou eel to ooor, tan he lead oil at once liom the positions of the 'aumi.s lideis when (he balance is in equilibrium, the sinker being mmcised in the luimcl. Thus the specific gravity of the liquid, he ndeis being disposed as m fig 4, is 1-374 rci erred to water at 5", and may be expressed as a density by means ol Table I, p 9. It happens on occaiuon that a detci mination o specific gravity s called Tot, and that no suitable instruments are at hand. It is vorth while knowing that an accurate icsult may be obtained with 10 moie elaborate apparatus than a wooden rod, which need not be inifoim, but should be uniformly graduated, and a few counter- Kusea of unknown weight, made 01 material unacted on by the liquid mder test. Suppose a knitting needle, or a piece of a small triangulai LIQUIDS AND GASES 13 ile, to be fixed in the rod to form a fulcrum (fig. 5) Suspend the veights W and Wx from the rod by loops of thread, and move Wj. Fig.s mtil the lod is level. If w be the weight of the rod acting at a hstance y from the fulcrum, we have WY + toy = f now, without disturbing W, we allow W x to hang in a beaker of rater at a temperature f, we have, if the point of balance be now hifted to #2, and W 2 be the apparent weight of W 1$ WY + wy = W 2 # 2 These equations give VV i VV 2 ^2 ~~" "~ *^i f a beaker of the liquid under test at a temperature t be substituted or the water and the balance point be now at x 3 , we have by similar easoning or w.-w, lence * = W nd the specific gravity, which may as before be reduced to density y means of Table I, p. 9, is given accurately in terms of lengths leasured along the rod The variation of density with temperature has been the subject f many investigations; most of the equations proposed to represent lis variation (under certain specified conditions) are applicable, r ith great exactness, over limited ranges only. A formula has, how- ler, been put forward recently, which gives the relation between rthobaric density and temperature with very considerable accuracy, ver the whole range of existence of the liquid phase. It is developed xus: 14 THE MECHANICAL PROPERTIES OF We shall see later that, for unassociated liquids, the relation between surface tension and reduced temperature (-m) is given by where n varies slightly from liquid to liquid, but does not deviate greatly from the value 1-2. Further, for any one such liquid the relation between surface tension and the densities of the liquid and vapour phases is * F V where p does not deviate greatly from the value 4. Eliminating y between these two equations, we find where B stands for the p tn root of y /C. If we assume that, at the absolute zero, the density of the super-cooled liquid is about four times the critical density, and that of the vapour is negligible, we have , , P* P = 4Pc(i -m) 03 , assuming constant values for n and for p. But it is well known that, if we take the mean of the orthobaric densities of liquid and vapour at any temperatuie, and plot these mean values against tem- perature, the result is, to a high degree of approximation, a straight line inclined at an obtuse angle to the temperature axis. That is, Pe H- Pv = P Q^2- The condition that, at the critical point (yn = i) we have p e = p v = p c , combined with the condition previously mentioned that for wt = Q we have p e = 4p c and p v = o, gives us Taking these equations for (p e -f- p v ) and (p B p v ), eliminating p v and dropping the subscript /, we find p = 2p e [(i - m) 03 + (i which is a reduced equation between density and temperature applicable to all unassociated substances. The equation may be tested by writing it in the form (i po or Y = X s , LIQUIDS AND GASES 15 vhere s may or may not be equal to 0-3 A logarithmic plot of Y gainst X shows, in general, very good straight lines, whose slope leviates very little from the value 0-3. But the lines do not pass hrough the origin. It follows then that Y=GX" p = 2 Pc [G(i - vrif* + (i - o sw)], /here G is a constant, whose value vanes from liquid to liquid. [lie variation is not great, and a mean value of G is about 0-91 This general form gives very satisfactory results, but, if very close grcement with the experimental figures is necessaiy, the values of j and of ^ special to the particular liquid must be chosen If we put wi = o in this equation, and take the value of G as 91, we see that the (absolute) zero density of the supercooled quid is about 3 82/3 c , probably a better approximation than the sual value 4p c Again, the equation enables us to compute a reasonably good alue for the critical density if the density at any one reduced smperature is known This, of course, involves a knowledge of tie critical temperature If the critical temperature is not known /e may make use of Guldberg's rule, that for unassociated liquids lie boiling-point under noimal pressure is very approximately wo-thirds of the cntical temperature Putting, therefore, m = f , fc have Pfc = 2/40 91(1- !) 03 -h(i-o 5 x*)], geneial relation between the density at the normal boilmg-pomt nd the critical density By eliminating p e instead of p v a similar formula may be derived 3 show the march with temperature of the density of the saturated apour. Compressibility We have seen that a perfect liquid is, as a matter of definition, icompressible that is, its bulk modulus is infinite. Liquids m ature are under ordinary circumstances very slightly compressible,* nd the determination of their compressibilities is, in effect, a de- * Constantmesco, p. 222,, if. THE MECHANICAL PROPERTIES OF FLUIDS teimination of then bulk moduli which, at any given pressure and tempeiature, is defined by the equation 8w /dp\ , = v(* \ v vWr where 8/> is the additional stiess (i e pressure per unit aiea) causing a da icase in volume Bv of a substance whose initial volume is v t and (^ j stands foi the uite of decrease oi piessure with volume \fa>A under isolheimal conditions (T constant). The compressibility, at any given pressure and tempeiatme, may be defined as the recipiocal of the bulk modulus, i e. the latio V \9/> Another definition of compicssibility is sometimes used, namely? /"i \ ( v \ ; in this case v is the volume of unit mass of the liquid. \9/>/T We shall heie confine out selves to a discussion of the com- pressibilities oi liquids those of gases and vapouis will be tieatcd later. It is clear that a complete study of the compressibility ot a liquid lesolvcs itsell into the drawing of a p, v, T suiface for the .substance in question, so that the volume of i gm of the substance is known at any piessuic and tcmpciatuic The impoitancc oi this knowledge can luudly be ovei estimated When we have diawn the />, v, T surface for any liquid we aie in a position completely to de- teunme its most impoitant thei modynamic piopertics In this connection the iccent woik of Bridgman f is pic-eminent in value, and we shall heic give a discussion, as brief as may be, of his woik, leaving the icadei to study details oi the oldei cxpcuments, il he be so minded, in other books The punciplcs involved are simple, but it must be icmembeied that the experimental difficulties, when the pleasures aic pushed up to the older ol 20,000 Kgm per square centimetre, aie very gicat. The substance under test is placed in a stiong chiome-vanadiurn steel cylinder, and the pressure is produced by the advance ^of a piston of known cross-section, the amount of advance of the piston * This notation means that /> is icRatdcd as n function of v and T, and T is kent constant m finding the clenvative '^ f Proc, Amcr, Acad Set,, 48, 309 (1912). LIQUIDS AND GASES 17 giving the change in volume. It would make the story too long were we to discuss in detail the method of packing of the piston to ensure freedom from leakage, and the manner of correction for the change in volume of the cylinder, but it may be of interest to iote that the pressure was measured by the change of electrical resistance of a coil of mangamn. The resistance of the coil was about 100 ohms, and it was constructed of wire, seasoned under pressure, of resistance 30 ohms per metre. For high-pressure measurements this forms a very simple and convenient form of gauge. It must, of couise, be calibrated, and Bndgman performed this by making, once for all, a series of measuiements of the change of electrical resistance oi the wiie with piessure, measuring the piessure by means of a specially constructed absolute gauge It was found that the change of resistance with pressure was so accurately linear up to pressures of 12,000 Kgm. per square centimetie that the readings could be extra- polated with confidence up to 20,000 Kgm. The changes of icsistance were measured on a specially constructed Caiey Foster bridge The whole apparatus was immersed in a theimostat, and series of pressure-volume readings were taken at diffeient temperatures From these readings Table II was drawn up, exhibiting the be- haviour of water up to 12,500 Kgm per square centimetre pressure, and 80 C. A caieful study of this table will show that we can extiact from it data which give veiy complete details of the thermodynamic pro- peities of water within the range considered. The reader is stiongly recommended to work out a few of these results, by so doing he will learn in an hour or two moie of the principles of thermodynamics and of the properties of water than he would gam from a week's reading of books where everything is painstakingly explained for him The hints given below should suffice to set him going, and should he have access to Biidgman's papers, it would be well to com- pare his results with the curves given by Bndgman * /) \ /) \ ( i ) Calculate the compressibility ( ) 01 - ( ) , and plot a curve \op/i v \9p/T between this quantity and p at any one temperature Repeat for vanous tempei atures f * Loc cit | The vauous thcimodynarmcal iclations given in (i) to (10) will be found in treatises on theunodynamics, e R Dictionary of Applied Physics, article "Thermo- dynamics ", Remember that T here stands foi absolute temperatuie. i8 THE MECHANICAL PROPERTIES OF FLUIDS (Ci \ / l \ - ) 01 - ( = ) , and plot it as Ol/p V \0i/p a function of the pressuie at various temperatures (3) The mechanical work done by the external pressure in com- pressing the liquid at constant temperature is given by between given limits, and is obtained by mechanical integialion (planimeter, square counting, or the like) of the cuives showing the relation between p and v at constant temperature (4) The total heat given out, Q, during an isothermal com pression is similarly derived by mechanical integration fiom 3T. using the results of (2) to plot the desired curve. (5) Knowing the mechanical work and the heat liberated in com- pression, we can find the drffeience between these, thus giving the change of internal eneigy along an isothermal, and can plot this against the pressure. (6) The pressure coefficient is given by idv\ = -W dT It can thus be determined with the aid of the results of (i) and (2), and can be plotted against the pi essure (7) The specific heat at constant pi essure may be obtained by mechanical integration from the equation This, of course, involves working out the second derivative from the (o \ ) in the same manner as the fiist deiivative is Oi/P worked out fiom the original tables. Values of the specific heat as a function of the tempeiature at atmospheric pressure may be taken, as Bridgman took them, from the steam tables of Marks and Davis, 1ATURE Pressure, Kgm 60 65 70 75. So". per cm a O IOI68 1-0195 i 0224 10255 I -0287 500 99 6 5 9992 I 'OO2O I 0049 1-0075 I,OOO 9791 9816 9842 9869 9896 1,500 9632 9657 9682 9707 9732 2,OOO 9489 95*3 '9537 9561 9585 2,500 9363 9386 9409 9433 9457 3,000 9247 9269 9292 '93 H 9337 3,5 9138 9160 9182 9204 9226 4,000 ( 937 9058 9080 9101 9123 4,500 8945 8965 8986 9008 9028 5,000 i 8858 8879 8899 8920 8940 5>5oo 8777 8798 8818 8838 8858 6,000 8702 8722 8742 8762 8781 6,500 8631 8650 8670 8689 -8709 7,000 8564 8583 8602 8621 8640 7,500 8499 8519 8538 8557 8575 8,000 8438 8457 '8477 8495 8513 8,500 8381 8400 8419 8437 8455 9,000 8327 8346 8364 8383 8401 9,500 8275 8294 8313 833i 8349 10,000 8226 8245 8264 i 8282 8300 10,500 8179 8198 8216 8235 8252 11,000 8i33 8152 8170 8188 8206 11,500 8088 8107 8125 8143 8160 12,000 8043 8062 8080 8098 8115 12,500 7999 8017 8036 8054 8071 [Facing p 18 LIQUIDS AND GASES tg (8) Knowing C^, we can determine C u from the equation '80V A9T/, (9) The rise in temperature accompanying an adiabatic change of ressure of i Kgm per square centimetre may be deduced with the ^lp of the equation = Z ( dv ' Cp\i I the quantities on the light-hand side of the equation being lown from the icsults of previous sections. <f> lefers to the itropy Finally (10): The difference between the adiabatic and isothermal impressibilities is given by f dv\ {dv\ __ T d may theiefore be calculated Bndgman has added to the value of this work by making similar idies oi twelve organic liquids. For details the student should nsult his original papei s * Intel estmg iclations exist between the compressibility of a liquid d ceitam other of its physical constants, these we shall discuss er undei other heads. Meantime we pass on to a consideiation of. Surface Tension We may take it as an expeimiental fact easily deduced fiom the )st ordinary obseivations that the surface of a liquid is in a state tension and is the seat of energy. The spherical shape of small ndrops or of small globules of mercuiy shows that the liquid face tends to become as small as possible in the circumstances, a sphere is that surface which for a given content has the smallest Derficial area. Again, the fact that the surface is the seat of eneigy illustrated by a simple experiment suggested by Cleik Maxwell, agme a large jar containing a mixtuie of oil and water well shaken * Proc. Amer. Acad, Set., 49, 3 (1913)- ao THE MECHANICAL PROPERTIES OF FLUIDS up, so that the oil is dispersed through the water in small globules. If the system be left for some time it will be found that the oil has " settled out ", and it is clear that the settlmg-out process has involved the motion of considerable masses of matter that is, a definite amount of work has been done. The only difference between the two states of the system is that before the settling out the surface-aiea of the oil-watei interface was considerably greater than the area of the interface in the final state. We conclude that the surface pos- sesses energy, and it will be seen shortly that an important i elation exists between surface energy and surface tension We assume, then, that across any line of length ds drawn in the surface of a liquid there is exerted a tension yds, the dnection of this tension being normal to the element ds and in the tangent plane to the surface. The quantity y is called the surface tension of the liquid, its dimensions aie clearly those of force length, and a surface tension is reckoned, m C.G.S. units, in dynes per centimetre, 01 in grammes per second per second. This tension differs fiom the tension in a sheet of stretched india-rubber, with which it is commonly compared, in that it is, within wide limits, independent of the area of the surface. It is constant at constant tempera- ture, but varies with the temperature, and the calculation of its temperature coefficient is a matter of great theoretical im- portance. Textbook writers usually give the relation between surface tension and temperature in the form a result of little value, holding good over a very limited range The small value attaching to the formula can be shown at once, if we remember that at the critical temperature the suiface tension vanishes, so that we must have t c = -. But values of t c calculated in this a way are wildly wrong, showing that the range of the foimula is exceedingly restricted. It can be shown that, for liquids which do not show molecular complexity, the relation between suiface tension and tempeiature is given by y = y (i - bt)\ where n varies from liquid to liquid, but in general has a value not differing very greatly from 1-2. This equation holds good from freezing point to critical temperature, and its accuracy may be tested LIQUIDS AND GASES 21 jy comparing the value of t c obtained fiom direct experiment with hat obtained from the relation * c = 7- The test is shown in Table III below. b Substance Ether Benzene Caibon tetrachlonde Methyl formate Propyl formate Propyl acetate TABLE III b I 248 0005155 I 218 o 003472 I 206 0-003553 I 210 0-004695 I 2 3 I o 003774 I 294 o 003623 t c Calcu- t c Ob- Differ- lated served ence + 02 -OS -16 i o + 01 -02 Cent Degrees. 194 Cent Degrees I 93 S 288 2885 281 5 2831 213 214-0 265 264 9 276 276 2 /e have previously leferred to the relation between suiface tension id surface energy The assumption that the suiface tension F a surface is equal to its suiface , *,, >,., lergy (per unit aiea) is anothei >mmon error. The "pi oof" given Biially follows these lines Imagine soap film stt etched over the veitical lie frame shown in fig 6, the bar CD nng movable If the bar be pulled Dwnwards through a distance 8x, the c oik done against the suiface forces 2yLSx (remember that the film has ro sui laces) But if A be the eneigy :i unit area, since the increase of rface is 2lSx, the inciease of super- ial eneigy is A.2/Sx Hence, equating these quantities, we have y = A. it this argument overlooks the important fact that surface tension mmishes with increasing temperature. Hence it follows from ermodynamic principles that, in oider to stretch the film isother- illy, heat must flow into the film to keep its temperature constant, Fig 6 22 THE MECHANICAL PROPERTIES OF FLUIDS and this heat goes to increase the surface energy. A simple thermo- dynamic argument shows that the relation between y and A is given by 7 ~ ^ A 8T' where T stands for absolute temperature, and only if the tempeiature coefficient of surface tension were zero would the simpler equation hold. Since we know the relation between surface tension and tem- peiature for unassociated substances we can easily work out, by substituting for y and 9y . , . . -~ in the equation just given, the relation be- tween surface energy and temperatuie. This relation is shown in fig 7, the dotted curve showing the variation of surface tension, the full curve the vaiiation of surface eneigy with temperature. The two ""--,. curves intersect each ~~~~ TEMP other and the axis of Fjg ? temperature at the critical temperature, showing that at that point both surface tension and surface energy vanish. But for lower temperatures the two quantities are in general very different in numerical magnitude, surface energy increasing much faster than surface tension with falling tem- perature. This important fact should carefully be borne m mind Many relations, empirical and otherwise, have been suggested connecting surface tension with other physical constants Thus, Macleod * has recently found that for any one liquid at different temperatures y = * Trans Faraday Soc , 1923. The present writer has also shown (Trans Faiaday Soc,, July, 1923) that the constant C may be expressed m the form C = AT C / M$pc*P, where M is the molecular weight, p c the critical density, and A a constant independent of the nature of the liquid LIQUIDS AND GASES 23 wheie C is a constant independent of the temperature, p t the density of the liquid, and p v that of the saturated vapour of the liquid. We should naturally expect surface tension and compressibility (f?) to stand in intimate relation, and expeiiment shows that liquids of high compiessibility have low surface tensions and conversely. Richards and Matthews * have examined the quantitative relation between these two constants, and find that, for a large number of unassociated substances, the product y^ is a constant quantity. A most important equation connecting surface tension, density, and temperature, is that proposed by E6tvos,f - T - 8), >vhere M is the molecular weight of the liquid, p its density, and 5 and K are constants for unassociated liquids, 8 being about 6 md K 2 12 The equation shows that a knowledge of the tempera- ui e variation of y enables us to calculate the molecular weight of he liquid under examination, and hence to determine whether its nolecules aie or are not associated In iccent yeais this test of association has been slightly altered. [nstead of examining the vanation of y( ) with temperature, the , , /M\ 1 , . , j , i variation of A( has been studied, wheie, as we have seen, \P/ = 7- IgFp 3ennett and Mitchell J have shown that for unassociated liquids his quantity, which we may call the total molecular surface energy, s constant over a fanly wide lange of temperatuie, and have used his constancy as a test of non-association We now tuin to the discussion of a problem of fundamental mportance that of the i elation between the pressure-excess (posi- ive or negative) on one side of a curved suiface and the tension in he suiface It is fanly cleai that the piessure just inside a curved iurface such as that of a spherical bubble is greater than the pressure ust outside the suiface, and the manner in which pressure- excess s connected with surface tension may be calculated as follows. * Zeit Phys Chem , 61, 49 (1908) | See Nernst, Theoretical Chemutry, p. 270 (1904). [ Zeit Phys Chem , 84, 475 (1913). 24 THE MECHANICAL PROPERTIES OF FLUIDS Imagine a cylm ducal suiiace whose axis is peipendicular to the plane of the paper, pait of the trace of the surface by the plane of the paper being the curve AB (fig. 8). Consider the equilibiium of a poition of this cylindrical surface of unit length perpendicular Fig. 8 to the plane of the paper, and of length ds in the plane ot the paper. If II and II -f p be the pressures at CD on the two sides of the cylinder, we have, resolving normally, n 2y sin- -)- lids -- (II -j- p)ds, 2 when 6 is the radian measure of the angle indicated m fig. 8, or, since 9 is small, .,0 _ p^ s ds But 9 ~ where R is the radius of curvature at C, and theiefoie ft - Z P ~ R* If the surface is one of double curvature, the effects aie additive and we have where R x and R 2 are the principal radii of curvature at the point in question. Thus for a spherical drop, or a spherical air-bubble in a liquid, we have 2 y P = P LIQUIDS AND GASES 21 where R is the radius of the drop or bubble. For a spherical soap- bubble, which has two surfaces, we should have p = R The use of the pressure-excess equation, combined with a know- ledge of the fact that a liquid meets a solid at a definite angle called the contact-angle, will suffice to solve many important surface-tension problems. Thus the rise or fall of liquids m capillary tubes is readily explained. Water, for example, meets glass at a zero contact Fig 9 angle, hence the suiface of water in a capillary tube must be sharply curved, and the nanower the tube the sharper must be the curvature m order that the liquid may meet the glass at the proper angle The state of affairs shown in fig 9 (i) is impossible, for the pressuie at A being atmosphenc, the pressuie at B must be less than atmosphenc 2v by -, where R is the radius of curvature of the meniscus at B But the pressure at C in a liquid at rest must be equal to that at B, and the pressure at C is clearly atmospheiic. Hence the liquid must rise in the tube until the additional pressure due to the head h just brings the pressure at B up to atmospheric value. We must have therefore - *. 26 THE MECHANICAL PROPERTIES OF FLUIDS If the tube be very narrow and the ciitenon of nanowness that is r - shall be small compared with unity the meniscus will be a segment of a sphere, and the contact angle being zeio we may put R = r, the radius of the tube, giving the well-known equation y = $gprh If -, though small, be not negligible compared with unity the meniscus will be flattened; a very close approximation to the truth may be obtained by treating the meridional curve as the outline of a semi- ellipse. Suppose the semi-axes of the ellipse to be r and b (fig 9 n). If we take the contact angle as zero, theie will be an upwaid pull of 2-nry on the liquid m the tube all round the line of contact of the liquid with the glass. Equating this to the weight of liquid raised (including the weight of that in the meniscus) we have ZTrry Trr^hpg -f- lvr z bpg or 20? = rh -f- ^rb, if for brevity we wiite a 2 for . But if R be the radius of curvatuie SP at O, we have accurately Piessuie-excess = gph = , or za 2 = Rh. Now R, the radius of curvature at the end of the semi-axis minor of r a an ellipse, is equal to . Hence 2,0 V fl and therefore 2a z r h -f- lr. , 2<2 2 or i2 4 6rhaP r^h = o. Solving this as a quadratic in aP and expanding the surd, we obtain O T In all practical cases - is small compared with unity, and the above LIQUIDS AND GASES 27 juation gives values for a z (and therefore for y) in close numerical agreement with those obtained from the equation 3 2 = ^/! _)_ ^ _ 0-1288^ + \ il (v btained by the late Lord Rayleigh * as the result of a lather complex id difficult analysis. The problem of the measurement of interfacial tensions has jcently assumed great technological importance, mainly on account " the rapid development of colloid chemistry and physics. Tanning, yeing, dairy chemistry, the chemistry of paints, oils, and varnishes, " gums and of gelatine are all concerned deeply with the properties * colloidal systems in which one phase is dispersed in very small irticles through the substance of another phase. There is conse- aently a relatively great extent of surface developed between the ro phases, and the interfacial tension at the surface of separation of ie phases may play an important, not to say decisive, part in deter- iining the behaviour of the system This tension may be measured by a modification of the capillary ibe experiment desciibed above For example, the tension at a mzene-water interface has been measured by surrounding the ipillary with a wider tube, and filling with benzene the space reviously occupied by air Exact determinations can conveniently be made by the di op- eight method, wherein a drop of liquid is formed at the end of id detached slowly fiom a veitical thick- walled capillary tube timeised in the second (and lighter) liquid The method can, of )urse, be used to deteimme liquid-au tensions So many erroneous atements have been made concerning the practice and theory of us method that it is woith while consideimg it in some detail or example, a common practice in physico-chemical works is to juate the weight of the detached diop to zirry, a procedure which, Lit for the fact that the di op-weight method is often used as a com- uative one, would give results about 100 per cent in error. Those, jam, who are alive to the error of writing mg = zirry 3t infrequently tell us that the constant ZTT must be replaced by lother constant of value 3 8, for no very apparent or adequate sason. Let us then investigate the problem as exactly as may be * Proc. Roy. Soc. 92 (A), 184 (1915). 2 8 THE MECHANICAL PROPERTIES OF FLUIDS m an elementary manner, and see if some justification exists for this procedure. Suppose for the moment that the drop is formed in air. If we assume that the diop is cylindrical at the level AB (fig. 10), then, II being the atmospheric pressure, the pressure at any point in the plane AB is II -f -. Consequently, resolving vertically for the forces acting on the portion of the drop below AB, we have mg -f- ?" 2-rrry II. .77T leading to mg Fig 10 exactly half the value of the weight of a drop as given by most of the textbooks. But the detachment of a drop is essentially a dynamical pheno- menon, and no statical treatment B can be complete. We can, how- ever, obtain some assistance from the theory of dimensions. As- sume that the mass of a detached drop depends on the surface tension and the density of the liquid, the radius of the tube, and the acceleration due to gravity. We may thus write m = Dimensionally [M] = [MT- 2 ]*[LT~ 2 f [ leading to x -j- ss = i, x + y = o, y 3* + > o. Solving for w, y, and # in terms of x we find T , yr m = K-- 8 or m ' LIQUIDS AND GASES 29 inhere F is some arbitrary function of the variable -. The late gpr* jord Rayleigh determined the weight of drops of water let fall slowly rom tubes of various external diameters. Knowing the surface snsion of water, he was enabled to tabulate the variation of the unction F with that of the independent variable -; for, as we see, hie function F is given by F = ^ yr' n this way the following table was drawn up. TABLE IV \ yr 2-58 4-13 1-16 397 o 708 3 80 o-44i 3 73 0-277 3-78 o 220 3 90 o 169 4 06 * will be seen that foi a considerable variation of the variable ~- gpr* -and this means a considerable variation of r the function F does not actuate seriously, and foi most purposes it is permissible to assume lat F is constant and equal to 3-8. Hence the reason for the luation o 1 mg = 3 8ry. The argument for mterfacial tensions follows identical lines, and ie reader should have no difficulty in working it out for himself, membering that the drop of density p, say, is now supposed to be sndent in a lighter liquid of density p v If we assume that the liquids with which we are dealing obey the 3wer law for the variation of surface tension with temperature, we ive ! sir < . i i 4 j* i * ' .a,% " |T) I *"** ^ " ? * I \vV ^~~~*^"' 30 THE MECHANICAL PROPERTIES OF FLUIDS where, for convenience, we express temperatures in the reduced form. The total surface energy A is given by = y m~ 9 am and, with this form for y, is A = y (i <m} n -' L {i + (n i)-m}, and we see that, contrary to some statements, there is no indication of a maximum value for A, the march of A with temperature following the curve shown in fig 7. We have seen that a reduced equation may be developed between orthobaric density and temperature which, in its simplest form, may be written p = 2p c [(i w) 03 -j- i It follows then that free molecular surface energy (e) defined as y(M/p)t and total molecular surface energy (E) defined as A(M//>)i, have their vanation with temperature at once determined on sub- stituting m these expressions the appropriate expressions for y, A and p The deduction of the equations showing how e and E vary with the temperature is left to the reader, but it may be noted that e is not a linear function of the temperature, nor is E independent of the temperature, although the variation at fairly low temperatures is very small, and the assumption of constancy over ordinal y ranges of temperature need lead to no serious error. Neveitheless it is worthy of note that, considering the whole range vn = o to wi = i, the quantity E rises very slowly to a not very pronounced maximum at a temperature about f of the critical value, thereafter falling rapidly to zero at the critical point It is interesting to see that this slight maximum is shown in the experimental figures, but was overlooked, as workers in the subject were looking rather for constancy than variation with temperature. Some time ago Katayama remarked that very considerable simplification resulted if the difference of the liquid and vapour densities were substituted for the liquid density in the definitions of E and e. We thus have / M \l . ,, ,/ M \f e = y [ ) and E = A ( - 1 , \Pc PJ \Pe PJ LIQUIDS AND GASES 31 and Katayama points out that, in these circumstances, e and E are linear functions of the temperature given by e = <? (i m], E = E (i + o'2iri). As the reader may easily convince himself, these results depend on the power law being followed with n equal to 1-2, and Macleod's law being obeyed with the index equal to 4 If we write this latter law in the more general form y = C( Pe - Pv y, and do not assume any special value for n in the expression for the power law, we readily find that E = E (i m} x {i + (n i)m}, where for brevity x is written for (n i znfep) If n 1-2 and p = 4, we have x = o, and Katayama's value for E results In no instances that we have examined is this exactly true The index x is small but positive, and the result is that E climbs by an almost linear ascent to a definite maximum, thereafter falling very rapidly to zero at the cutical point behaviour much more consonant with our usual conception of surface energy than that given by Kata- yama's equation, which gives E its highest value at the critical point To establish these results is not difficult If we have, for a sub- stance whose critical temperature is known, a series of values of surface tensions determined over a wide temperature range, a logarithmic plot of (i m) and y serves to test the power law, and to determine the value of n where the power law is followed The values of A, e and E at different temperatures may then readily be computed E 0) the zero value of the total molecular surface energy, is readily deduced, and it may be remarked that this quantity varies in very mteiestmg and icgular fashion with varia- tion in chemical constitution The quantity My i /(p e />) (where M stands for molecular weight) has been named the parachor. It provides us with a number which measures the molecular volume of a liquid at a temperature at which the surface tension is unity, and therefore gives a most valuable means of comparing molecular volumes under corresponding conditions. Viscosity We have seen that a perfect fluid is one in which tangential stresses do not exist, whether the fluid be at rest, or whether its different portions be m motion relative to each other. Such stresses do, 32 THE MECHANICAL PROPERTIES OF FLUIDS howevei, appear in all known fluids when lelative motion exists, and the fluid may be looked upon as yielding under the stress, different fluids yielding at very different rates. The most obvious effect of the existence of such tangential stresses between different parts of the fluid is the tendency to damp out relative mption. Thus, if we have a layer of liquid flowing over a plane solid surface, the flow taking place in parallel horizontal layers, the layer of liquid m contact with the surface will be at rest, and there will be a steady increase, with increase of height above the solid suiface, in the horizontal velocity of the successive layers. Considering the surface of separation between any two layers, the tangential stress existing there will tend to retard the faster moving upper layer, and to accelerate the slower moving lower layer. The magnitude of the tangential stiess may be written down if we assume, following Newton, that the tangential stress is propoitional to the velocity gradient, so that, if the horizontal velocity is v at a vertical distance y from the fixed surface, we have dv , dv S oc , i e. equals p -, ay ay where //, is a constant called the coefficient of viscosity of the fluid. If - is unity, then S = ju. Hence we are led to Maxwell's well- ay known definition of p " The viscosity of a substance is measured by the tangential force on unit area of either of two horizontal planes of indefinite extent at unit distance apart, one of which is fixed, while the other moves with unit velocity, the space between being filled with the viscous substance " The dimensions of // are those of stress divided by velocity gradient; this works out to M = [ML-'T-'J, so that a coefficient of viscosity in C.G.S. units is correctly given as x gm. per centimetre per second. If in fig. i (c), p 4, we put Aa dx, AD = dy, we see that the ligidity modulus (N) is given by S = N^. dy d"V Comparing this with S = ft , it is clear that the dimensions of iy LIQUIDS AND GASES 33 .cosity differ from those of rigidity by the time unit, in the same y as the dimensions of length differ from those of velocity. In t, the rigidity modulus of a solid determines the amount of the ain set up by a given tangential stress, and the viscosity modulus a fluid determines the rate at which the fluid yields to the stress. The fall of a sphere through a viscous fluid aptly illustrates r eral interesting physical phenomena; we shall therefore study 5 problem in some little detail. Considerable assistance is given an application of the theory of dimensions. Suppose that the istance (R) experienced by the sphere depends on its radius (a), velocity (v), and the density (/>) and viscosity (/u,) of the surrounding id, we then have R = ka x p y ^v M , I as the right-hand side must have the dimensions of a force, we ain, by equating the exponents of M, L, and T, x zv, y = w -- i, z = 2 w, so that R = fof--v \ p> I P low velocities we may assume that the resistance is proportional he velocity. Putting therefore zu = i, we find R = lore complex analysis, originally given by Stokes,* shows that R = tiowever, we assume that for high velocities the resistance varies be square of the velocity, we have, putting zv = 2, R = kv*a z p, viscosity does not enter into the question energy is expended, in overcoming viscous resistance, but in producing turbulent ion in the liquid. Returning to the problem of low velocities, let us write down the ition of motion of a sphere falling vertically through an infinite n of fluid. The forces acting are the weight (W) of the spheie * Lamb, Hydrodynamics, p. 532 (1895). (D312) 3 34 11IK MECHANICAL PROPERTIES OF FLUIDS (downwards), the lesistancc (R), and the buoyancy (B) of the displaced lluul (upwaids). This gives W - (B + R) = mf, whcic m is the mass and /the downward acceleration of the sphere Hut us the velocity of the spheie incieases, R increases pan passu so that the acceleration steadily ^_ diminishes, until when R has increased to such an extent that W - B = R, / becomes zero, and the sphere henceforward falls with a con- stant velocity known as the " terminal velocity ". Calling this velocity V, the density and radius of the sphere p and a respectively, and the density of the fluid p > we have jiTtt s g(p PQ) = leading to UDC I'Jtf I r 9 V ' Clear ly, measurement of the terminal velocity V enables us to determine the viscosity of a liquid The method is peculiarly suited for the mea- surement of the viscosities of very viscous hquids such as heavy oils or syrups, and has been much used ol late yeais. The simple apparatus required is shown m %. n The outci cylinder icpiesents a thermostat; the inner cylinder contains the liquid under expeiiment. The sphere steel ball bearings 0-15 cm in diameter are suitable An liquids having viscosities comparable with that of castor oil is diopped centrally through the tube AB, and its velocity is measured over the suifacc CD, which icpiesents one-third of the total depth of the liquid. LIQUIDS AND GASES 35 Two important corrections are necessary one for " wall-effect ", e for " end-effect "; for it must be remembered that the simple sory given above applies only to slow motion through an infinite 'an of fluid. These corrections have been investigated by Ladenburg,* who s shown that in order to correct for wall-effect we must write - V ., iere V is the observed velocity, V^ the corresponding velocity an infinite medium, and - the ratio of the radius of the sphere to it of the cylinder containing the liquid. Similarly for the end-effect Lere h represents the total height of the liquid which is supposed be divided into three equal poitions, V representing the mean ocity over the middle third. Introducing these corrections into }kes's formula, we obtain 4- r ^ O 7 , The method has been much used dining the war period for the asurement oi /x for liquids of high viscosity, and is fully desciibed a papei by Gibson and Jacobs f A commercial viscometcr has iccently come into use, which is of ;eedmgly simple type, and gives fairly ichable icsults A steel 1 m. in diameter is placed inside a hemisphencal steel cup of jhtly larger dimensions. The cup cairies on its internal surface ee small pi ejections in length about 0-002 in. A little of the oil der examination is poured into the cup, and the ball placed in sition inside the cup. The ball is pressed down on to a table, ; cup being uppermost, and at a given instant cup and ball are ed clear of the table. The time taken for the ball to detach itself measured, and this gives a measure of the viscosity of the oil.| * Ann der Physik (IV), 23, 9 and 447 (1907) f>r. Chem Soc , 117, 473 (1930). j For fuller description see Chapter III, p. 119. 36 THE MECHANICAL PROPERTIES OF FLUIDS Comparative measurements only can be made, and the instru- ment must be standardized by means of a liquid of known viscosity Viscometeis for use with oidmaiy liquids usually depend on measurements of the flow of a liquid through a horizontal or vertical capillary tube. The solution of the pioblem for a horizontal capillary affords an interesting application of the general equations of hydro- dynamics, and we shall attack the problem from that side. The reader may or may not be able to follow the arguments by which these equations are established he may study them at leisure in the tieatises of Lamb, of Bassett, or of Webster what is important is that he should see clearly their physical significance and obtain piac- tice in handling them. This is best done by a caieful study of one or two of theii applications. The equations of motion of an incompressible fluid aie: * Du d , , D a , a , a , 9 where = -f u -j- v -f- w~, ut ot ox ay oz v* = !L 4- !L + *. dx* dy z a* 2 ' p is the pressure at any point, X, Y, Z the components of the external force per unit mass, u,v,w the velocity components. To apply these equations to the steady flow of a liquid thiough a horizontal capillary tube, we take the axis of the tube as ^-axis, and assume the flow everywhere parallel to this axis Then u = v = o, and from (a) and (J3) we have dp dp^ dx dy so that the mean pressure over any section of the tube is uniform. * See Chapter II, p. 83. LIQUIDS AND GASES Also from the equation of continuity, du . dv . dw _|_ _ _j = o, dx dy dz , dw we have = o. 9# fence (a) and (/3) vanish, and (y) becomes, assuming no extraneous >rces, dp ince w varies only with t, x, and y, and p only with #, then = constant = c (say), and therefore f, i, i j_ /e can now obtain the equation known as Poiseuille's equation, for the motion is unaccelerated - = o, and dt "ransforming to polar co-ordinates,* id z w i dw _, i A 1 \ h o i ~ ^r~ ~r ~^ r, w being independent of 9, d z w 'o~T or z r or ""his may be written ,/ 9 2 2y . 9zw\ a 9 q r + ) = C r- = c. r\ or* or/ r or \ or \ )= / ntegrating, we have dw c r 2 . . V _ . - . _ _ _|_ f\* ' ^5^ i^ - tl 9r /u, 2 * See p. 53 et seq. 3 8 THE MECHANICAL PROPERTIES OF FLUIDS and integrating a second time 10 = I : - r z 4- A losr r 4- B, *JL JLC ' O ' ' where A and B are constants of integration When r = o (on the axis of the tube), w is finite. Consequently A must be zeio, and w i"[I" r + Also, if there is no slipping of the liquid at the walls of the tube, when r = a,w = o, and consequently If V is the volume of liquid which escapes from the tube in a time T the volume issuing in unit time is given by Y = f T I'* *" / / > P\ J TT U = c (a r*} rar - c 4ft V ; 8 /* Ifpi and /> 2 are the pressures at the entrance to and exit horn the dp pipe, then remembeiing that stands for the late oi inucase of p with z, we have whei e / is the length of the pipe. Hence 8 V / If the liquid is supplied to the tube under a constant head //, and escapes into the air at a low velocity, we have __ TT a*T gph p _ _._ T . This equation is known as Poiseuille's equation. All the quantities on the light-hand side may be determined expei mien tally, and hence ju may be evaluated. LIQUIDS AND GASES 39 Comparative measurements by this method are usually made using twald's viscometer (fig. 12). The bulb C is filled with the liquid der examination, which is then drawn up by suction until it fills 5 bulb D. The pressure is then released, and the time of transit L ween the marks A and B observed. The pressure head is varying oughout the fall, and clearly we cannot apply Poiseuille's equation it stands. But, noting that for liquids of equal densities and Ferent viscosities the times will be proportional to the viscosities, i that for liquids of equal viscosities and different densities the les will be inversely proportional to the densities, have in general, t = K or \i = cot, ere c is a constant for the apparatus to be deter- tied by using a liquid of known viscosity. A viscometer of dimensions suitable for the detei- nation of the viscosity of water is not suited for use h heavy oils But if we have a series A, B, C, viscometers of gradually increasing bore, calibrate by using water, and then use the most viscous aid suitable for A m order to calibrate B, continuing s process as far as necessary, we are provided with ham of viscometers which can be used over a veiy ie lange. Since the viscosity of liquids decreases rapidly h increase of temperature, it is of vital necessity t the apparatus be enclosed in some form of thei- Fig J2 stat and that the temperature of experiment be en and recorded. This rapid change of viscosity with temperature kes it very difficult to obtain relations between the viscosities of Ferent chemically related substances, as it is by no means easy to tie the temperature of comparison. It has been found, however, t consistent results may be obtained if viscosities are compaied temperatures of equal slope that is at temperatures for which ~ CIL he same. Using this standard it has been shown, for example, t the molecular viscosities * of a homologous series increase by a istant amount for each addition of CH 2 * The molecular viscosity of a liquid is defined as /J.(Mv)$, where M is the ecular weight and v the specific volume of the fluid concerned, 40 THE MECHANICAL PROPERTIES OF FLUIDS There are many important piactical problems which depend foi their solution on a knowledge of friction in fluids The viscosities of mixtures of liquids, the viscosities of gases, the theoiy of lubiication the discussion of turbulent motion, to mention but a few, piesen important and most inteiestmg aspects. These matters are full) discussed in Chapter III One interesting problem may be mentioned m passing the suspension of clouds in air, where we have the appaient paiadox oi a fluid of specific gravity unity suspended in a fluid of specific gravity 0-0013. The paradox is cleared up by an application oi Stokes' formula, V = - (^ZJ ^ 3 . 9 P- Taking the viscosity of air as 0-00017 m C.G S units, the leader is recommended to calculate the teiminal velocities of spheres of water say o-i, o-oi, . . . cm in radius The terminal velocities ol minute drops will be found to be surprisingly small The kinetic theory of liquid viscosity has not icceived a great deal of serious attention, and formulse developed to show, for example, the dependence of liquid viscosity on tempeiature have usually a purely empirical basis Of these, one proposed by Porter may be specially noted. Suppose that the variation of viscosity with tem- perature has been experimentally investigated foi two liquids Take a temperature T x at which one liquid has a viscosity ?j , Find the temperature T 2 at which the second liquid has the viscosity 77 1 Repeat foi different values of T x . Then Tj/Tg is a hneai function ofTj. Recently Andrade has put forward a kinetic theory in which he assumes " that the viscosity is due to a communication of momen- tum from layer to layer, as in Maxwell's theory of gaseous viscosity but that this communication of momentum is not effected to uny appreciable extent by a movement of the equilibrium position oJ molecules from one layer to another, but by a tempoiary union al the periphery of molecules in adjacent layeis, due to their large amplitudes of vibration." This assumption leads to a formula connecting viscosity and temperature of the form r, = a formula which had been put forward previously on an empirical LIQUIDS AND GASES 41 isis. Porter's relation, as the reader may verify, follows at once om this equation. In the deduction of this equation, variation of volume with tnperature has been neglected and, taking this factor into account, idrade deduces a second formula, iere v is the specific volume. For a great many organic liquids is formula gives a very good fit, though, as was to be expected, iter and the tertiary alcohols show abnormalities. Equations of State Much labour has been expended on the problem of devising uations which shall represent accurately the pressure- volume- nperature relations of a substance in its liquid and its gaseous ases It may be said at once that it is impossible to devise an nation which shall be accurate over such a range without being possibly cumbrous Nevertheless the simpler equations have, we shall see, considerable value in giving a fairly adequate )resentation of the general behaviour of a homogeneous fluid, IL FIR 13 A very simple type of such a fluid is a gas, considered as an emblage of material points which are in rapid and random ition, and which do not exert any attractive forces on each other nsider a given quantity of such a gas enclosed m a cube of side md let the component velocities of any one particle be u, v, w shown (fig. 13). The pressures on the faces of the cube are due to the impacts of particles thereon. At any impact on, say, the face A the velocity nponent normal to that face will be reversed, and the change of menturn of the particle consequent on the rebound will be mu ( mu} = zmu. (D312) 3* 42 THE MECHANICAL PROPERTIES OF FLUIDS 2/ The time of tiavel fiom A to B and back is sec.; the frequency u u of the impacts on the face A is ,; hence for any one particle the change of momentum at A per second will be u mu z zmu x - = =-, 2/ / and similarly for the other particles The force on the face A due to molecular bombardment is, therefore, -~u*. V Ifp is the pressure on the face A, ^ o v po p = _ _ = -Zir = -Z,r, where V is the volume of the cube. Similarly, foi the pressures on the faces perpendicular to A, we have the expressions ~Zv*, 1 ~Sw\ But these pressures are equal, and therefore p = ^(a -I- where U 2 = 2 -f v z + ro 2 Now let us define a mean velocity U by the relation f? _ jff U - 2, N , N being the total number of particles in the cube. We then have and wN being the mass of the gas, we have, if /> be its density, , p = and /> = Hence Boyle's Law. If we assume that U 2 is pioportional to the absolute temperature, LIQUIDS AND GASES 43 i have Charles's law, and can write as the characteristic equation of ir " perfect " gas pV = RT. V stands for the volume of unit mass of the gas, R will be different r different substances. A simple deduction from our fundamental uation shows, however, that the gramme-molecular volume* is the me for all gases. Consider two different gases for which and the pressures and volumes are the same, the temperatures are equal, then, assuming that the mean kinetic jrgies are the same, that Nj = N 2 at is, equal volumes of two gases, under the same conditions of iperature and pressure, contain the same number of molecules This is the formal statement of Avogadro's hypothesis. It ows, therefore, that the weights of these equal volumes are pro- tional to the molecular weights of the gases, and hence that the mme-molecular volumes of all gases, measured under the same iditions of tempeiature and pressure, are the same. The giamme-moleculai volume, measured at o C and 760 mm. Hg, is 22-38 litres. If then V stands for this volume, the constant vill be the same for all gases. Its value should be calculated by reader. But no gas behaves in this simple manner, although for moderate ssures and high temperatures the equation is accurate enough for inary computations, as far as the more permanent gases are cerned. Suppose that we study experimentally the p - v relations of erent fluids, drawing the isothermals for various different tem- itures (fig. 14). Starting at a sufficiently low temperature we find the volume steadily diminishes with increase of pressure up to a ain point at which the fluid separates into two phases liquid and ;ous. The pressure then remains constant until the gaseous I e the volume occupied by M gm of a gas at normal temperature and sure, where M is the molecular weight. PRESS 44 THE MECHANICAL PROPERTIES OF FLUIDS phase has completely disappeared, when further mciease of pressure causes but small diminution in volume. If we now repeat the experiment at a higher tempeiature, we find that the hoiizontal poition AB of the curve, representing the period of tiansition from the gaseous to the liquid phase, is shoitei, and shortens steadily with increasing temperature until the isothermal for a certain temperature exhibits a point of in- flexion with a hori- zontal tangent, run- ning for a moment parallel to the volume- axis, and then turning upwards again. The tempeiature for which this isothermal is drawn is called the critical temperature, and the point G the critical point * Above this temperature no amount of pressure causes a separation into two distinct phases The experimental determination of these curves, over a wide range of pressure and temperature, is a matter of no small difficulty. Once a suitable pleasure gauge has been devised we have seen that the change of electrical lesistance of manganin may be utilized obseivations are fairly straightfoiwaid, but the cali- bration of such a gauge demands experimental work on a hcioic scale Amagat, for example, peiformed a Boyle's Law experiment in which nitrogen was compressed in the closed (shorter) limb of a U-tube, the open lirnb being installed on the side of a shaft 327 m. deep. The p-v relation for nitrogen being known, this gas may then be used as a standard m studying the behaviour of othci * Foi a description of the physical state of the fluid at the cntical point, consult any of the standaid treatises on heat, e g Poyntmg and Thomson, or Pieston. _ VOL Fig 14 LIQUIDS AND GASES 4S ases, or in calibiating a different type of piessure gauge. Let us now examine briefly the character of the curves obtained om experiments o this type. It is convenient to plot pv against , as this procedure exhibits very clearly the departure of the gas mcemed from the " perfect " state. Some of the results obtained e shown in fig. 15 The reason for the difference between the isothermals for an 'eal and for an imperfect fluid is not far to seek. The equation pV = RT kes no account of the forces of attraction between the molecules, )r of the volume occupied by the molecules themselves. It makes '} pv "Perfect Gas Carbon. DuDxUie Fig 15 zeio when p becomes indefinitely great, and it is cleaily more in cordance with the properties of fluids to put p(V - b) = RT, that asp increases indefinitely, V tends to the limit b, b represents, eiefoie, the smallest volume into which the molecules can be eked. Further, the mutual atti action of the molecules will result m the oduction of a capillary pressure at the fluid surface, the intensity the molecular bombardment will be diminished, and the pressure the surface of the containing vessel correspondingly decreased, it, theiefore, (p + w)(V - b) = RT. ithout discussing the matter very closely, we can determine the lue of cu from consideration of the fact that the attraction between o elementary portions of the fluid is jointly proportional to their 46 THE MECHANICAL PROPERTIES OF FLUIDS masses that is, in a homogeneous fluid, to the square of the density, or inversely as the square of the volume. We see, then, reasons for writing co = -, and the equation of state becomes the form originally pioposed by van der Waals This equation is a cubic m v, and if the isothermals are plotted for different values of 0, we obtain cmves whose general shape is that of the curve HAEDCBK of fig. 14. It will be observed that for temperatures below the ciitical tempeiature a hoiizontal constant pressure line cuts any given isothermal either in one point or in three points corresponding to the roots of the van der Waals cubic Taking an isothermal nearer to the critical tempeiatuie, we see that the three real roots are more nearly coincident, and at G, the ciitical point itself, the roots coincide. Above the critical point, a hoiizontal line cuts any given isothermal in one point only two ol the loots of the cubic are imaginary. If we write down the condition that the three roots shall be coincident, we easily ainve at values ol the ciitical constants in terms of the constants of van dei Waals' equation These are rn I * c But it is preferable to write down the condition that at the ciitical point the isothermal has a point of inflexion with a horizontal tangent If we therefore differentiate van der Waals' equation with icspcct dp 3 2 /> to v, put ?j- and -^ equal to zero, the resulting equations, combined with the original equation of state, serve to deteimine p c , v c , and T c The work is left as an exercise to the reader. This method is pieferable, since it is perfectly general and may be applied to characteristic equations which are not cubics in v, and to which, therefore, the " equal-root " method, beloved ol wiitcis on physical chemistiy, is not applicable It will be observed that the equation tells us nothing concerning the straight line AB 3 which lepresents the actual passage observed m nature from the vapour to the liquid phase. The position of this line on any given isothermal can, however, be obtained from the simple consideration that the areas DBCD and AEDA must be LIQUIDS AND GASES 47 qual,* and the line must be drawn to fulfil this condition. Fig 1 5 shows that the pv-p curves in general exhibit a minimum alue for pv, and that the locus of these points lies on a definite curve. 'he equation to this curve may be obtained by writing pv as y and as x in the characteristic equation, and expressing the condition lat y should have a minimum value. Of the other characteristic equations that have from time to time sen proposed we may cite, naming them by their authors: Clausms (a): Clausius (b). Dieterici (a): L + * \ (V - b) = RT, Dieterici (6). />(V b) = RTe he deduction of the cutical constants from these equations is left the reader. The value of a chaiactenstic equation which shall closely repiesent e pressure- volume relations of a fluid over a wide range of pressuies id temperatuies, is obvious We have seen, in the section on impossibility, that many important physical constants may be pressed in terms of theimodynamic equations involving certain ffeiential coefficients and integrals The values of these physical instants may be worked out by substituting, in the appropriate ermodynamic equations, the values of the differential coefficients * Foi in any reversible cycle ( /) -~ o If the cycle be isothermal, (M9 = 1 (f) dQ, and therefore (l)dQ = o. it for any cycle ( /) (dQ -I- dW) = o, )cfW = o So that, if we take it mass of the substance leversibly round the cycle AEDCBDA (fig. 14), the >rk done, lepresented by the sum of the positive and negative areas AEDA and 3BD, must be zero. Hence the two areas are equal (For critical remarks on is pioof see Preston, Theory of Heat, 479 (1904), or Jeans, Dynamical Theory oj uses, p. 159.) 48 THE MECHANICAL PROPERTIES OF FLUIDS obtained from the differentiation of the equation of state Un- fortunately, no equation yet pioposed covers the whole giound satisfactorily. An equation which fits the experimental figuies at one end of the scale is usually unsatisfactory at the other end, and con- versely. A few simple tests may be suggested by which the fitness of any given equation may be roughly examined. The expenments oi RT Piofessor Young show that, while the value of the latio - c vanes PPc slightly from substance to substance, its mean value may be taken as about 3-75 Now the equation of state of an ideal gas gives unity for this ratio, and is cleaily veiy far out of it. Van der Waals' equation gives RT C / Sa \ / a \ , ____ c = (R x O ( : X 36) = 267, p e v e \ 2 7 R/;/ \2 7 & 2 6 I h and is not a veiy good approximation to the truth Similarly the (a) and (b) equations of Clausms give for this ratio the values 2-67 and 3 oo respectively, and the corresponding equations of Dieterici give 3-75 and 3-69, if in the (a) equation we take the value of k as -* Another test may be derived from the expenmental fact that the critical specific volume v c is about four times the liquid volume Now the constant b, which represents the least volume into which the molecules can be packed, cannot be seiiously drfleient from the liquid volume. Accordingly we find, on woikmg out the values lor v c) that foi the (a) and (b} equations of Clausms the values ol v c ai e 3& and 4/; i espectively,* whrle the corresponding equations ol Dretencr give the values $ and 2&. Foi van dei Waals' equation the value is, as we have seen, 36. But the subject may be studied from a different point of view Instead of attempting to devise an equation which shall leprescnt the properties of a substance over a wide lange a process which usually lesults in a cumbrous foimula we may tiy to anive at an equation which shall be simple and manageable in foim, so that the vanous physical constants of the fluid may be leadily worked out from the corresponding thermodynamic relations, while at the same time the equation shall represent a very close approximation to the tiuth over a limited lange, the range chosen being one of practical impoitance. Whether such a formula can, 01 cannot, be extrapolated * If m the (6) equation we put zC = b. LIQUIDS AND GASES 49 yond the limits of the range is a mattei of secondary interest what important is that the formula should be as exact as may be within 2se limits. Let us then investigate the form which such an equation as the equation of Clausms assumes for moderate pressures. Re- iting the equation as see that at model ate pressures, when the volume is large, we shall t be seriously in error if we write (v+V = ^ ( a pp roximatel y)- 1 thus have p(V - b) = RT - a TV I again, putting, m the small term, V - RT T' find, on rearranging the equation, V = *I-^ + a, p T 2 ' 2re c is put for . If we replace T 2 by the more general form T", wheie n varies from stance to substance, we have - , p T" ch is the form known as Calendar's equation This equation has been applied very successfully to elucidate pioperties of steam over a lange of pressure from o to 34 atmos- res; the value of n appropriate to steam is --3- Space will not nit us to discuss at length this important equation Indeed, the ussion lies within the piovince of thermodynamics, and the ler desirous of further information should consult the articles hermodynamics " and " Vaporization " in the Encyclopedia 'anmca, or a textbook such as Ewing's Thermodynamics for ineers, 50 THE MECHANICAL PROPERTIES OF FLUIDS Osmotic Pressure If we throw a handful of currants or raisins into water and leave them for a while, we find that the fiuits, originally shiunken and wrinkled, have swelled out and become smooth.* Water has passed through the skin of the fruit, while the dissolved substances inside cannot pass out or at least do not stream out so freely as the water streams in. This unilateral passage of a substance thiough a mem- brane is termed osmosis.^ In the limiting case, when we have a solution on one side of a membrane and pure solvent on the other, the membrane is called a semi-permeable membrane if it freely admits of the passage of the solvent, but is strictly impervious to the dissolved substance tt has been asserted that no such membranes exist in nature, but, as far as experiment can show, a membrane of copper ferrocyanide forms a true semi-permeable membrane to a solution of sugar in water Suppose such a membrane, prepared with due precautions J and it is not so easy as one would imagine to prepare a thoroughly resistant membrane to be deposited on the inside of a cylindncal porous pot. The pot is filled with a sugar solution, closed, and attached to a suitable manometer which shall measuie the pressuie inside the pot It is then placed in a vessel containing pine water We shall find that the pressure in the pot uses, finally i caching a maximum stable value The maximum value of this pressure, assuming that the membrane is truly semi-permeable, is called the osmotic pressuie of the solution. It will thus be seen that osmotic pressure is defined in teims of a semi-permeable membrane. It is only in very loose phiaseology that one can speak of the osmotic pressure of a solution without reference to the existence of a semi-permeable membiane. A solu- tion, qua solution, has no osmotic pressure. If this definition be consistently followed, a great deal of vague and loose reasoning of the sopoiific-power-of-opium variety will be swept away. It is all the more needful to emphasize this point as there has arisen, in biological (and even in engineering) circles, a tendency to ascribe to " osmotic pressure " a power and potency * Imbibition of water by the dried tissues will also play a part m the smoothing process f From iSoy*6s, a raie Greek noun meaning " thrusting " or " pushing through ". t Morse, Jour. Amer Chem, Soc., 45, 91 (1911). LIQUIDS AND GASES 51 ich is almost proportional to the vagueness with which the chanism of that pressure is conceived. Consider, for example, the common remark that osmotic pressure cts the wrong way " that is, causes motion from a region of lower lottc pressure to a region of higher osmotic pressure. It only uires a little consideration of the definition of osmotic pressure y to realize that the argument involves a varrepov irporepov, it is clear that it is osmosts which produces osmotic pressure, not lotic pressure which produces osmosis. The quantitative laws of osmotic pressure were first studied by ffer, whose figures show that for dilute solutions the pressure, constant temperature, is proportional to the concentration, and at constant concentration the pressure is proportional to the )lute temperature. We may therefore write PV = KT, it has been shown by van't Hoff that the constant K has the e value as the gas constant R Hence it follows that the osmotic ssure of the solution is the same as that which would be exerted the dissolved substance were it dispersed in the form of a gas >ugh a volume equal to that occupied by the solution, [f we desire to correct this simple gas law, we find it necessary ook at the matter from a different angle. We define an ideal tion as containing two completely miscible unassociated com- ents, of such a nature that there occurs no change of volume on ing, and that the heat of dilution is negligible For such a solution it can be shown that the osmotic pressure P ^^ P^ 2 RT f 1 ( P + 0- = -^-| ~ log a (l - re /? denotes the compressibility of the solvent, V its molecular me, and x the ratio of the numbei of molecules of the dissolved stance to the total number present. If we neglect /?, which is illy small, and expand the logarithmic term, we have the con- entform RT/r ^ ^ v P = -^r-U-h- + -+ ) V \ 2 3 / 3 equation holds good for any concentration. Despite a large amount of criticism, the Idnetic theory of osmotic sure still holds the field as the only one which gives values of the 52 THE MECHANICAL PROPERTIES OF FLUIDS pressure calculated from theory.* The properties of the membiane, which play a large part in some theories, whilst of great inteiest and value, are distinctly of secondary importance in the kinetic theory. It is the thermal agitation of the molecules of the solute which is effective in producing osmotic pressure, and the magnitude of the pressure calculated from the agitation of the molecules is equal to the value obtained by experiment, " Any other theoiy put forwaid to account for osmosis must fulfil, then, a double duty, not only must it be competent to explain osmosis, but it must also explain away the effects that we have the light to expect from the molecular agitation of the solute."! NOTE ON TRANSFORMATION OF CO-ORDINATES Perhaps the simplest way of passing from one foirn to the other is to consider the concentiation of fluid, reckoned pei unit volume per second, at a point If u, v, w are the component velocities at a point P, the fluid leaving the elemental volume SxSySst m time 8t is readily found, by the method given below for a more difficult case, to exceed that which enters it, by an amount ,~ ~ (mi . ov />(;r- + \ox where p is the density, i.e the concentration, icckoned in mass pei unit volume per second, is r . OU Q- oy Taking now the element of volume shown m fig. 16, rS6 Fig. 1 6 * Porter, Trans Far Soc , 13, 10 (1917) The reader desiring more information on the subject should consult this valuable discussion f Porter, loc, cit,, p. 8, LIQUIDS AND GASES 53 ind letting u' be the ladial velocity and / the tangential velocity at P-, ve sret u'rBdSzBt for the radial flow in, and u' + ~-o>) (r80 -f l(rS0)8fW* for the radial flow out. \ or / \ or / The latter exceeds the former by OU , / r -- L u ' or The tangential flow in is nd the tangential flow out, r hich exceeds the former expression by e the total flow out is , . du' . 3w'\ s /j<j r> o. u + r- + -^ )808r88f, 9r 30 / e the total flow //o the element is this expiession taken with the unus sign The elemental volume is rWSzftr , cnce the concentration is in' , du' . i . . f\\ L _L P\-~ I -5" r ~ \r or r The concentration mass per unit volume per second must be the mie whatever co-ordinates we use, hence du , dv u' du' . i d^ ,-, ^.^ nzn 4 *i lil /-\ I -\ I *-i I O/l ^ ' 9^; oy r or r ou If the fluid motion has a velocity potential, the component velocity i any direction is the gradient of potential in that direction, i e 3c4 3<i ' 11 = - L. rv J ^ o > 3^ 9y , 8(4 / 1 3(4 M' I rri' : 7" "X" ' 'vi' 3r r 30 54 THE MECHANICAL PROPERTIES OF FLUIDS for the element of length perpendicular to r, i.e. in the tangentia direction, is rd&\ hence by substituting the values of w, w', w, e;', in equation (i). We have proved the transformation when the dependent variable is the velocity potential <, but as the transformation is a purely analytical one in its nature, the form must be equally true whatever be the physical nature of (j>; for instance, if (f> = w, the # component of 2 the form which actually occurs (p. 33), and, in geneial, at every point in a plane where U is a single-valued function of the co-ordinates of the point and possesses finite derivatives up to those of the second order there. CHAPTER II Mathematical Theory of Fluid Motion It is assumed throughout this chaptei that the fluid with which we deal may be regarded as incompressible. This only means that changes of pressuie aie propagated in it instantaneously, instead of with the (very great) velocity of sound. Since the velocity of sound in air is only about four times less than in water, it is cleai that many of our lesults will be equally applicable to gaseous fluids, the influence of compressibility being negligible except in the case of veiy rapid differential motions. It is further assumed in the first instance that the fluid isfnction- less, i e. that it presses perpendicularly on any surface with which it is in contact, whether it be the surface of an adjacent poition of fluid, or of a solid boundary This hypothesis of the absence of all tangential stress is not in accordance with fact, but it greatly simplifies the mathematics of the subject, and there are, moreover, many cases of motion in which the influence of friction is only secondary It follows from this assumption that the state of stiess at any point P of the fluid may be specified by a single quan- tity p, called the " pressuie-intensity " or simply the " piessme ", which measures the force per unit area excited on any suiface through P, whatever its aspect. This is in fact the cardinal pioposition ol hydrostatics. It is convenient here to prove, once ior all, that the icsultant of the pressures exerted on the boundary of any small volume Q of fluid is a force whose component in the direction of any line element s is where dpfis is the gradient of p in the direction of 8$. Take first the case of a columnar portion of fluid whose length $x is parallel to MATHEMATICAL THEORY OF FLUID MOTION 57 he axis of #, and suppose that the dimensions of the cross-section w) are small compared with Bx. The pressure-intensities at the wo ends may then be lenoted by pa} _ + Q ;)+( P <.$BS X)Q) p and p + /Sx, * Flg , ox o that the component parallel to the length of the pressure on the olumn is he pressures on the sides being at right angles to Bx Since 5 the volume of the column, the formula (i) is m this case verified. iince, moreover, any small volume Q may be conceived as built up f columnar portions of the above kind, and since there is nothing pecial to the dn action Ox, the result is seen to be general. Stream -line Motion i Bernoulli's Equation A state of steady 01 stream-line motion is one in which the stream- nes, i e the actual paths of the particles, preseive their confirmation nchanged The most obvious examples are where a stream flows ast a stationary solid, and the designation is naturally extended ) cases where a solid moves umfoimly m a straight line, without station, through a surrounding fluid, piovided the superposition of uniform velocity equal and opposite to that of the solid reduces ic case to one of steady motion m the formei sense This super- osed velocity does not of couise make any difleience to the dynamics f the question The stream-lines drawn through the contoui of any small area ill mark out a tube, which we may call a stream-tube. Since the ime volume of fluid must ti averse each cioss-section in the same me, we have heie co!, o> 2 are the areas of any two cross-sections at P L , P 2 , and L , # 2 the corresponding velocities in the direction (say) from P x > P 2 . Now consider the region included between these two sections. i a short time dt a volume Q = oj^dt will have entered it at 58 THE MECHANICAL PROPERTIES OF FLUIDS P 1} and an equal volume Q = o> z q 2 dt will have left it at P 2 The work done by hydrostatic pressure in this time on the mass of fluid which originally occupied the space PjPg will be Pi^iqidt or piQi, at P lf and p z a) z q z dt or p z Q at P 2 . The same mass will have gained kinetic energy of amount ^pQ(q^ #i 2 ), where j"/"^ ^X/^ P ' 1S tne density, i.e. the mass per unit volume. If V denotes the poten- tial energy of unit mass, the gam of potential energy will be pQ(V 2 Vj) Hence, equating the woik done on the mass to the total increment of energy we have or Pi Hence along any stream-line P + W + /v = c, ............. (2) where C is a constant for that particular line, but may vaiy from one stream-line to another. This equation is due to D. Bernoulli (1738) and was proved by him substantially in the above manner The formula has many applications For instance, in the case of water issuing from a small orifice in the wall of an open vessel we have at the upper surface p = p (the atmospheric piessure), and q = o, approximately Again, the value of V at the upper surface exceeds that at the orifice by gsi, where a is the diflcrence ol level and g the acceleiation due to gravity Hence if q be the velocity at the surface of the issuing jet, Po -I- gpz = A + lpq z , ^ r (? 2 = 2gZ, ........... (3) r a foimula due to Tomcelli (1643). If S' be the section of the jet at the " vena contracta ", where it is sensibly paiallel, the piessure over S' will be p Q The velocity will therefore be given Jg 3 by the above value of q, and the discharge per unit time will be pqS' The ratio of S' to the area S of the orifice is called the " coefficient of contraction ". It is not easy to determine this coefficient theoretically, but a MATHEMATICAL THEORY OF FLUID MOTION S9 r ery simple argument shows that in the case of an orifice in a hm wall it must exceed f. Take, for instance, an orifice in a r ertical wall. In every second a mass of pqS' escapes with the elocity q, and carries with it a momentum pq z S f . This represents he horizontal force exerted by the vessel on the fluid. There must e a contrary reaction of this amount on the vessel. On the opposite rail of the vessel, where the velocity is isignificant, the pressure has sensibly the tatical value due to the depth, and if this /ere also the case on the wall containing c" tie orifice there would be an unbalanced > 3rce gpzS urging the vessel backwards r. Actually, owing to the appreciable elocity, the pressure near the orifice will e somewhat less, so that the reaction ex- Flff 4 2eds gpzS . Hence pq z S' > gpzS , or (since * = 2 gs) S' > |S In a particular case, where the fluid escapes y a tube projecting inwards in the manner shown in the figure, the atical pressure obtains practically over the walls, and S' = -|S sactly. This arrangement is known as " Borda's mouthpiece " Another application of (3), much used in aeronautics and engi- eering, is to the measurement of the velocity of a stream, e g of ic relative wind in an iioplme. The quantities / ' i i ^ and p -\- Ipq* are mea- > ired independently, and icir diffeience detei mines A fine tube, closed at ( le end and connected with pressure-gauge at the F ' s her, points up the stieam ) as to interfere as little as possible with the motion, and contains few minute holes in its side, at a little distance from the closed id; the gauge therefore gives the value of p On the other hand, i open tube diawn out at the end almost to a point, and >nnected to a second gauge, will give the value of p -f %pq z at a tort distance ahead of the vertex. For if p' be the pressure at the *tex itself, where the velocity is arrested, we have p + %pq z p', for points on the same stream-line. The two contrivances e often united (as in the figure) in a single appliance known as a Pitot and static pressure tube". 60 THE MECHANICAL PROPERTIES OP FLUIDS 2. Two-dimensional Motion, Stream-function There are two types oi stream-line motion which ate specially simple and important. We take first the two-dimensional type, where the motion is m a system of parallel planes, and the velocity has the same magnitude and direction at all points of any common normal. It is sufficient then to confine oui attention to what takes place in one of these planes. Any line drawn in it may be taken to represent the portion of the cylindrical surface, of which it is a cross-section, included between this plane and a parallel plane at unit distance from it. By the " flux " across the line we understand the volume of fluid which in unit time crosses the surface thus defined Now taking an arbitrarily fixed point A and a vaiiable point P, the flux (say from right to left) will be the same across any two lines drawn from A to P, provided the space be- tween them is wholly occupied by fluid This is m virtue of the assumed constancy of volume The flux will theiefoie be a function only of the position of P, it is usually denoted Fig o by the lettei ?//. It is evident at once from the definition that the value of ^ will not altei as the point P describes a stream-line, and theiefore that the equation i/r = constant . . (4) will define a sti earn- line For this icason ;/ is called the stt cam- function. If P' be any point adjacent to P, the flux across AP' will differ from that across AP by the flux acioss PP', whence, wilting PP' = 8s, we have 8i/r q n Ss, where q n is the component velocity normal to PP', to the left. Thus where dfi/ds is the gradient of ^ in the direction of 8s. This leads to expressions for the component velocities w, v parallel to rectangulai co-ordinate axes. If we take 8s parallel to Oy we have q n = w, whilst if it be taken parallel to Ox we have q n = v. Thus %? 5 These satisfy the relation du . dv MATHEMATICAL THEORY OF FLUID MOTION 61 which is called the equation of continuity. It may be derived other- wise by expressing that the total flux across the boundary of an elementary area BxBy is zero. Again, if we use polar co-ordinates ', 6, and take Bs (= Br) along the radius vector, we have q n = v, where v denotes the transverse velocity; whilst if Ss (== rS6) be at ight angles to r, q n = w, the radial velocity. Hence difj di[t /o\ * = ~m* v " fr ( } t follows that f v 0V f \ "f\" V "/ 1^ /"v/i ~~"~~ > * * i**o** \;7/ 9r 30 vhich is another form of the equation of continuity. It is to be remarked that the above definition of /r is purely ,eometncal, and is merely a consequence of the assumed incom- tressibihty of the fluid. If we make any assumption whatever as the form of this function, the formulas (6) or (8) will give us a lossible type of motion; but it by no means follows that it will >e a possible type of permanent or steady motion. To ascertain the ondition which must be fulfilled in order that this may be the ase we must have recourse to dynamics, but before doing this it is onvenient to introduce the notions of circulation and vorticity. The circulation round any closed line, or circuit, in the fluid is tie line-integral of the tangential velocity taken round the curve 1 a preset ibed sense In symbols it is ds, . . .(10) /here x is the angle which the direction ot the velocity q makes with tiat of the line-element 8s In lectangulai co-oidmates, resolving and v in the direction of Bs, we have dx , dy q cosx = u + v-+, ds ds o that the circulation is Kdx . dy\j [/ j , j \ t \ u~ + v~- )ds, or (udx + vdy) (i i) A ds] i t will appear that the circulation round the contour of an infinitely mall area is ultimately proportional to the area. The ratio which bears to the area measures (m the present two-dimensional case) ic vorticity] we denote it by . Its value, in terms of rectangular 62 THE MECHANICAL PROPERTIES OF FLUIDS co-ordinates, is found by calculating the circulation round an elementary rectangle PQRS whose sides aie $x and Sy. The por- tions of the line-integral (n) due to PQ and RS S * R aie together equal to the difference in the cor- icspondmg values of uSx, i.e. to -~SySx. \ I The portions due to QR and SP are in like manner equal to SxBy. Equating the sum to p >- Q x Fig. 7 &*%> we have c. dv __ du , ^ dx 9y' or, from (6), f\e* i r\n t o O^/r O^ift The value of at any point P is related to the aveiage lotation relative to P of the particles in the immediate neighbourhood To examine this, we calculate the circulation round a circle of small radius r having P as centre The velocity at any point Q on the circum- ference may be regarded as made up of a general velocity equal to that of P, and the velocity relative to P. The former of these contri- butes nothing to the required circulation The latter gives a tan- gential component cur, where cu is the angular velocity ol QP The circulation is therefore fZtr rZt I corrdO = r 2 / ./ o J o where 6 is the angular co-ordinate of Q. Since the same thing is expressed by nr 2 , we have TTJ which is twice the average value of co on the circumference. For this reason a type of motion in which is everywhere zero, i.e in which the ratio of the circulation in every infinitesimal circuit to the area within the circuit vanishes, is called irrotational. MATHEMATICAL THEORY OF FLUID MOTION 63 3. Condition for Steady Motion We can now ascertain the dynamical conditions which must e satisfied in order that a given state of motion may be steady, 'or this purpose we consider the forces acting on an element, 'QQT', of fluid included be- tfeen two adjacent stream-lines tid two adjacent normals. The itter meet at the centre of cur- ature C. We write PQ = &, P' = Bn, PC = R. The mass f the element is therefore p$s8n. 'he forces acting on it may be ssolved in the direction of the ngent and normal, respectively, 1 the stream-line PQ. Tan- in tial resolution would merely ad us again, after integration, Bernoulli's equation (2). ormal resolution e help of (i), gives, with Fig 8 SsSn 3V dn here dp fin and 3V /dn are the gradients of p and V in the direction - Hence a z lie cnculation lound the cucuit PQQ'P' will be equal to Ss8n calculating this cnculation, we may neglect the sides PP', QQ' >ce they aie at right angles to the velocity The contributions of e remaining sides are q$s and - lere $s r = P'Q'. Now from the figure we have 8/ = CP' ^ R - Sn Ss ~ CP ~ " R ' that the circulation is 64 THE MECHANICAL PROPERTIES OF FLUIDS omitting teims of higher oidei than those retained Hence =,, _^ + l .......... ( I6 ) if <-\ I Tpk \ / dn R The formula (15) may now be written -pql ..... (I?) Comparing this with (2), we have 8C = pqt$n, .... . . (18) where C is the quantuy which was proved bcloie to be constant along any stream-line, but will in general vary fiom one stieam-lme to another. If we fix our attention on two consecutive stieam-lmes, SC will be a constant, and q$n will also obviously be constant The dynamical condition for steady motion is therefoie that the vorticity should be constant along any stream-line. When it is fulfilled, the distribution of pressure is given by (2) and (17) We may expiess the result otherwise by saying that any fluid element letains its voiticity unchanged as it moves along. This is a paiticular case oi a theoiem in vortex-motion to be proved later An obvious example is that of fluid rotating with uniform angulai velocity co about a vertical axis, and subject to giavity The law ot distribution of pressure may be deduced from (17), or moie simply from first principles. If r be the ladms of the ciicular path of a small volume Q, the resultant force upon it must be radial, of amount pQo>V. Hence, and since there is no veitical acceleiation, we have 2 ty ^/> / \ *"' = <>=--' ...... (-9) the positive direction of % being that of the upward veitical. It follows that p = %pa> z r z gpz -\- constant ......... (20) The free surface (p p Q ) is therefore the parabola x = - if the origin of x is where the free surface meets the axis (r o). MATHEMATICAL THEORY OF FLUID MOTION 65 If we imagine the fluid contained within the cylindiical surface = a, rotating in the above manner, to be surrounded by fluid loving irrotationally, we have in the latter region dq/dr + q/r = o, om (16), or x #r a= constant = w^ ernoulli's equation then gives p = constant gpz \p he equation to the free surface is therefore ry _L ~ o ~ (22) (23) g here the additive constant has been chosen so as to agree with i) when r = a It appears that these equations also give the Fig 9 ne value of dz/dr for r = a Putting r = oo in (24), we find it the depth of the dimple formed on the free surface is cuPaP/g. 4. Irrotational Motion We proceed now to consider more particularly the case of irro- ional motion. The condition for steady motion is fulfilled auto- itically if = o everywhere, provided, of course, the necessary undary conditions are satisfied, as they are in the case of the w of a liquid past a stationary solid. The geometrical condition i) reduces to , 2 , , 2 , (D312) 66 THE MECHANICAL PROPERTIES OF FLUIDS or, in polar co-ordinates, ?! -1- I ^ -t- I ^ = o (26) dr z ^ r $r r 2 d6* ............ V ' and the pressure- distribution is such that P 4- |p# 2 -f pV = constant ....... (27) throughout the fluid. The particular value of the constant is for most purposes unimportant, since the addition of a unifoim pres- sure throughout does not alter the resultant force on any small element of fluid, or on an immersed solid. Some simple solutions of (25) or (26) aie easily obtained. Thus = \] y = Ur smS ...... (28) gives a uniform flow with velocity U from left to right. Again, take the case of symmetiical radial flow outwards from the ongm. The stream-lines are evidently the radii, so that ifi is a function of Q only. Since the total flux outwards across any circle r constant must be the same, we have froin (8) 9.4 - = constant = m, say, rod , m n t -. or = 0. ..... (29) 27T If m be positive we have here the fictitious conception of a line-source which emits fluid at a given rate If m be negative we have a sink Since (29) would make the velocity infinite at the origin, these imaginaiy sources and sinks must be external to the legion occu- pied by the fluid. The formula (29), for instance, would be realized by the expansion of a circular cylinder whose axis passes through O. Again, since the differential equations (25) and (26) are linear, they are satisfied by the sum of any number of sepaiate solutions. For instance, the combination of a source at A, and an equal sink at a point B to the left of A, gives = --&-**), ............... (30) 27T wheie 1} a are the angles which the lines drawn from A and B to any point P make with the direction BA. Since 1 2 = APB, the lines ^ = constant are a system of circles through A, B. This MATHEMATICAL THEORY OF FLUID MOTION 67 ind of motion would involve infinite velocities at A and B, but if e combine (28) with (30) we get the flow past an oval cylinder hich encloses the imaginary source and sink. If the points A and be made to approach one another, whilst m increases so that the roduct mAE is constant, we have ultimately t Z APB = B sm#/r. We thus get the form ombining this with (28), we have / - Ur + - (32) Fig 10 ie stream-line j/r = o now consists of the radii 6 = o, 6 = d the circle r a, provided C = Ua z . The formula a d, = - \j (r- \ -\smO j (33) jrefore gives the flow past a circular cylinder. The normal velocity the surface is of course zero, whilst the tangential velocity is ^ = - U/i + *\in0 dr \ r 1 / (34) cutting the external forces, if any, represented by V, which have rely an effect analogous to buoyancy, the pressure at the cylinder p = constant 2/>U 2 sin 2 ............. (35) 68 THE MECHANICAL PROPERTIES OF FLUIDS Since this is unaltered when is replaced by 0, or by + (TT 6}, it is evident that the stream exerts no resultant force on the cylinder. Some qualification of this result will be given presently. Meantime we note that if we superpose a general velocity U from light to left, we get the case of a cylinder moving with uniform velocity (and zero resistance) through a fluid which is at rest at infinity. The stream-function then has the form Fig n so that the relative sti cam-lines are portions o the circles r = Csinfl, which touch the axis of x at the origin. If we calculate the square of the velocity from (36), we find (^ l\ ' Oijj\' to) W ' \rW/ V* The total kinetic energy of the fluid is therefore (37) where M' is the mass of fluid displaced by the cylinder. The effect MATHEMATICAL THEORY OF FLUID MOTION 6g f the presence of the fluid is therefore virtually to increase the tertia of the latter by M'. Another simple type of motion is where the fluid moves in mcentric circles about O. The velocity is then a function of r ily If the motion is irrotational we must have, by (22), qr nr UJ. constant Or K __ , 27T iere K is the constant value of the circulation round 0. Thus additive constant being without effect. This corresponds to the se of a concentrated hne-vortex at the origin, and would give mite velocity there. For this icason (38) can only relate to cases iere the origin is external to the space occupied by the fluid The combination of (33) and (38) makes - U r -} sind + -1 logr r J 2TT (39) Fig 12 ! tangential velocity at the cylinder is now = - 2U of 2ira 70 THE MECHANICAL PROPERTIES OF FLUIDS whence p = constant 2pU 2 sin 2 + /> sm0 ...... (40) TtCL The last term is the only one which contributes to a resultant force. Since it is the same for 9 and -n 0, there is on the whole no force parallel to Ox. The force parallel to Oy is r J - p/clL. .(41) This resultant effect is due to the fact that (if K be positive) the circulation diminishes the velocity above the cylinder and increases it below, and that a smaller velocity implies (other things being the same) a greater pressure. It may be shown that the icsult is the same for a cylinder of any form of section, as might be expected from the fact that it does not depend on the radius a. This theoiem is the basis of Prandti's theory of the lift of an aeroplane. 5. Velocity-potential We may imagine any area occupied by fluid to be divided by a double series of lines ciossing it into infinitesimal elements. The circulation round the boundary of the area will be equal to the sum of the circulations round the various elements, piovided these circulations be estimated in a con- sistent sense. For, in this sum, a Fig 13 Fig 14 side common to two adjacent elements contributes amounts which cancel. Hence if the motion be irrotational the circulation round the boundary of any area wholly occupied by fluid will be zero. We have here assumed the boundary to consist of a single closed MATHEMATICAL THEORY OF FLUID MOTION 71 urve. If it consists of two such cuives, what is proved is that the urn of the circulations round these in opposite senses is zero. In ther words, in irrotational motion the circulation in the same sense s the same for any two circuits which can by continuous modifica- ion be made to coincide without passing out of the region occupied y the fluid. For example, in the case to which (38) refers, the cir- ulation in any circuit which embraces the cylinder is /c, whilst that i any other circuit is zero. This leads to the introduction of the function called the velocity- otentzal, in terms of which problems of irrotational motion are often iscussed. This is defined by the integral & == I (udx -f- vdy) . (42) J A A ken along a line diawn from A to P. The integral has the same due for any two such lines, such as ABP, ACP in the figure, pro- ded the space between them is fully :cupied by fluid. For, reversing the rection of one of these lines, the iths ABP, PCA together form a osed ciicuit, round which the cncu- tion is zero It follows that so long A is fixed, cf) will be a function of e position of P only. If P' be any Fig 15 >int adjacent to P, the mciement of in passing from P to P' is S</> = qfis, where q t is the component locity m the direction PP', and PP' = 8s. Hence lere d(f>/ds is the gradient of <j> in the direction PP'. For instance, rectangular co-ordinates, putting first Ss = Sx, and then = Sy, ' have dA d< / . . v U=. __L i} = ~ . . (44) dx' dy nilarly, the radial and transverse velocities in polar co-ordinates ; given by d<f) d<f> f \ u = ~ x, v -^ (45) dr red 72 THE MECHANICAL PROPERTIES OF FLUIDS From (7) and (44) we deduce 8V . 8V sl + iy ' ....... ......... whilst in polar co-ordinates, from (9) and (45) 9 / 8fv . i 8V , . (47) It is the similarity between these iclations and those met with in the theories of attractions and electrostatics that has suggested the name " velocity-potential ". For the same reason the cuives foi which <j) is constant are called equipotential lines If in (43) Ss be taken along such a line we have q s = o, showing that the equipotential lines cut the stream-lines at light angles If on the other hand Sn be the perpendicular distance between two adjacent equipotential lines, we have 80 == q$n If, therefoie, we imagine a whole system of such lines to be drawn for equal small increments S(f>, the perpendicular distance between consecutive lines will be evciy- where inversely propoitional to the velocity. If, further, we suppose the stream-lines to be drawn for intervals Si/f each equal to S</>, we have Si/r = #3$', where Ss' is the interval between consecutive stream-lines of the system. Hence 8s' = 8n, showing that the stream - lines and equipotential lines drawn for equal inciemcnts of the functions will divide the region occupied by the fluid into infinitesimal squares. The functions and fi are connected by the relations =, = - dx 9y' dy fix in which the equations (25) and (46), expressing the incompressi- bihty and the absence of vorticity, are implied. If we write For this makes dzo .- . dw = a -f zy, ........... (49) are satisfied by any assumpti =/(*)... ....... (50) where i +/( i), the relations (48) are satisfied by any assumption of the form MATHEMATICAL THEORY OF FLUID MOTION 73 vhence, substituting the value of w, and equating sepaiately real and maginary paits, we repioduce (48). For example, if zo Uz, we have I = ~U*, = -Uy, ............ (52) xpressing a uniform flow paiallel to Ox. Again, if w ~ CJz - i sin0) ..... (53) This corresponds to (36), if C = U<z 2 , and shows that in the ase referred to (54) r A moie general assumption is to = Cs", or $ + lift = C(x + y) B = Cr n (cosn6 The stream-function is now = O" /hich vanishes both for 6 o and 6 a, provided n rr/a. Baking these lines as fixed boundaries we have the flow in an angle, r round a salient, according as a > TT. The radial and transverse elocities are, by (8), nCr n ~ l cosnd and nCr n ~ l smnd, espectively If a < TT, n > i , and these expressions vanish at the ertex where r = o If, on the other hand O.>TT, n<i, and tie velocity theie is infinite Even if the salient be rounded off, tie velocity may be very great, with the result that the pressure falls luch below the value at a distance It is otherwise obvious that if tie fluid is to be guided round a sharp curve there must be a lapid icrease of pressuie outwards to balance the centrifugal force If bis is not sufficient a vacuum is formed and " cavitation " ensues. If w = C log#, where C is real, ^ -f njj = C log(x + ty) = C logre' 9 = C logr + iC6. (55) Tiis represents a line-souice of strength m, if to agree with (29) re put C WZ/27T. The corresponding value of ^ is (D312) 4 * 74 THE MECHANICAL PROPERTIES OF FLUIDS If on the other hand C is a pure imaginary, zA, say, ^ -j- ^ = ~ AO + tA logr (57) This repiesents the case of the line-vortex to which (38) refers, if we put A = K/ZTT, and so make (58) The function <{> has an important dynamical inteipietation. Any state of motion in which there is no circulation in any circuit, and in which, theiefore, cj> has a definite value at every point, could be generated instantaneously from rest by a proper application of impulsive pressures over the boundary. For the icquisite condition for this is that the resultant of the impulsive pressures (oi) on the surface of any small volume Q should be equivalent to the momen- tum acquired by this. Hence if q s is the component velocity in the direction of any linear element Ss we must have which is satisfied if & = P^ .............. (59) Hence <f> determines the impulsive pressure requisite to stait the actual motion in the above manner. As an example, we may take the case of a cylinder moving through a laige mass of liquid, without circulation, to which the foimula (54) refers. The resultant of the impulsive piessuies on the surface of the cylinder is paiallel to Ox, of amount = M'U, . .(60) if M' = 7r/>a 2 as befoie. The total impulse which must be given to the cylinder to start the motion is therefore (M + M')U. This confirms the former result that the inertia of the cylinder is viitually increased by the amount M'. MATHEMATICAL THEORY OF FLUID MOTION 75 6. Motion with Axial Symmetry. Sources and Sinks The second type of motion to which reference was made on 54 is where the flow takes place in a system of planes passing rough an axis, which we take as axis of x, and is the same in each ch plane. We denote by x^y the co-oidmates in one of these ines, by r distance from the origin, and by 6 the angle which r ikes with Ox. The conditions for steady motion are obtained by e previous process. Resolving along a stream-line we should be i to Bernoulli's equation (2); whilst the normal resolution in an ial plane yields equations of the same form as (15) and (17), pro- led now denotes the vorticity in that plane The inference to the distribution of vorticity is however altered. The space tween two consecutive stream-lines now represents a section of thin shell, of revolution about Ox, and the flux in this is accord- *ly q 27ryS Comparing with (18), w.e see that along any stream- e must vary as y. We may conceive the fluid as made up of nular filaments having Ox as a common axis. The section of such filament, as it moves along, will vary inversely as y, hence the oduct of the vorticity into the cross-section must remain constant. us is a particular case of a general theorem that the strength of a rtex-filament (in this case a vortex-ring) lemams unaltered as it If = o, the aigument for the existence of a velocity-potential 11 hold as befoie One or two simple cases may be noticed If : imagine a point-source at O, the flux outwaids across any con- Qtnc sphencal surface of radius r must be equal to the output ') per unit time whence d<{> . m i ., , -^ ATT?-" = m, or d> = -- ..... (61) or 47? r We may apply this solution to the collapse of a spherical bubble. R be the radius at time t, we have rh R2 ^ R /< x #- T -^ - - (62) ice this makes 90 Jdr dR/dt for r R. The corresponding netic energy of the fluid is y6 THE MECHANICAL PROPERTIES OF FLUIDS If pQ be the pressure at a distance, the rate at which work is being done on the fluid enclosed in a spherical surface of laige radius r is -p q4<7rr* = - 4 77/> R a ~, (64) the pressure inside the bubble being neglected. Equating the rate of increase of the energy to the work done, dt whence W-* n / W-J.X \ o A'ft/TTfc Q T^ ON //" /" \ R3( \ = fP(R 3 R 3 ), (66) where R is the initial radius of the cavity It is not easy to integrate this further in a practical form, but the time of collapse happens to be ascertainable, it is ' (6?) Thus if pQ be the atmospheric pressure, and R = i cm., r is less than the thousandth part of a second. The total eneigy lost, or lathci converted into other foims is, from (63) and (66), irf> R 8 . . . ..(68) 3 In the particular case referred to, this is 4 19 X io 6 eigs, 01 o 312 of a foot-pound The expansion of a spherical cavity owing to the piessuie of an included gas can be treated in a similar way This illustrates, at all events qualitatively, the early stages of a submanne explosion The potential energy of a gas compressed under the adiabatic con- dition to volume v and pressure p is pvf(y i), wheie y is the ratio of the two specific heats. If p be the internal picssure when the radius of the cavity is R, and p its initial value, we have by the adiabatic law . /~ v 3V The potential energy is therefore MATHEMATICAL THEORY OF FLUID MOTION 77 xpressmg that the total energy is constant, we have here /* ^^; ../ f *f\ /n ) 1^72 I his quantity c , which is of the dimensions of a velocity, is a easure of the rapidity with which the changes take place. It is not isy to carry the solution further except in the particular case of = -. If we write 3 R/R = i + ,, (73) 5 have then . at R tience bis gives the time taken by the radius of the cavity to attain y assigned value R The following table gives a few examples. R/RO C O */R O I 1 O 2 2 64 3 6-27 4 11-76 5 1942 3 a concrete illustiation, suppose the initial diameter of the cavity be i m., and the initial pressure p Q to be 1000 atmospheres, so at c = 3-16 X io 4 cm /sec. We find that the radius is doubled -rrhj- sec., and multiplied five-fold m about -$V sec. It must be membered that m this investigation, as in the preceding one, the iter has been assumed to be incompressible With an initial ternal piessure of the order of 10,000 atmospheres, we obtain lues of dR/dt compaiable with the velocity of sound m water. tie influence of compressibility then ceases to be negligible. 78 THE MECHANICAL PROPERTIES OF FLUIDS The combination of a somce m at a point A and a correspond- ing sink m at B gives (76) If we imagine the points A and B to approach one another, whilst the product mBA is constant (= /A), we have ultimately r z r^ = AB cos#, and n , u. cose/ / x < = ~ -o- .......... 77) 47? r 2 We have here the conception of a double-source. If we combine this with a uniform flow $ = U# = U> cos# parallel to Ox we have , = - ( \ 477T 2 / cos0 This makes d(/>]dr = o for r = a, provided fi = The formula (78) therefore gives the steady flow past a sphere of radius a. The tan- gential velocity at the surface is = - rod and the pressure is accordingly p = constant fpU 2 sm a # .......... (79) Since this is the same when 6 is replaced by TT 6, the resultant effect on the sphere is ml. If we superpose a general velocity U, we get the case wheie the sphere is in motion with velocity U in the negative direction of x; thus ............ (80) If we imagine this motion to be produced instantaneously liora rest, the impulsive pressure of the fluid on the sphere, in the direc- tion of ^-negative, is / R r I m cosd.ZTTa sindadd I pfi cosd 2ira s'mOadd Jo Jo (81) MATHEMATICAL THEORY OF FLUID MOTION 79 or |MTJ, where M' is the mass of fluid displaced. The impulse which must be given to the sphere to counteract this is fM'U, and the total impulse in the direction of the velocity is (M + $M')U, where M is the mass of the sphere itself. It is a proposition in Dynamics that the kinetic energy due to a system of impulses is got by multiplying each constituent of the impulse by the velocity produced in its direction, and taking half the sum of such products. In the present case this gives $(M -f |M')U 2 . The case is analogous to that of the cylinder, already treated, except that the virtual addition to the mass is $M' instead of M'. This result, viz. that the effect of a frictionless liquid on a body moving through it without rotation consists merely in an addition to its inertia, is quite general. Whatever the form of the body, the impulsive pressme necessary to start the actual motion of the fluid instantaneously from rest will evidently be proportional to the velocity (U), and the reaction on the body in the direction of motion will therefore be &M'U, where k is some numerical coefficient. The impulse necessary to be given to the solid is therefore (M + M')U. A similar conclusion would follow from the consideration of the energy produced The value of k will, of course, depend on the form of the solid and the direction of its motion. The following table gives values for the case of a prolate ellipsoid, the ratio c\a being that of the longer to the shorter semi- diameter The column under " 7^ " relates to motion " end-on ", and that under " k z " to motion " bioadside-on " c\a &! k z i (sphere) | | 1-5 0305 0621 20 o 209 o 702 30 O 122 O 803 40 o 082 o 860 50 o 059 o 895 60 o 045 o 918 70 0036 0933 8 o 0-029 "945 90 o 024 o 954 10 o 0-021 0-960 oo (cylmdei) o i j Any line AP drawn in a plane through the axis represents an 8o THE MECHANICAL PROPERTIES OF FLUIDS annular portion of a surface of revolution about Ox The flux across this portion, say from light to left, will be the same for any two lines from A to P, provided the space between them is occupied by fluid. If A be fixed, this flux will therefore be a function only of the position of P; we denote it by zmft. If PP' be a linear element 8s, drawn in any direction, the flux across the surface geneiated by its revolution about Ox will be ,(82) where q n is the velocity normal to Ss Hence i 9 ** y * It was to simplify this formula that the factor 2-77 was introduced in the definition of $. As paiticular cases of (79), the component velocities parallel and perpendicular to Ox are = V = - y (83) The lines for which Fig. 1 6 is constant are stream-lines, and /r is called the stream-function To find for the case of a point-source, we calculate the flux act oss the segment of a spherical suiface, with OP as radius, cut off by a plane * through P perpendiculai to Ox The radial velocity across this segment is mf^.7rr z , and the aiea is 27rr(r A?), wheie r = OP, x ON. Hence, omitting an additive constant, the flux in the desired sense is , or tft m qrr cos#. (84) The combination of an equal source and sink at A and B gives m iff = 4" COS0 2 )>. MATHEMATICAL THEORY OF FLUID MOTION 81 whilst if A and B are made to approach coincidence in such a way hat wAB = ft, we have ultimately 8 (cos0) = sm0 80 = sin0 (AB sin0)/r, and therefore $ = (86) 477 r 'or a uniform flow parallel to Ox, we have 2$ = Uy 2 , and if re supeipose this on (85) or (86) we get stream-line forms, one of rhich may be taken as the profile of a stationary solid in the stream, 'or instance, combining with (86), and putting fj, = - = - 4:2 _ f 2 \ r ""he line iff o now consists of the circle r = a and of the portions f the axis of x external to it. If we now remove the uniform flow r e get the lines of motion due to the sphere moving in the direction f ^-negative with velocity U. The process just indicated admits of great extension By taking senes of sources and sinks, not necessarily concentrated in points, long the axis of x, subject to the proviso that the aggregate output < zero, and superposing a uniform flow, we may obtain a variety of urves which may serve as the profile of a moving solid This pro- sdure was originated by Rankine from the point of view of naval rchitecturc, and has recently been applied to devise profiles which mtate those of airships. Since the motion of the fluid is known, ic pressure distribution over the suiface can be calculated and ompaied with model experiments 7 Tracing of Stream-lines Theie aie vanous methods by which diawmgs of systems of tream- lines can be constructed. For example, suppose that the trcam-function consists of two parts ^ 15 /r 2 , which are themselves eadily tiaced Drawing the curves J/T! = ma, ifj z = na, fheie. m, n aie integeis, and a is some convenient constant (the mailer the better), these will divide the plane of the drawing into curvilinear) quadnlaterals. The cuives * == na 82 THE MECHANICAL PROPERTIES OF FLUIDS will form the diagonals of these quadrilaterals, and are accordingly easily traced if the compartments are small enough. For instance, in the case of (33), where we may put U = i, a = i, without any effect except on the scale of the diagram, we should trace the straight lines . n r sine? = ma, which aie parallel to the axis of x and equidistant, and the circles r = sin# no. Another method is to write the equation (as above modified) in the form and to tabulate the function i/(i i/r 2 ) for a series of equidistant values of r, beginning with unity. This is easily done with the help of Barlow's tables. The values of y where a particulai stream- line crosses the corresponding circles are then given by Giving i]i in succession such values as 0*1, 0-2, 0-3, ... a system of stream-lines is leadily drawn The same numerical work comes in useful in the case of (39) A similar process can be applied to tracing the stieam-lmes past a sphere, to which (87) refers A more difficult example is presented by equation (99) later. Nothing is altered except the scale if we write this in the form ? i 2 ^z whence (x -f- i ) -|- y .^ i \2 i 2 c > and therefoie r 5 -f- i = 2/ , ,, + J = 2,sc coth^-llr. The hyperbolic function on the right-hand has been tabulated, so that we can calculate the values of r (the distance from the origin) MATHEMATICAL THEORY OF FLUID MOTION s 3 at the points where any given stream-line curve cuts the lines x = constant. 8. General Equations of Motion The general equations of hydrodynamics have so far not been required. To obtain them in their full three-dimensional form we denote by u, v, w the component velocities parallel to rectangular axes at the point (, y t #) at the time t. They are therefore functions Df the four independent variables x, y t #, t. If we fix our attention m a particular instant t , their values would gives us a picture Df the instantaneous state of motion throughout the field. If on he other hand we fix our attention on a particular point (a? , y , # ) n the field, their values as functions of t would give us the history )f what takes place at that chosen point. We introduce a symbol D/Dt to denote a differentiation of any property of the fluid con- sidered as belonging to a particular particle. Thus D/D* denotes the component acceleration of a particle parallel to Ox; this is to be dis- inguished from Bu/dt, which is the rate at which u varies at a parti- cular place. The dynamical equations are obtained by equating the ate of change of momentum of a given small portion of the fluid o the forces acting on it. Considering the portion which at time t iccupies a rectangular element SacSyS*, we have, resolving parallel o Ox P S%S* D " - - L)t ox vhere the first term on the right hand is the effect of the fluid pres- ures on the boundary of the element, as determined by (i), whilst he second term is due to extianeous forces (such, for example, as gravity) which aie supposed to be conservative, V being the potential nergy per unit mass. Thus we find, DM dp __ 8V > P D* " dx P bc' Dv dp 9V p Dt = -5TV Dzy _ __dp_ 9V P Dt ~~ dz P 8F; To find expressions for Du/Dt, &c., let P, P' be the positions iccupied by a particle at two successive instants %, t z . Let %, / 84 THE MECHANICAL PROPERTIES OF FLUIDS be the values of u at the points P, P', icspectively, at time t it and u &) # 2 ' the corresponding values at time t z . The average acceleration of the particle parallel to Ox in the interval t z ^ is theiefore u z ' ~ Wj u z HI . u% u z The limiting value of the left-hand side is Dw/Dtf; that of the first term on the right is dujdt, the rate of change of u at P. Again, u z ~ w a i s tne difference of simultaneous component velocities at the points P' and P, so that , du TVTJ/ du , . .v u 2 ' - U2 = _.PP' == _(/ 2 ~ tj, where # is the resultant velocity x/(tt a -f v 2 + w 2 ), and 9#/3.s is the gradient of u in the direction PP'. Thus n a Dt at os Now if I, m, n be the direction cosines of Ss, dti ,du , du . du / - -f- ?W ~ + W ~5 oi 1 ox ay az and/g 1 = u, mq = ^, raw = ;. Hence, finally, DM 3^ . du . du . du , x D* = -s +I fe + ^; + "5 ........ (90) Similar expressions are obtained for DvjDt, Dw/Dt Substituting in (88) we get the dynamical equations in their classical foim To these must be added a kinematical i elation, which expi esses that the total flux outwards across the boundary of the element SflSyS^ is zero. The two faces perpendicular to O^; give uBySx, on and (u -f ;r-8#)SyS# respectively, the sum of which is dujdxBxSySz C/JC Taking account in like mannei of the flux across the remaining faces, and equating the total to zero, we have the equation of con- tinuity &< &> 8 of which (7) is a particular case. MATHEMATICAL THEORY OF FLUID MOTION 85 When the motion is irrotational we have dy dydx dx* ind similar relations, so that (90) becomes 92 = -J!. + j?, ............ (92) Dt dxdt ' dx ' Vhen this is substituted in (88), it is seen that the dynamical equations lave the integral , P - = f-W-V + FW, ....... (93) p ul inhere F(f) denotes a function of t only which is to be detei mined >y the boundary conditions, but has no effect on the motion. It is vident beforehand that a piessure uniform throughout the liquid, ven if it varies with the time, is without effect The occurrence f F(t) m the present case is merely a consequence of the fact already lentioned that m an absolutely incompressible fluid changes of ressure are transmitted instantaneously. The equation of continuity (91) now takes the form ?* + ?V + ?X = o 8^8/^ 8* 2 i steady motion d^/dt o, and (93) i educes to our foimer result '.?) Vortex Motion i Persistence of Vortices Turning now to the consideration of voitcx motion, the funda- icntal theorem in the subject is that the circulation in any circuit >ovmg with the fluid (i e. one which consists always of the same irticles) does not altei with the time. For, consider any element >x of the mtegial r I (udx + vdy + wdz}, hich expresses the circulation. We have , N (95) 86 THE MECHANICAL PROPERTIES OF FLUIDS Now D(Bx)jDt is the rate at which the projection on the axis of #, of the line joining two adjacent pai tides, is increasing, and is therefore equal to SM. Hence, Dt Dt p dx dx and therefore = - 8 + V - ftf 2 ) (96) When this is integrated round the circuit, the result is zero. Hence l(udx + vdy -f- wdsi) = o. . . ... (97) It is important to notice the restrictions under which this is proved. It is assumed that the density is uniform, that the fluid is frictionless, and that the external forces have a potential. The fitst of these assumptions is violated, for instance, when convection currents are produced by unequal heating of a mass of water, owing to the variation of density. The second assumption fails when the influence of viscosity becomes sensible. Irrotational motion is characterized by the property that the circulation is zero in every infinitesimal circuit We now have a general proof that if this holds for a particular portion of fluid at any one instant, it will (under the conditions stated) continue to hold for that particular portion, whether there be rotational motion in other parts of the mass or not. Again, in two-dimensional motion we have seen that the circulation round any small area is equal to the product of the vorticity into the area Since the area occupied by any portion of fluid remains constant as it moves along, we infer that the vorticity also is constant. This has already been pioved otherwise in the case of steady motion. The value of is, of couise, constant along a line drawn normal to the planes of motion. Such a line is a vortex-line according to a general definition to be given presently, and the vortex-lines passing through any small contour enclose what is called a vortex- filament, or simply a vortex. The strength of a vortex is defined by the product of the voiticity into the cross-section, i.e. by the circulation immediately round it. Still keeping for the moment to the case of two- dimensions, we have seen that the circulation round the boundary of any area MATHEMATICAL THEORY OF FLUID MOTION 87 occupied by the fluid is equal to the sum of the circulations round the various elements into which it may be divided, provided these be estimated in a consistent sense. In virtue of the above definitions an equivalent statement is that the circulation in any circuit is equal to the sum of the strengths of all the vortices which it embraces. 2. Isolated Vortices The stream- and velocity-functions due to an isolated rectilinear vortex of strength K have already been met with in (38) and (58) The velocity distributions due to two or more parallel rectilinear cortices may be superposed. Suppose, for instance, we o^ j- Oj- lave a vortex-pair composed A B )f two vortices A, B of equal Flg I7 ind opposite strengths K tach produces in the other a velocity K/27ra, where a is the listance apart, at light angles to AB. The pair advances herefore with this constant velocity, the distance apart being un- iltered. The lines of flow are given by ^-log- 1 , (98) 27r r z vhere r^, r z are the distances of A, B respectively from the point P o which i/r icfeis The lines for which the ratio r^\r z has the same alue are co-axial circles having A, B as limiting points If we super- )ose a uniform flow K/z-na in the direction of y negative, the case s reduced to one of steady motion, and the stream-function is now 277 , . (99) The stream-line i/r = o consists paitly of the axis of y, where r { = r z nd x o, and paitly of a closed curve which surrounds always he same mass of the fluid. This portion is canied forward by the ortex-pair in the original form of the problem. If a flat blade, e g a paper-knife, held vertically, be dipped into rater, and moved at light angles to its breadth for a short distance, nd then rapidly withdiawn, a vortex-pair will be produced by liction at the edges, and will be seen to advance in accordance with lie preceding theory. The positions of the vortices are maiked by 88 THE MECHANICAL PROPERTIES OF FLUIDS the dimples produced on the watei suiface In this way the action of one vortex-pair on another may be studied. The detailed study of vortex motion in three dimensions would lead us too far, but a brief sketch of the fundamental relations may be given. It is necessary in the first place to introduce the notion of vorticity as a vector. Through any point P we draw three lines PA, PB, PC parallel to the co-oidinate axes, meeting any plane drawn infinitely near to P in the points A, B, C. It is evident at once from the figure that the circulation round ABC is equal to the sum of the circulation round the triangles PBC, PCA, PAB, pro- vided the positive diiection of the circulations be light-handed as regards the positive directions of the co-ordinate axes. Now, if Fig 18 /, m, n be the diiection-cosines of the normal drawn horn P to the plane ABC, and A the area ABC, the areas of the above tuangles are M, mA, nA, icspectively. Hence if f, 77, be the vorticities in these planes, i.e. the ratios of these circulations to the respective areas, the ciiculation round ABC will be ( -f )7 + )J (100) We may regard f, 77, as the components of a vector w, and the expression (100) is then equal to co cos# A where 9 is the angle which the normal to A makes with the direction of to. In other words, the voiticity in any plane is equal to the component of o> along the normal to that plane. The value of has been given in (12). Writing down the corresponding formulae for and 07, we have altogether .. dw dv du dzo v dv du , . = _ rj = _ =5 r (lOl) ay 02 8ar 9 fix dy MATHEMATICAL THEORY OF FLUID MOTION 89 We have, of course, A line diawn from point to point always in the direction of the vector (, 77, ) is called a vortex-line. The vortex-lines which meet any given curve generate a surface such that the circulation in every circuit drawn on it is zero. If the curve in question be closed, and infinitely small, the fluid enclosed by the surface constitutes a vortex-filament, or c 1 (^ ^ B 1 simply a vortex. Consider a circuit such A ' as ABCAA'C'B'A'A in the figure, drawn on the wall of the filament. Since the circulation in it is zero, and since the por- tions due to AA' and A' A cancel, the circu- lation round ABC is equal to that round A'B'C'. Supposing the planes of these two curves to be cross-sections of the filament, we learn that the product of the resultant vorticity into the cross-section has the same value along the vortex. This product is called the strength of the vortex. The dynamical theorem above proved shows that under the conditions postulated the strength of a vortex does not vary with the time. The constancy of the strength of a vortex-ring has already been proved in the case of steady motion. The argument by which the circulation m a plane circuit was, under a certain condition, proved to be equal to the sum of the strengths of all the vortices which it embraces, is easily extended (under a similai condition) to the geneial case. The most familiar instance of isolated vortices is that of smoke rings, which are generated in the first instance by viscosity, but retain a certain degree of persistence. A voitex-nng at a distance from other vortices, or from the boundanes of the fluid, advances along its axis with uniform velocity. The mutual influence of vortex- rings is closely analogous to that of vortex-pairs. Wave Motion I. Canal Waves Water-waves are by no means the simplest type of wave-motion met with in Mechanics, and the general theory is necessarily some- what intricate, even when we restrict ourselves to oscillations of go THE MECHANICAL PROPERTIES OF FLUIDS small amplitude. The only exception is in the case of what are variously called long waves, or tidal waves, or canal waves, the charac- teristic feature being that the wave-length is long compared with the depth, and the velocity of the fluid particles therefoie sensibly uniform from top to bottom, Taking this case first, we mquiie under what condition a wave can travel without change of form, and therefore with a definite velocity. Supposing this velocity to be c, from left to right, we may superpose a general velocity cm the opposite direction and so reduce the problem to one of steady motion. The theory is now the same as for the flow through a pipe of gradually varying section, except that the upper boundary is now a free suiface, instead of a rigid wall. If h be the original depth, the velocity where the surface-elevation is 77 will be ch f ^ *=-*+} .......... (103) The pressure along the wave-profile, which is now a stream-line, is given by Bernoulli's equation P / \ _ t> I TI \ constant f# 2 g*rj = constant ik 2 ( i -f- ~) g-q \ nj = constant $c a (i ^J gr), . (104) \ fa f approximately, if we neglect the square of 77 [h. This pressuie will be independent of 77, provided c = *J(gh) ...... (105) The required wave-velocity is therefore that which would be acquired by a particle falling vertically under gravity, from rest, thiough a space equal to half the depth. If we now restore the original form of the problem, by imposing a velocity c in the positive direction, we have ch 77 , ,. q = c - - = c-L ........... (106) h + T] h approximately. The velocity of the water itself is therefore forward or backward, according as 77 is positive or negative, i.e. it is forward where there is an elevation, and backward where there is a depression. The potential energy per unit area of the surface is Igprf, and the corresponding kinetic energy is \pfh = %pc*v) 2 /h. Since these are MATHEMATICAL THEORY OF FLUID MOTION 91 qual by (106), the energy of a progressive wave is half-potential nd half-kinetic. The condition for permanence of form has not, of course, been xactly fulfilled in the above calculation. A closer approximation o fact is evidently obtained if in (105) we replace h by h -j- if)', this rill give us the velocity of the wave-form relative to the water in he neighbourhood, which is itself moving with the velocity given >y (106), if 7]//j is small. The elevation ^ is therefore propagated n space with the velocity ippioximately. The more elevated portions therefore move the faster, with the result that the profile of an elevation tends to become steeper in front and more gradual in slope behind. 2. Deep-water Waves Proceeding to the more general case, we will assume that the motion takes place m a series of parallel vertical planes, and is the same in each of these, so that the ridges and furrows are rectilinear. Fixing our attention on one of these planes, we take rectangular axes Ox, Oy, the former being horizontal, and the later vertical with the positive direction upwards. The problem being reduced to one of steady motion as before, the stream-function will be j^ = cy -J- j/fj, . .(108) wheie ^ is supposed to be small By Bernoulli's equation P L ^ i (/ P 9 ( Ai \ 2 , /9<Ai\ 2 l = constant - gy - M (c + 'M -|- (-^) \ p l\ cy J \dx/ j O I = constant gy c~, (IOQ) ay if we neglect small terms of the second order. We assume the motion to have been originated somehow by the operation of ordinary forces, and therefore to be irrotational, so that We further assume, in the first instarj.ee, that the depth is very great 9 2 THE MECHANICAL PROPERTIES OF FLUIDS compared with the other lineal magnitudes with which we are con- cerned. The simplest solution of (no) which is periodic with respect to x, and vanishes for y oo , is ^ = Ce^sinkx (m) If we take the origin O at the mean level of the suiface, the con- dition that the wave-profile may be a stream-line is, by (108), C C . = _ e ky s'mkx = smkx, (112) c c if we neglect an eiroi of the second order in C. We have still to secuie that this is a line of constant pressure. Substituting in (109), the result will be independent of x, provided g- - kcC = o, or c 2 = f , (113; C R to our order of approximation. The wave-length, i.e. the distance between successive crests or hollows is A = 27rjk, so that //#A\ , N Vw ^ II4 ^ This gives the wave-velocity relative to still water The original form of the problem is restored if we omit the first term in (108), and replace x by x ct. Thus, if we denote the sur- face amplitude C/c by a we have tjs = ace ky smk(ct To find the motion of the individual particles, we may with consistent approximation write ___ w _______ o sm (ct #) , Dv dib , kv ,, . I ^ _^: = v = _L = &0c g'y. cosk(ct X Q ), Dt ox where (^ , ^ ) is the mean position of the particle referred to. Inte- grating with respect to t, and recalling (113), we have x = X Q a e ky " cosk(ct x), MATHEMATICAL THEORY OF FLUID MOTION 93 ie particles therefore describe circles whose radius a e ky diminishes m the surface downwards. At a depth of a wave-length, y = A, = e~ zv = 0-00187. The preceding investigation is therefore ictically valid for depths of the order of A, or even less. For smaller depths, provided they are uniform, the solution (in) to be replaced by ^ = C smh/e(jy + h) sinkx, .......... (118) ice this makes v = o for y = h. We should now find c* = - tanhM = tanh ....... (119) k 27T A r small values of A/A this gives c = \/(gh}> and so verifies the -mer theory of long waves As /z/A increases, tanh/z tends to ity as a limit, and we reproduce the result (114). The paths of the dividual particles are ellipses whose semi-axes coshk(y + h) smhk(y ~r h) smhM ' smhkh e horizontal and vertical, icspectively The eneigy, per unit aiea of the surface, of deep-water waves is und as follows The potential energy is 2 sm z k(ct x), . (120) e mean value of which is {gpa z The kinetic energy is ..(121) r (115) Since c 2 = gjk the energy is, on the whole, half-potential id half-kinetic The total energy per wave-length (zTr/k) is TrpaV. his is equal to the work which would be required to raise a stratum the fluid, of thickness , through a height |. The theory of waves on the common boundary of two supei- )sed liquids, both of great depth, is treated in a similar manner. he formulas (108), (109), (in) may be retained as applicable to ie lower fluid. For the upper fluid (of density p') we write 'A = cy + </V> ............ (122) id 0/ = C'e~ ky smkx ........... (123) 94 THE MECHANICAL PROPERTIES OF FLUIDS since i/r/ must vanish when y is very great. This makes C' = smkx, c We have also - = constant c~- ( I2 >5) The two values of p will be equal provided C C' gp- kcpC gp (- hcp'C' (126) c c By comparison of (112) and (124) we have C = C', and therefoie If (p p')/(p + />') is small, as in the case of oil over water, the oscillations are comparatively slow, owing to the relative smallness of the potential energy involved in a given deformation of the common surface. A icmaikable case in point is where there is a stiatum of fresh water over salt, as in some of the Noiwegian fiords, where an exceptional wave-resistance due to this cause is sometimes experienced The preceding theory of surface- waves is restricted to the case of a simple-harmonic profile It is true that any othei form can be resolved into simple-harmonic constituents of different wave-lengths, and that it is legitimate, so far as our approximation extends, to superpose the results. But the formula (114) shows that each con- stituent will travel with its own velocity, so that the form of the profile continually changes as it advances. The only exception is when the wave-lengths which are present with sensible amplitude are all large compared with the depth, in which case there is a common wave-velocity \/(gh) as found above. 3. Group Velocity One consequence of the dependence of wave-velocity on wave- length is that a gioup of waves of approximately simple-harmonic type often appears to advance with a velocity less than that of the individual waves. The simplest illustration is furnished by the MATHEMATICAL THEORY OF FLUID MOTION 95 combination of two simple-haimonic trains of equal amplitude but slightly different wave-lengths, thus T) = a cosk(x ct) + a cosk'(x c'l) 'k - k' kc - k'c'\ X t}( 2 2 / If k and k' are nearly equal, the fiist trigonometrical factor oscillates very slowly between -f- i and i as x is varied, whilst the second factor represents waves travelling with velocity (kc + k'c')l(k -|- /'). The surface has therefore the appearance of a series of groups of waves separated by bands of nearly smooth water. It is evident then that the motion of each group will be practically independent of the rest. The centre of one of the groups is determined by k - k' kc - k'c' x t o; 2 2 the group as a whole is therefore propagated with the velocity U = = , (120) 7 if jj * v y / k k dk m the limit This is called the group-velocity If c is constant, as when the wave-length is laige compaied with the depth, we have U = c. On the other hand, lor waves on deep water, c 2 gjk, by (113), so that 2 dc _ i c dk k' whence U = \c, . . . ..(130) or the group- velocity is only one-half the wave-velocity The geneial foimula, obtained fiom (119), is U , . kh This expression diminishes from i to | as kh increases from o to oo. The group-velocity U determines the rate of propagation of energy across a vertical plane. To take the case of deep-water waves as simplest, the rate at which work is done on the fluid to the right 96 THE MECHANICAL PROPERTIES OF FLUIDS of a plane through the oiigin perpendiculai to the axis of x is /o I pudy .......... (132) J _co The value of p is given by Bernoulli's equation provided we put q* = ( c -|- uf -f- v 2 c 2 zcu, to our order of approxima- tion. The only term in the resulting value of p which varies with the time is pcu. Now f u z dy = pW sm z kct t e* ky dy = %pka z c* sitfkcl. . . (133) J 00 * 00 pc The work done in a complete period (zTrjkc) is therefore which is half the energy of the waves which pass the above plane in the same time. The apparent paradox disappears if we lemember that the conception of an infinitely extended tram is an aitificial one. In the case of a finite tram, generated by some peiiodic action at the origin which has only been in operation for a finite time, the profile will cease to be approximately umfoim in chaiactei and sinusoidal near the front, there will be a gradual diminution of amplitude, and increase of wave-length, by which the tiansition to smooth water is effected. We infer from the piecedmg aigument that the approximately simple-harmonic portion of the tiain is lengthened only by half a wave-length in each period ol the originating force. The principle that U lather than c detei mines the rate of pro- pagation of energy holds also, not only in the case of waves on water of finite depth, but in all cases of wave-motion m Physics. Some further results of theory must be merely stated in geneial terms. A localized disturbance travelling over still watei with velocity c leaves behind it a train of waves whose length (2ir/k) is related to c by the formula (113) or (119), as the case may be. In the same way a stationary disturbance in a stream pioduccs a liam of waves on the down-stream side. In the former case the encigy spent in producing the train measures the wave-resistance expencnced by the disturbing agency. If E be the mean energy per unit length of the wave-train, the space in front of the disturbance gains in unit time the energy cE, whilst the energy transmitted is UE, where U is the group-velocity. The wave-resistance R is therefore given by Rc= (c-U)E ................ (134) MATHEMATICAL THEORY OF FLUID MOTION 97 "lie value of E has been found to be $gpa 2 , but unfortunately the alue of a can be predicted only in a few rather artificial cases. A curious point arises in the case of finite depths. It appeal's -om (119) that the wave-velocity cannot exceed \/(gh). The above tatements do not apply, therefore, if the speed of the travelling isturbance exceeds this limit The effect is then purely local, and L = o. A considerable diminution in resistance was in fact observed y Scott Russell when the speed of a canal boat was increased m lis way; and an analogous phenomenon has been noticed m the ise of torpedo boats moving m shallow water. Viscosity I. General Equations The subject of viscosity is treated in Chapter III, which deals lainly with cases of steady motion wheie this influence is pie- aimnant The general equations of motion of a viscous fluid ive the forms Dti dp 3V . -_ , , 'Di"- -'to* ' iV " l/ ' ..... (I35) ith two similar equations m (v, y) and (w, z), where v 2 = a 2 /a*: 2 + a 2 /^ 2 + a 2 /as 2 . he formal proof must be passed over, but an mteipietation of the ]uations, which differ only from (88) by the terms at the ends, n be given as follows. Considering any function of the position a point, let F be its value at P, whose co-oidmates are (x, y, #). s value at an adjacent point (x -[- a, y + ft, z + y} will exceed its lue at P by the amount r J_ 9F L 9F -a + ft + y dy 3# iproximately. If we mtegiate this over the volume of a sphere small radius r having P as centre, the first three terms give a ro result owing to the cancelling of positive and negative values a, 0, y. The terms containing j5y, ya, a, also disappear for a nilar reason. The mean value of a a or p* O r y 2 on the other hand (D812) 5 98 THE MECHANICAL PROPERTIES OF FLUIDS is -i-r 2 , by the theory of moments of ineitia. The mean value over the sphere of the aforesaid excess is therefore T y- 2 V 2 F. The reason why this should vary with the radius of the sphere is obvious. It is also clear that the expression V 2 F gives a measure of the degree to which the value of the function F in the immediate neighbourhood of P deviates from its value at P. In particular V 2 M measures the extent to which the a: - component of the velocity in the neighbourhood of P exceeds the component at P. The first of the equations (135) accordingly asset ts that in addition to the forces previously investigated there is a force propoitional to this measure. An excess of velocity about P contributes a force tending to drag the matter at P in the direction of this excess. The coefficient //. in (135) is called the coefficient of viscosity. In cases of varying motion we are often concerned not so much by the viscosity itself as by the ratio which it bears to the inertia of the fluid. It is then convenient to introduce a symbol (v) for the ratio pip This is called the kinematic viscosity. An important conclusion bearing on the comparison of model- and full-scale experiments can be drawn from the mere form of these equations Omitting the term repiesenting extraneous force, the first equation is in full du . du . du . du I dp , , ^ _]_ u j_ v + w = __ _ Jl 4. v ^ u . . .(136) ot ox oy o% p ox Now consider another state of motion which is exactly similar except for the altered scales of space and time. Distinguishing this by accented letters, a comparison of corresponding terms in the respec- tive equations shows that we must have u' . u __ u' 2 . w 2 __ p' , p __ v'u' vu ' The equality of the first two ratios requires that f t/V VV u :u = -7 : -, as was evident beforehand. The equality of the second and fourth ratios requires , ^r,'- <'3) use ux A necessary condition for the similarity of the two motions is there- fore that VJ?/i> should have the same value in both, where V is any MATHEMATICAL THEORY OF FLUID MOTION 99 racteristic velocity, and / any linear dimension involved. The o of corresponding stresses is then JL/ ',.,'2 p p U p pu z It is to be noted that the viscous terms disappear from the ations (135) if the motion is irrotational, since we then have = o, and therefore V 2 u = o, V 2 # = o, V 3 z# o. But it is ,eneral impossible to reconcile the existence of irrotational motion i. the condition of no slipping at the boundary, which is well blished experimentally. The above remark suggests, however, , when the motion is staited, vorticity originates at the boundary- is only gradually diffused into the interior of the fluid. ^~ 2. Two-dimensional Cases The diffusion of voiticity is most easily followed in the two- ensional case The equations may be written, in virtue of , in the forms where X = - + l(u* + ^ 2 ) -f- V, ........ (141) icntiating the second of equation (140) with respect to x, and irst with lespect to y, and subtracting and making use of the tion of continuity (7), we have, finally a-*'* ................ (I43) is exactly the equation of conduction of heat, with the vorticity place of the temperature, and the kinematic viscosity v( /n//>) ace of the thermometric conductivity. Consequently, various r n results in the theory of conduction can be at once utilized e present connection. ioo THE MECHANICAL PROPERTIES OF FLUIDS For instance, the known solution for the diffusion of heat from an initially heated straight wire into a surrounding medium can be applied to trace the gradual decay of a line vortex initially concen- trated in the axis of #. Since there is symmetry about Os the equation (143) takes the form dt as may be seen by a comparison of the left-hand membeis of (25) and (26) It is easily verified by differentiation that this equation is satisfied by which vanishes for t = o except at the origin. Moreover, this gives for the circulation in a circle of radius r 2-jrrdr = K(I e~ t ~'^" t ) ( 1 4 ( ->) As t increases from o to o , this sinks from K to o. The value of , on the other hand, at any given distance r increases from zeio to a maximum and then falls asymptotically to zero. A comparatively simple application of the equations of motion is to the case of " laminar " flow in parallel planes, or of smooth rectilinear flow m pipes, but the results have only a restricted application to actual phenomena To take an example due to Helmholtz, consider the flow of a hypothetical atmosphere of uniform density, and height H, over a horizontal plane If it is subject to meitia and viscosity alone, the equation of motion is "du d 2 u = v } . (147) with the conditions that u = o for y = o and du/dy o for y = H. These are all satisfied by u = Ac-"* 1 ' sinky (148) provided cos/di = o, or k = (zn -f 1)^5, (149) where n is an integer. By addition of such solutions with different values of n and suitable values of the coefficients A we can represent MATHEMATICAL THEORY OF FLUID MOTION ror the effect of any initial state, e.g. one of uniform velocity. The most persistent constituent in the result is that for which n = o. This will have fallen to one-half its original value when v&t - loga, or t = ........ ( IS o) Putting v = 0-134 (air), H = 8026 metres, this makes t = 305,000 yeais! The fact is that in such a case the laminar motion would be unstable, turbulent motion would ensue, by which fresh masses of fluid moving with considerable velocity are continually brought into contact with the boundary, so that the influence of viscosity is enormously increased. CHAPTER III Viscosity and Lubrication A. VISCOSITY All motions of actual fluids, as distinguished from the " peifect fluid " of the mathematician, are accompanied by internal forces which resist the relative movements and are theiefore analogous to frictional forces between solid bodies The origin of the frictional resistances is in all cases referred to the property of viscosity, common in varying degree to all fluids, which has already been defined in general terms in Chapter I. The present chapter is devoted to a fuller explanation of the theory of this property and to discussions of some of its direct applications, one of the chief of these being to the theory of lubrication There aie other direct applications of the theoiy of viscosity which are of importance to engineers, most though not all of which relate to the motion of fluids in narrow channels or in thin layers between solid surfaces, and these applications are met with in all branches of engineering. The fluid frictions, however, which chiefly concern hydraulic and other engineers, who deal with fluids such as water or air in large volumes, though physically referable in origin to viscosity, cannot be directly calculated by means of its theory The appropriate methods applicable to such cases are discussed in Chapters IV and V. In the meantime it may be said of the direct applications of the theory, in Rayleigh's words (30, p 159),* that m these cases "we may anticipate that our calculations will correspond pretty closely to what actually happens more than can be said of some branches of hydrodynamics ". * Arabic numerals in brackets after names of authors refer to the short biblio- graphy at the end of thia chapter. 102 VISCOSITY AND LUBRICATION ro3 Laminar Motion The law of viscous resistance is most clearly conceived in the se of laminar motion, which may be defined as a state of motion of body of fluid m which the direction of the motion of the particles the same at all points and the velocity is the same throughout ch of a series of planes parallel to one another and to the direction motion A volume of fluid in laminar motion can thus be roughly garded as a series of very thin pers of solid material, sliding one ^ )on another in a common direc- / / / / m Quantitatively, if the face k JP u p , By of the rectangular element , 8y, S# (fig i) is parallel to the &!/ X* iiinse, and if the laminar motion in the direction of X, the velo- y of flow, u, at any point P, 11 depend only on the distance, of the point P from the plane Y If the element is sufficiently lall, u may be taken as varying uniformly with z over the small stance Sz, so that if w , % are respectively the velocities of the rimas which form the lower and upper faces ol the element = z/ -f- Bz, in which can be regarded as constant over dz dz e small distance $z In a viscous fluid there will then be exerted a shearing force, or iction, parallel to X, between the portions of the element of fluid ove and below any section of the element parallel to the face By, tending to retard the portion which is moving with the higher locity, and the magnitude of this force will be S = pBxBy, (i) oz ou being a quantity, independent of x, y, z, u, and , known as the Efficient of viscosity. 104 THE MECHANICAL PROPERTIES OF FLUIDS Coefficient of Viscosity The value of the quantity ^ varies greatly from one fluid to another, and in any one fluid it changes with the temperature, and to a smaller extent with the pressure, of the fluid. Its value is in general much higher for liquids than for gases Liquids m which the value of p is low are said to be " limpid ", " thin ", or " light ", while those in which it is comparatively great aie said to be " vis- cous ", " thick " or " heavy ". There is however no necessary, or general, correspondence between the density of a liquid and its viscosity. Thus mercury, the heaviest of known liquids at atmos- pheric temperatures, is one of the least viscous. The fact that the coefficient of viscosity, for a given liquid at constant temperature, is independent of the rate of shear was fiist experimentally proved with great accuracy by Poiseuille (i), not, howevei, by duect measmement of plane laminar flow, but by in- vestigation of the flow of water in small cylindrical tubes The flow of fluids in such tubes, as well as the motion of viscous fluids in many other cases which are of practical inteiesl, is closely analogous to plane laminar flow. The importance of the coefficient of viscosity /A, however, anscs from the fact that it is the sole physical constant connecting the internal frictional resistances of fluids with their iclative motions, not only in the case of such simple types of motion, but of all kinds of fluid motion however complicated, provided that they aie not discontinuous or unstable The explanation of this unique prop city of the coefficient of viscosity icqmres some analysis of the types of deformation ot which a fluid element is susceptible This analysis is given briefly in the following paragraphs, from which it will be seen that the iclations between the internal motions and stresses in a fluid aie similar to, but essentially simpler than, those between the deformations and stresses in an elastic body The failuie, already icf cried to, of the law of viscosity when fluid motions become discontinuous or unstable may be regarded as analogous to the failure of the laws of elasticity in solids when fracture takes place or the " yield-point " is exceeded. In such cases the conditions which result are no longer amenable to theoretical calculation. We proceed to show that, when no such discontinuities exist, there is in fluids only one kind of internal resistance and only one coefficient of viscosity. VISCOSITY AND LUBRICATION Relative Velocities 105 If M, v, w (see fig 2) are the components of the velocity parallel to the rectangular axes X, Y, Z of n par- , tide of the fluid at the point x, y, #, the z Ar - correspondmg components for the neigh- bouring point x -f- Bx, y + By, 2 + Bz are , . du <j . du j, . du <,, n = M -(- ojc -f- dy -f- o#, C/JC OV O*^ v = and the components of the velocity of the second point relatively to the first are u' u,v' v, w' w, or \JtAi j"^ (3) dx ' 3j 3^; Of the derivatives in these expressions it is clear from inspection c r i ;. 3 dv , dw , . , of iig 2 that , , and - repiesent rates of stretching or elongation ox ay ds of the element in the directions of X, Y, and Z icspectively, while by the pairs of sums of denvatives* dzv . dv dy 3V du dw dz dx' dv . du dx dy 1 are icpiesented respectively rates of change of the angles between the edges By and Bz, Bs and S#, and Bx and By of the element Thus by means of these six expressions any deformation of the element can be expressed. As a hypothesis which is suggested as probable by the experi- mental law proved by Poiseuille, but which depends for its real justifi cation on the consistent correspondence of the results of theory with experience, it is assumed that the frictional forces arise from (D312) 5* o6 THE MECHANICAL PROPERTIES OF FLUIDS he rates of deformation of the elements of the fluid, and are linear unctions of these lates. ~ As to the three rates of elongation , , , it is a well-known ox oy oss heorem that they can be resolved into a rate of dilatation or com- >ression of the elementary volume, uniform in all three directions, ombined with three lates of shearing deformation respectively in he directions of the diagonals of the faces of the element supposed ubical.* As there is no experimental evidence of any internal lesistances, ither in liquids or gases, depending on rates of change of volume >y dilatations or compressions equal in all directions, resistance to o n elongation, such as -- Sx, can only arise from its shearing com- ox >onents. Such resistances are theiefore of the same kind as those vhich depend on the purely shearing deformations whose rates are w . dv _L ATP I n ) tx ' 1 - / y as: In the applications which follow, the axes X, Y, Z will be so hosen that the rates of elongation, such as --, and consequently also ox heir component rates of shear, are everywhere small compared to tie rates of shear lepresented by + , &c. oy dz Now in a homogeneous liquid or gas there is no physical difference i the properties depending on the direction of the co-ordinates, onsequently (the fnctional forces being linear functions of the rates f shear) the only foices that will arise may be expressed as: o , dv^ dw ; Bu\ -.dx dyj 'j ivolving the single coefficient ft By comparison with fig. i, and *Cf the similar theorem for stresses (Morley, Strength of 'Materials, 2nded ,p 12). f In this notation the first subscript indicates the direction of the noinial to the ane on which the foice acts, the second the direction in which the force acts us Syz is the sheaung stress on a plane a normal of which is parallel to Oy and <g acts in the O# direction VISCOSITY AND LUBRICATION 107 equation (i) with the second of the above equations (4), in r\ ich is taken as zero, it is seen that this constant is the same dx the coefficient p introduced in the special case of laminar ition. Conditions at the Bounding Surfaces of Fluids Before the laws of fluid friction can be applied to fluids as we ually have to deal with them, account must be taken of the be- dour of the fluid where it is in contact with the solid bodies which itam it. In the case of liquids the condition of a free upper face, usually a surface of contact with air at atmospheric pressure, also to be considered. It is clear in the first place that the presence of a boundaiy in- ves that on the bounding surface the itive velocity of the fluid normal to t suiface is zero. The noimal velo- r will furtheimoie be very small at points near the bounding surface. , let W, fig 3, be the fixed bound- suiface, and W a suiface in the d parallel, and very near, to W. Fig 3 simplicity W and W may be sidered plane. Let the average velocity towards W over a le of radius R in the plane W be v, the normal distance between and W being Sn. Then a volume of fluid rrR 2 v flows through circular area in unit time In the same time a volume 27rRw.cr re outward between the surfaces past the circumference of the le, o- being the mean outward ladial velocity parallel to the sur- s. Thus Sn t \ v = 2o-~, (5) he noimal velocity is veiy small compared to the velocity parallel lie surface. [n the case of a solid boundary it will be seen from the next igraph that the velocity a is itself very small close to the surface, hat in this case the normal velocity v is a small quantity of the nd order. 10 8 THE MECHANICAL PROPERTIES OF FLUIDS Motion Parallel to Bounding Surfaces With regard to the motion of fluids parallel to solid walls w which they are m contact, there is strong evidence that in the a of liquids at least the relative tangential velocity a at the wall is ze Some of the evidence will be referred to later in connection with t flow of liquid through tubes under great pressure, and in the d cussion of the theory of lubrication. Even when the mutual molecular attraction of a liquid and so appears to be compaiatively small, so that the liquid does not tend spread over, or " wet " the suiface of the solid, as is the case w mercury and glass, there is no observable sliding or slipping of 1 fluid over the solid at theii common surface. If the tangential tractional force between liquid and solid, a consequently the late of shear in the liquid neai the suiface, ; finite, the relative tangential velocity, being zero at the suiface, mi be still small at all points of the liquid near the surface, as was , serted in the last paragraph In gases, the same rule as to the relative velocity being zero a soHd surface is found to apply under ordinary circumstances, least as a very close approximation When, however, a gas is at su low pressure that its molecules aie at distances apart comparable w the dimensions of the volume of gas which is being dealt wi phenomena are observed which can be regarded as ansing from appreciable velocity of slipping of the gas over the solid surfa According to Maxwell,* the motion of the gas is very nearly the sa as if a stratum, of depth equal to twice the mean fiee path of the j molecules, had been removed from the solid and filled with the g there being no slipping between the gas and the new solid suiface. At free surfaces, which, of course, can only exist in liquids, 1 normal velocity relative to the surface is again obviously zeio. 1 liquid surface may, however, have a tangential velocity, and it usual to assume that the law of viscous shear holds up to the suif; and that either the tangential traction there becomes zeio, or, if 1 liquid surface is exposed to a stream of air, that the traction is c only to the rate of shear in the air near the common surface. Expc ments by Rayleighf and others have shown that, at least in the ca of water with an uncontammated surface and of oils and other Hqu which are capable of dissolving solid grease films, theie are fnctional resistances peculiar to the surface film * Collected Papers, Vol II, p 708 f Collected Papers, Vol. Ill, p 363. VISCOSITY AND LUBRICATION 109 Viscous Flow in Tubes On the principles which have been explained, we can proceed calculate the flow of viscous fluids in various cases which are piactical inteiest Take first the case of a uniform tube of cular section of winch the diameter is small compared to the gth of the tube. A fluid flows through the tube as the icsult a constant difference of pressuie between its two ends. The ition, except veiy neai the ends, will be sensibly parallel to the s of the tube, and the piessuie (and consequently the density) I be sensibly umfoim over every noimal section. By symmetry, any one section the velocity must be the ae at all points at any given radius r from axis. If w be the velocity (upwards in fig. 4) this radius, p the density, and p the ssure at any section, the radius of the e of the tube being a, the mass dis- 1 1-1 i 1 rged per unit time, which must be the le for all sections, is f fl I pzo 2rrrdr = m, constant . . (6) J o Fig 4 2 axis of the tube is taken as the axis Z, and is supposed to be so rly straight that effects due to its curvature can be neglected, and the fiist instance the motion will be supposed so slow that the 2tic energy of the fluid is inappreciable. The fluid may be either quid or gas. The effect of gravity is disregarded, or, if included, - gpz is to be written instead of p. From equations (4), p. 106, since the velocity w varies radially r he rate -, but not circumferentially, there is a ti action in the Of Action of Z on each unit of area of the cylindrical body of i inside radius r of amount 8U=/ ........... (7) isidering a section of this cylinder of length 8^, the total traction its cylindrical surface, whose area is 2ur8#, must be equal to the io THE MECHANICAL PROPERTIES OF FLUIDS ifference of the total pressures on its upper and lower ends, so that ~z = 7rr*-~-Sz t dr dz dw r dp or -=-, dr 2u dz and therefore w = -- -- -- ) . / Since w = o when r = a, C = , and 2 i which the negative sign expresses the obvious fact that the direc- on of flow is opposite to the direction of increase of pressure. Now fiom (6) and (8) = / pw.2Trrdr = / J o J o4/A -- dz In the case of a liquid, p and [Jt, may usually be taken as constant, dp that is constant along the length of the tube, being equal to _ 2 Fl where p lt p z are the pressures at the lower and uppei ends f the tube whose length is /. m, TTCflp pi p 9 , N Then m = -^ ^ 1 ra .......... (io) OjU. / "he limits of application of this formula will be more fully explained i a later chapter For the present it may be stated to be applicable ) the flow of all liquids through " capillary " tubes (that is to say, ibes whose diameter is only a fraction of a millimetre), unless the * In all numerical applications of this and other formulse thioughout this lapter all quantities must be expressed m the C.G.S. or other absolute system of uts. VISCOSITY AND LUBRICATION in lifference of pressures, p p z , is greater than is ordinarily met with i engineering practice, provided that proper correction is made for he disturbing effects of the ends of the tube. In the case of viscous lubricating oils, the formula is applicable, vith certain restrictions, to their flow through ordinary lines of )ipmg, but it must be regarded as subject to correction, or even wholly inapplicable, to the flow of the less viscous oils especially inder considerable pressures.* In the case of a gas, p = -^- , T being the absolute temperature R 1 md R a constant Thus from (9) If T and ju, can be regai ded as constant throughout the length of he tube, integrating (n) we have is the equation connecting the flow and the fall of piessure. In the preceding discussion the kinetic eneigy of the fluid has Deen assumed to be negligible All the foimulas given, however, -emain coirect for the case of a liquid even when the kinetic eneigy s appicciable, provided that they aie applied only to the middle portion of the tube and not to its end poitions where the flow is iffected by the acceleiation and letardation of the fluid which occur near the inlet and outlet It is well known that the kinetic energy which a fluid acquires in enteung an orifice is not wholly lestored is pressuie eneigy at its discharge There is theiefore a resistance to the flow aiising fiom the acceleration and retardation at the inlet and outlet of a tube, additional to the factional losses within the tube itself. In the case of a square-ended tube opening into large vessels at each end, the loss of pressure is approximately 1-12 X u z J2g, where u is the mean velocity at the outlet. \ There aie further sources of resistance not taken into account in our calculations, arising from viscous friction between the streams at the ends, where the lines of flow aie not parallel to the axis of the tube. Fig 5! shows the course of the particles of fluid at the inlet and outlet of a square-ended tube when the kinetic energy is appreciable and both ends of the tube are immersed in the fluid, * See e.g. (13), p 159 t See Hoskmg, Phil Mag , April, 1909, Schiller, Zeits Math, u. Meek , Bond, Proc. Phys, Soc , 34, IV. j Fiom (10), p. 158. 12 THE MECHANICAL PROPERTIES OF FLUIDS Fig. 6 illustrates the condition which occurs when the outlet of such a tube is not immersed but discharges the fluid in a series of drops. In this case there is another resistance to the flow, due to the excess of internal pressuie which is necessary to extend the surfaces of the drops during their formation. OuHel vessel [ J Fig 5 Fig 6 The calculation of the resistances due to these disturbing effects is rathei uncertain, and on this account an accurate correspondence between the results of calculation and those of expenment can only be expected when the tubes are very long compared to then diameteis Use of Capillary Tubes as Viscometers The experimental determination of coefficients of viscosity is earned out by instmments of vaiious kinds, known as " viscometeis " or " glischiorneteis " These are divided into two classes, namely " absolute " viscometers, by the use of which the coefficient of viscosity can be determined in absolute measure dnectly iiom the dimensions of the instrument itself (combined with measurement of a time interval), and " secondary " or " commercial " viscometers, which require to be calibrated by comparison of then results with those of an " absolute " viscometer. The best absolute viscometers, for liquids at least, depend on the measurement of flow through capillaiy tubes, //, being determined from the equation (10) given on p. no, after instrumental measure- ment of the other quantities involved. The appaiatus by which Poiseuille made the first accurate determinations of the viscosity of water was of this. class. The tubes which he used varied in diameter from o-ooi to 0-014 cm., their lengths being a few centimetres, and the pressure was applied by a column of mercury up to 77 cm. in height. Such instruments are capable of very considerable accuracy VISCOSITY AND LUBRICATION FiS 7 Stone's Absolute Viscometer hen used with proper precautions, and when the necessary cor- 'Ctions are applied for the various disturbing factors. The viscosity "water, for instance, at atmospheric temperatures is probably known ithin Tilth of i pei cent of its true value.* See(u),p. 158. ii4 THE MECHANICAL PROPERTIES OF FLUIDS The consistency to this order of accuracy of determinations made with different instruments and under different conditions is con- clusive evidence of the correctness of the basic assumption of the linear connection of traction with shear, and of the absence of slipping of the fluid over the walls of tubes The puncipal precaution which has to be taken in the use of the capillary viscometer, in addition to the elimination of (01, in so far as that is not possible, the correction foi) the end-disturbances which have been pointed out, is the accurate determination of the tempera- ture of the fluid undei test. The latter requiiernent is usually met by sin rounding the capillary tube with a water-jacket, means being provided for warming or cooling the water, and measuring its temperature. The most convenient form of " absolute " capil- lary viscometer for liquids is that descubed by W. Stone (18, p. 159). In this instrument the pressure is applied by a column of mercuiy of which the height is automatically maintained constant, and other devices aie provided which further simplify the manipulation of the instrument and the calculation of the results from the observations The Stone viscometer is illustrated in fig. 7, the capillaiy tube and its attachments being shown separately in fig 8. The following is an abbreviation of the designer's descuption cited above. The instrument consists of thiee essential ele- ments, viz the viscometer burette, the adjustable constant-head apparatus, and the piessuie-gaugc. The viscometer burette consists of two glass vessels A and B (fig 8), of equal internal diameters and suitable lengths, connected at their lower ends by means of a wide-boie tube C, and of a capillary tube D of suitable dimensions for the desired purpose. The three portions of the burette are held together by the brass clips and tension-rods R. Several interchangeable tubes D may be provided for fluids of different viscosities. The measuring vessel A is provided with two platinum wires sealed into its wall, and so bent that the inner end of each wire lies on the axis of the tube. The capacity of the vessel between the two platinum points can be thus accurately measured. A glass tap T R Fig 8 VISCOSITY AND LUBRICATION 113 is provided on the inlet to the burette to control the staitmg of a test. The whole of the burette is immersed in water contained in a glass tube (see fig. 7) having a brass bottom. A brass cover is also fitted having a slot for the insertion of a stilling rod and a ther- mometer A Bunsen burner selves to heat the water. The adjustable constant-head apparatus consists of two glass vessels, the lower one F being furnished with a tap V at the top and the upper one G suspended by a spring from a hook attached to a sliding clip H which can be clamped to the standard S at any desired height. Through the outer end of the clip a glass siphon pipe passes to the bottom of the vessel G when the latter is at its highest point, i.e against the clip H The siphon is connected to the lower vessel F by means of a rubber tube. The strength of the spring is so adjusted that as the mercury flows from G to F, the former, being thereby lightened, will rise so as to maintain the surface of the mercuiy m it at constant height above that of the mercury in F. The pressuie-gauge K is of the oidmary U pattern, with meicury as the woiking fluid A thiee-bianch pipe P connects the burette, pressure-gauge, and constant-head appaiatus. The instrument must be set up veitical. As the liquid to be tested is fed into the burette A (fig 8), the vessel F is removed from the socket J and raised to a sufficient height above G to reduce the lir-pressure in B (fig 8) and thus draw the liquid under test into it, lowering the suiface in A below the lowei platinum gauge-point. The glass tap T is then closed and the pressure apparatus adjusted to the deshed piessure. Then the tap is opened, and the time slapsing between the moments of contact of the liquid surface with the gauge-points in A is taken by means of a stop-watch or suitable :hionogiaph. By the use of this instrument the viscosity of a sample of oil 2an be determined at ten or twelve different temperatures within an houi The pressure can be varied fiom about 5 to 50 cm of mercuiy in order to give (without changing the tube D) convenient intervals D time for measurement according to variations in the viscosity of the oil Vanous other forms of apparatus have been used for the absolute deteimination of viscosities, their action depending, for instance, Dn the torsional oscillations of a disc or cylinder (a method which is convenient for measurement of the viscosity of gases, on account of the accuracy with which the very small forces involved may be ii6 THE MECHANICAL PROPERTIES OF FLUIDS measuied by this means), the continuous rotation of a cylinder or disc or sphere, or the fiee fall of a sphere in a body of fluid Foi geneial purposes, however, no other method is so convenient 01 accurate for absolute measurements of viscosity as that of Poiseuille. Secondary or Commercial Viscometers Tube viscometeis are also commonly employed foi making practical or commercial measurements of viscosity In ordei to reduce the time occupied by the measurements, and to simplify the apparatus and to reduce its delicacy, much shoiter tubes are used in these instiuments than are admissible for absolute instruments. In the Redwood viscometer, for instance, the tube is appioxi- mately 1-7 mm. in diameter, and 12 mm m length, being a hole drilled through an agate plug fixed in the bottom of a vessel which is ananged to contain a measured quantity of the liquid to be tested. The liquid flows out of the hole under the force of gravity, the time of efflux of the measured quantity being taken by a stop- watch. Means are provided for warming or cooling the liquid to any temperature at which it is desired to make the test, but the determination of the actual tempeiature of the fluid as it is passing through the hole is one of the chief difficulties in the use of this and similar instruments. In some of these the " tube " Is so much i educed in length as to become a meie orifice. It will be readily undei stood that the conections for the end effects of the tube, which have been pointed out as necessary in connection with all cases ol viscous flow in tubes, become i datively much rnoie consideiable in the case of such shoit-tube instiuments. In these, except for the moie viscous liquids, the times of efflux are no longer propoi- tional to the viscosity of the fluid. It is therefoie necessary, in order to obtain reasonably accurate results, that such instruments should be calibrated over the range of their intended application by comparison with an absolute viscometer. Such a system of cahbiations not having been generally adopted, an unfortunate practice has become common of expressing viscosities, not in terms of physical or engineering units (by which alone the value of the unit can be applied in calculations), but by the number of seconds or minutes required for the efflux of a certain volume through VISCOSITY AND LUBRICATION 117 ic or cither of the best-known forms of commercial viscometers, heie aie thus in use as many arbitrary, irreconcilable, and dynami- illy meaningless units of viscosity as there are manufactureis of >mmercial viscometeis. A different type of secondary viscometer recently intioduced the cup-and-ball viscometei The action of this instiument epends on the viscous flow of the fluid, not in a tube, but between vo nearly parallel and closely adjacent surfaces. The instiument id its mode of operation will be moie fully described below, after iscussion of the theoiy of that type of viscous motion. Coefficients of Viscosity of Various Fluids In Table I (p. 118) aie given values of the viscosity constant of a few of the fluids which aie of chief interest to engmeeis, specially in connection with lubrication The table contains also ^proximate numeiical data, for the same fluids, of certain other hysical properties, the significance of which, as affecting the utility f the fluids as lubricants, will be made more appaient by the later ortions of this chapter. The constants are expiessed in all cases i C G S units. The value of ju for instance is the ratio of a stress leasmed in dynes pci sqiuic centimetie to a late of sheai measuied i centimeties pci second pei centimetre The values oi /x aie given lor various tempeiaturcs between o ad 1 00 C The other constants, which for the most pait do not 01 y lapidly with temperatuic, aie stated for atmospheric tempeia- ues m the neighbouihood ot 15 C. The mle which is apparent om the table as to the values ot ^ for liquids, namely that the value 31 each liquid diminishes as the temperature uses, is tme geneially. t will be noticed that the late of vaiiation is much less lapid for icicury and carbon bisulphide than foi the other liquids In all ases, as in au, on the other hand the viscosity mci eases with the cmperatuie. Variation of Viscosity with Pressure The viscosity of both liquids and gases varies veiy little with raiiations of piessure over a range from many times less, to many lines greater, than atmospheric piessure. At pressures, however, )f the order oi intensity of hundieds of atmospheres most liquids ippear to have greatly increased coefficients of viscosity. pqpqpq COCO PH X XX X u S a K CO O VO COOO O co xfJ> O O N in M O O O O O ^ , ooo a X w ON o w PQ n oo ON o o o o O vO o o O O O I> CO N o o o oo o r-. ON O Th J" 1 * {** *T" M H O >O CO O O O <* H I 000 M --o o oo M X i o -a D4 O Kt W^- ^T5 tu JJ S O J3- <us p i>iis 118 <1 VISCOSITY AND LUBRICATION 119 The following table (Table II) from Hyde (21, p. 159) shows how the viscosities of a few lubricating oils vaiy with pressures of this order Although such pressures do not usually exist m ordinary bearings, there are cases in the application of the theory of viscosity, as will be seen later in this chapter, in which the changes of the viscous constant by mciease of pressure cannot be neglected. Within very wide limits, the viscosity of gases is independent of pressure, the viscosity of air for instance being practically invariable from a piessure of a few millimetres of mercury up to pressures of many atmospheres This law, originally predicted by Maxwell from the kinetic theory of gases, has been confirmed by numerous experi- ments TABLE II VISCOSITIES OF VARIOUS LUBRICATING OILS AT VARYING PRESSURES. TEMPERATURE 40 C Absti acted from a table by J H Hyde Proc Roy Sot, , A 97 Pressme,Kilo- Mineral Oil Troltei Oil Rape Oil ~ 1Tn n , ^"^ <" Bayonne ) (Animal) (Vegetable) S P eim Ol1 Coefficient of Viscosity, M, C G S o o 47 o 344 o 375 o 154 157-5 062 o 413 0422 0190 3150 092 0550 0539 0236 472 5 i 32 o 686 o 703 o 299 630 o i 86 o 824 o 880 o 368 787 5 2 51 i 089 945 3 6 5 I 21 7 l 3 10 1 102 5 5 32 i 578 o 619 1260 755 1731 Viscous Flow between Parallel Planes As one of the typical conditions of flow met with m problems of lubrication and other practical applications of the theoiy of viscosity, it is convenient to consider m detail the flow of viscous liquid between two paiallel and closely adjacent plane walls, supposed fixed In rectangular co-oidinates, let z = o, and z = h be the parallel planes, h being small compared to their dimensions m the X and Y directions, as indicated in fig 9 For the reasons already explained the components of velocity I2O THE MECHANICAL PROPERTIES OF FLUIDS normal to the planes must be eveiywhei e negligible In other words , the rates of shear and the momentum in the Z diicction are very small; consequently the fluid pressuie p does not vary in that direction but is a function of x and y only. Also the rates of change from the values of the finite velocity components, u, v, in the fluid to their values, known to be zero, on the walls are lapid compared to their rates of change m the X and Y diiections. Thus, considei- mg a rectangular element, as in fig 10, anywheie between the Fig 9 Fig 10 planes % = o and g = h, the viscous ti actions on its lower face in the directions in which x and y increase, aie. -- dz and 03 The corresponding tractions on the uppei face aie * \ o ' ' \dz //liri and The sums of these pairs of tractions added to the differences of the fluid pressures on the faces parallel to the YZ and ZX planes are respectively equal to the rates of increase of the momentum of the element m the X and Y directions, thus -T.* dt VISCOSITY AND LUBRICATION 121 d z u dp . du or = and similarly /* ? = o being the density of the liquid The rates of increase of velocity -^, _? are of the order of the du dv dt dt products UTT, and v~~, and are thus, if, as we assume, u and v are ox ay small, of the order of squares of small quantities. These momentum terms will therefore be neglected in this and the following discus- sions. With this stipulation the equations (13) reduce to d z u dp dz z dx' , d 2 v dp and LL = --. as 2 dy These can be directly integrated, since p is independent of #, and hus du i dp, , ^ . _(V _J_ ( i o o V 6 i ^IJ) G% jU. OX and u = --( + C,# + C 2 ), M QX\ 2 / and similarly v = ( -f- D,^ + D, ) udv\2 / Now since ?^ and v arc zero on the plane z = o, the integration constants C 2 and D 2 are each zero, and since u and v are also zero Dn the plane z = A- A 2 A 2 2. + dA = - + D^ = o, 2 2 so that Cj. = D x = --. 2 i dp z(z h] " j and ' rp. Thus a 122 THE MECHANICAL PROPERTIES OF FLUIDS and the resultant velocity of the fluid at any point is being in the diiection of, and proportional to, the most rapid fall oi pressure, and varying accoiding to a paiabolic law along each normal from one plane to the other, having its maximum value midway be- tween them. The total flow across a width dy (see fig 9) from plane z = o to plane % = h (m the direction of x increasing) is * TT Q { h j ^y^P[ h f 2 r\j S-yU = $y f ud% ~ I (^ %h)d% J o 2ft OXJ o _ ... A 8 3? or U = --*!.* (I7 ) Similarly the flow pei unit width in the y direction ^ 7 h z "dp , ON is V = -- ~. . . ...... (18) I2[JiOy Thus the total flow in any diiection across a unit width perpendiculai to that diiection is equal to the rate of decrease of piessure in that h 3 direction multiplied by the constant - I2/A The same relation evidently holds for the flow of a viscous liquid in the space between two concentric, fixed cylinders, in either the axial or the circumfeiential direction, provided that the ladn of the cylinders are so nearly equal that their difference can be neglected compared with either of them. In both of these cases, as well as in all othei cases of flow between parallel surfaces plane or curved, it is evident, considering any small rectangular element S#, Sy, which extends in the z direction from one surface to the other, that since the same amount of fluid must flow out of, as flows into, the element in unit time, = * d s t> 2 or from (17) and (18) VISCOSITY AND LUBRICATION 123 it being lemembeied that the surfaces # = o and % = h, are assumed to be fixed. Flow between Parallel Planes having Relative Motion If the plane % = h is moving parallel to the plane % = o, with components of velocity % and z^ in the X and Y directions, uniform rates of shear ~ and -~ in these two directions will be superimposed h h on the fluid velocities u and v of (15) and (i$d). The components of velocity at # will become , i dp z(z h) . % ___ '_ ~L n 2 h' . . i dp z(z A) i % and v' = -- --- - + iy \Loy 2 h but neither the pressures nor the relation ^ + ?* = o d x z dy z will be affected If, on the othei hand, the plane % = h is caused to move nor- mally away from the plane % o, with velocity -, so that the at distance between the planes continually mci eases at this rate, it is evident that an excess of inflow over outflow must take place thiough the sides (at light angles to the planes) of the elemental y volume hSxSy to supply the additional volume which is continually being added to the element at the late - SxSy dt Expressing this equality in symbols, , au, 8y 8; J dx c -)- S# By = 3y - ,;8*8y, dt au av __ dh 01 3^v 8y 7t' and consequently from (17) and (18), = dx* d z dt i2 4 THE MECHANICAL PROPERTIES OF FLUIDS If we take the planes as being circular, of radius a, and suppose that the fluid between them at this radius is in direct communication with a large volume of the same fluid at constant pressure IT, it is evident from symmetry that the flow will be everywhere radially inwaids and that the pressure will diminish from II at radius a to a minimum at the centre. Taking, instead of a rectangular element, a cylindrical ele- ment extending from one plane to the other and contained between radii r and r + Sr as well as between two radial planes at a small angle Sa apart, its rate of in- crease of volume, see fig. n, will be This must be equal to the rate of increase of the Fig ii dh dt Sr.rSa. inward ladial flow as r increases by Sr, so that from (17), p. 123, or Integrating, s <j or roa dt dt dh , a / k 5 dp dr \i2ju, 9r Sr. J ' ~ dr\ drJ' But from (17), since the radial velocity is zero at the centre, dp ~ = o when r = o, or r A dp 6/x. dh ^ = o and = -- r, dr h* dt ' so that integrating again 3/z dh P = f-^ h 3 at When r = <z, = II ^ 2 + C, /t 3 dt so that ^> = II ^ ~( 2 r 2 ) ( 2 i) The force, P, necessary to move the plane at % h, against the VISCOSITY AND LUBRICATION 125 viscous resistance is equal and opposite to the difference of pressure p II integrated over the whole circle, or . h* df /a* _ a* ~ 4" } (22) Gup-and-ball Viscometer The type of viscous motion which has just been discussed is that on which is based the action of the cup-and-ball viscometer already mentioned on p. 19, and illustrated in figs 12 and 13. In the actual instrument, however, as illustrated m fig. 12, the two parallel surfaces which are drawn apart aie not planes but segments of two spheres, one concave and the other convex. The fixed surface is the concave lower surface of a metal cup, to which is attached a hollow handle by which the instrument is suspended In the cup fits a steel ball, but its surface is prevented from making actual contact with the sphencal surface of the cup by three very small projections (j, fig. 12) horn the cup's spherical surface The two spheri- cal sui faces are thus maintained parallel and about o 01 mm apart, when the ball rests on the pro- jections The narrow interspace is filled with the liquid to be tested, and in addition a groove G formed aiound the edge of the cup, and having a capacity of a few cubic millimetres, is also filled with the fluid, which is held in both the groove and the interspace by capillary tension The groove forms the reservoir at constant pressure II from which the interspace is fed with fluid when the two surfaces are drawn apart, as in the preceding calculations. The force P employed to draw the surfaces apart is the weight Fig 12 126 THE MECHANICAL PROPERTIES OF FLUIDS of the ball, which is usually of steel and I inch in diameter The method of making a test is merely, after placing sufficient liquid m f- f 1 -'.*--* v_ -.,, TTJBMW. \*"ff- r . ? ^'.iTT^CaSNi;?/ Fig 13 Cup-and-ball Viscometer the cup to fill the groove and interspace, and pressing the ball home, to suspend the whole instrument and note the time by stop-watch which the ball takes to detach itself. The temperature of the instrument, which, on account of the good conductivity of the VISCOSITY AND LUBRICATION 127 netal and the very small mass of liquid, is also veiy nearly the ernperature of the latter, is observed at the same time by means of L thermometer inserted in the hollow handle. The time of fall, the dimensions of the instrument being given, an be calculated approximately from formula (22), the spherical egments concerned, which m the actual instrument are compara- ively flat, being treated as circular planes of the same area. dt __ dh ~ J h '''^TTjLtfl* dh ___ 37TjllflY I __ I 2 hJ n which t is the time of fall of the ball, of weight P, from its initial listance h^ to a final distance h z from the surface of the cup. This fall is to be consideied to be complete when the volume of luid drawn into the interspace is equal to the volume initially con- amed in the groove, i e /here S is the sectional area of the groove, thus ho = nd the time of the complete fall is +17 7 L 2S thus h z = //!]_ -| 8 S(ah l -j- S) P nd if S is laige compared to ah 1} as it should be, 77TU<2 4 , APh, 2 t , . t = 3 r and u= 3 i- ........ (23) v ^ It will be seen from the formula (22) that the velocity of fall - varies as the cube of the distance fallen through. It is thus very b mall at first, but increases very rapidly in the later stages, and there ; no difficulty in practice in deciding the moment when the fall is irtually complete. 128 THE MECHANICAL PROPERTIES OF FLUIDS Although the action of the cup-and-bali viscometer can be cal- culated with sufficient accuracy when its dimensions, including the initial thickness of the fluid film, are known, the determination of this thickness, that is to say the height of the three projections in the cup, with sufficient accuracy would be so difficult that in practice the mstiument is employed as a secondary viscometer only. Each mstiument requires, however, only a single calibration test, which suffices to determine a single constant for the instrument, applicable over its whole range. The corrections for the momentum of the fluid and for capillarity are negligible, the foimei because the velocity of the fluid is exceedingly low and the latter because the radius of curvature of the meniscus of the liquid in the groove is very laige compared to the thickness of the liquid film sublet to viscous traction. B. LUBRICATION The Connection between Lubrication and Viscosity Although viscous liquids and plastic solids have been used from the earliest times to dimmish friction between solid bodies moving in contact with one another, and although the practice of thus " lubri- cating " the bearings of machines has doubtless been universal since machines were first constructed, no rational explanation of the action of the lubricant was known until Osboine Reynolds (5), in 1886, gave a clear interpietation of the phenomena in teirns of the theory of viscosity. Reynolds' explanation was only complete in a quantitative sense in the case of journal beanngs furnished with special, and at that date unusual, means for supplying ample quan- tities of lubricant. He showed that in such cases the solid surfaces aie completely separated from one another by fluid films of ap- preciable thickness, and that such films are maintained and enabled to support the pressure imposed on them quite automatically by the relative motion of the parts. The theory has since been extended to bearings of other kinds than journal bearings, and by its application new types of bearings have been devised for various pui poses which have proved far more efficient than the foims which they were designed to replace. While this " viscosity theory " of bearing lubrication is not quantitatively complete in all cases, and while there are probably other modes of lubrication in which viscosity does not play an essential part, it is at present true that all the most efficient known VISCOSITY AND LUBRICATION 129 ypes of bearings which operate with sliding, as distinguished from oiling, contact utilize the principle of lubrication which was dis- :overed by Reynolds. The expenmental and theoretical work by vhich the principle has been developed may be followed in the >apers quoted in the bibliography attached to the end of this chapter, t is only possible in the present chapter to give an outline of the heory and a few of the leading results which have been established, vith examples of the practical forms of bearings in which the theory las been utilized The feature common to all the bearings to which Reynolds' heory can be applied is that the surfaces of the relatively moving tarts are not exactly parallel but slightly inclined to one another. ? or instance, in ordei that a journal bearing of the usual type may >e lubricated according to Reynolds' principle, it is necessary that he journal shall be slightly eccentric in the bearing, so that the ilm of lubricant shall be of a thickness varying around the ournal Similarly, for the proper lubrication of a slipper moving rela- ively to a plane surface, it is necessary that the suiface of the lipper, if plane, shall be slightly inclined to the plane surface >ver which it moves. Essential Condition of Viscous Lubrication The explanation of this essential condition is readily given as n extension of the calculations contained in the first part of this hapter. Usually in the bearings to which the theory is applicable one of le surfaces can be considered as 2 ontinuous or unlimited in dimen- lon in the direction of the relative lotion (as for instance the surface f a journal, or a thrust collar, or n engine cylinder), while the sur- ice of the other member is essen- lally limited or discontinuous in tie same direction (as the surfaces ' FIB 14 f the corresponding bearing-brass, tirust-bearing shoe, or engine piston). In fig. 14, let XY be axes f co-ordinates (straight or curved) in directions at light angles to ach other along the surface of the continuous element, and Z the (D812) 6 i3o THE MECHANICAL PROPERTIES OF FLUIDS co-oi dinate axis normal to this surface, i e. in the direction of the thickness of the film, and as before let u, v, w be the components of the velocity of the fluid at any point in these three directions The surfaces of the continuous and discontinuous elements are assumed to be nearly parallel, and the distance between them h to be small compared to their radii of curvature. The discontinuous surface is supposed to move with components of velocity u lt % in the X and Y directions, parallel to the continuous surface, at scy. The problem of finding the motions and pressures of a viscous film between the surfaces is the same as that discussed on p 123, except that the sur- faces are not now parallel. Considering, as before, the rate of change of volume and the flow of fluid into and out of an element extending from one surface to the other and standing on the base 8x, Sy, it is seen from equations (17), (18), that the rate of increase of volume of fluid m the element due to the rates of change of piessuie and of film-thickness, in the X and Y directions is 8 / h* 8p\ s a . 3 / W 9p\ a s , v / - ~ Y>xBy -f ( ; Sy8tf , ...... (24) ox\ 1 2ju ox/ dy\ while the rate at which fluid passes out of the element in consequence of the shearing deformation due to the movement of the upper surface over the lower is is $ , is % , ^ _1 _8#8y -f - 1 - Sj/Stf ........... (25) 2 ox 2 oy The volume of the element is however diminishing, in conse- quence of the movement of the upper plane, at the rate 8/L ^ 9/z~ -, u^oxoy -j- Vj_- oybx, QOC oy consequently 3 / W dp\ , 9 / h 3 dp\ (u v dh , v, dh\ / dh , dh\ _ ( __ _ ) _L_ _ ( _ * J | _ f _ JL. _ i - j I 11 _ _j_ <y _, j dx\i2pdx/ dy \i2ju- dy/ \2 dx z dy/ \ dx dy/' 9/ 7 o9*\ , 9/ 78 9p\ . , / dh . dh\ , ,. or A 3 -i- ) H- 7r + 6w u^~- + > 17r - = o (26) dx\ dx/ 2y\ dy/ ^\ l dx l dy/ v ; This is the general differential equation determining the value of p at every point, being solved by integration for each particular case when h is given as a function of x and y (thus defining the forms of the surfaces), and when the velocities %, ^ are assigned. The complete solution is often not practicable, but exact or approximate VISCOSITY AND LUBRICATION 131 lutions can be obtained in a number of the simpler cases which n be regarded as sufficiently close approximations to the actual nditions of various types of bearings. Inclined Planes Unlimited in one Direction Take the case of two plane surfaces, the lower, # = o, being ilimited in the directions of X and Y, while the upper, also un- nited in the direction of Y,* extends only from x = a^ to x = a z , d intersects the plane ss = o on the line x = o. Thus the dis- ace between the planes, erywhere small, is proper- Z tnal to x, so that h = ex, icie c is the tangent of the lall angle between the planes ;e fig. 15). Let us assume at the upper plane moves er the lower with velocity v/ , in the direction of X, v mg zero, and that the lole is immersed in fluid, so that the piessure both in front and behind the moving plane is H and is constant. Ob- Dusly none of the conditions vaiy in the direction of Y, so that and ~ are both zero. Thus equation (26) becomes oy 'X Fig 15 _ h , 6 u = dx\ dx/ 1 dx and therefore dp z ~ ox -J- = o, .(2?) > being the value of h where = o, that is to say at a point, ox ? x 1} where p has a maximum or minimum value rpi dp f Thus - OW dx ^ ice -- is positive when h<h 1) and negative when h>h lt it is seen *The dimension of a beanng in the direction of the motion will in all cases be sried to as its length, and the transverse dimension as its width, regardless of ich of these is the greater. I 3 2 THE MECHANICAL PROPERTIES OF FLUIDS that p has a maximum value (at x = x ) : between x == a lt and Integrating, p = -^(- -~ - C), (28) but sincej> = II, both when x = %, and when x = a z , H = ^ from which 2(2 1 <3:o / \ or ^ = - ; " ..... . ....... (29) &i + a* (the point of maximum pressure thus being nearer to a^ than to <7 2 ), and - n C 2 C 2 [a a. 2V 2 = n Thus by substitution for ^ 15 and C in (28), * = n c 6^% /% + a ^1^2 T \ /, n \ = 1J - "" , , v ( ~~ 5- ~~ I ) ---- (3) c 2 ^ + a 2 ) \ x X* ) this equation deteimining the pressures at all points between the two planes. The total upwaid pressure on the upper plane per unit width in the direction Y is 2^ 3 (<2 1 H- #2 VISCOSITY AND LUBRICATION 133 icing dependent only on the ratio of #., to a v for a given value of c, nd the mean pressure is P 6uMi i a* 2 ] f N log,- 2 - f ...... (32) Jso the total frictional resistance to the motion of the upper plane er unit width is F/ **1 J I f*' M '1. J f \ - I -t-sjtM . I I */7'V* I 1 ^ 1 i r(, ^ _ i . w^ *, . , I -S { J 117 f yin-i/i ^*^ *'* = ^ log,, (33^) c ^i ependent, like P, only on the ratio a 2 : a and c; and the ratio f traction to load, or " coefficient of friction " is F lo g fi " f = _ = f ^ f-7^ J p /- U't/ log,,-^ 2 X 2 Iso the position of the centre of the upward piessuie on the upper ane is given by ~ i [ a ~ 6ww r rtj ( ci ci \ * = pj - ] l ) xdx = p 6 2^_L Q J |( fl l + 2 ) - -^ ~ # j<& ^ o 1 "2 <2o" <2i 2^7i (2 2 1O&. *s i i j o^ - 1 --- (35) 2 / \ i / i ing independent of . Applications to Actual Bearings The solution of the problem in viscous motion illustrated in ;. 15 has been worked out in some detail because it affords m a igle case a general view of the nature of Reynolds' theory of lubri- tion. If we imagine the lower plane % o replaced by the surface of :ylinder whose axis is parallel to the Y axis of co-ordinates, and I 3 4 T HE MECHANICAL PROPERTIES OF FLUIDS the upper plane, extending from x = %, to x = a z , leplaced by a cuived surface which, at every point of co-ordinates x, y, measured respectively circumferentially from a generating line of the cylinder corresponding to x = o, and axially from a circumferential cncle of the cylinder corresponding to y = o, is at the same normal distance h from the cylinder as are the two planes from one another, the results which have been obtained will still apply This ideal form of a cylindrical journal beanng is illustrated in fig. 16 The cylinder can be regarded as the journal of an axle, and the upper surface as the bearing surface of the beai ing-brass of the axle. The lesults as to the fluid pressure which have been calculated above evidently remain true if, instead of the bearing-brass moving in the direction x with linear velocity w ls the journal revolves in the opposite direction with the same sur- face velocity. Actual bearings are, of course, not of unlimited width, but for the middle por- tions of a beanng whose dimension in the Fis l6 direction transverse to the relative motion is not less than two or thice times that in the direction of motion, the calculated results apply with fair accuracy. In such middle portions of the bearing the oil will flow in lines approximately at right angles to the geneiatmg lines of the cylinder. In the lateral portions of the bearing, on the othei hand, the oil being under pressure will tend to flow towaids the nearest side, and the theoietical conditions will on this account be depaited from If, however, the sides of the beanng be closed by some ar- rangement, such as a stuffing-box, preventing the escape of oil, the flow of oil will be everywhere, except within distances fiom the closed sides compaiable with h lt circumferential, and the conditions assumed for unlimited surfaces will be precisely icalized, piovided always, of course, that the bearing-brass is of such a form that h = ex, which is tiuc only to a first approximation for the form which is usually given to such brasses. The calculations apply more accurately to the case of a conical sleeve moving longitudinally on a cylindrical rod as illustiated in fig. 17. In this figure the axis of X is a generating line of the cylin- drical surface of the rod, the axis of Y is a circumferential circle, and that of Z as befoie is normal to the surface. As before, we assume that the normal distance between the surfaces is given by h = cx t VISCOSITY AND LUBRICATION 135 that the conical and cylindrical surfaces, which are coaxial, inter- ct at x = o. The sleeve extends from sc = <% to x = a%, and supposed to move parallel to the axis of X with velocity u^. From symmetry the motion of the fluid must be everywhere rallel to the axis of X, and as the cone and film of fluid have no undaries in the direction of Y, the solution given above will hold curately provided that the thickness of the film is very small mpared to the radius r^ and length, a z a^ of the cone. Thus, for ample, the resistance to the motion of the cone, from (33^), p. 133, is le curve /> x in fig. 18 shows the mode in which the fluid pressure r \j Fig 17 Fig 1 8 ween the surfaces of figs 15, 16, and 17 vanes m the direction x for the particular case in which > = za . It will be seen that ' maximum pressure occurs at x 1 = -a 1 , or at one-third of the 3 gth of the sleeve or bearing-brass fiom its rear end, and, as may seen by wilting 2% for a 2 in (3^), the resultant piessure urs at a = i'43i 1 , or 0-431 of the length from the same end. Table III, p. 130, shows the actual numerical results in j.S units for a moving surface carrying a resultant pressure of Cgm. with a lubricating fluid of viscosity i C G.S The surface issumed to be i cm long in the direction of motion (i e. a 2 % i cm), and the results are expressed for i cm. of width in the tisverse direction The quantities tabulated are' h lt the thickness of film at x = a lt unit io~ 3 cm.; h z , the thickness of film at x = a 2 , unit io~ 3 cm.; - flp the distance of the centre of pressure from the trailing end. 136 THE MECHANICAL PROPERTIES OF FLUIDS Unit, i cm", f, the effective coefficient of friction, = F x the tractive force in kilograms. The independent variable in the fiist column of the table is the a* ratio - 2 = TABLE III h z a i- / I o o o 5000 00 1-2 02775 0333 04818 3-285 X 10-3 1-4 03465 o 4851 o 4664 2-428 1-6 03793 o 6079 04532 2 065 1-8 03955 o 7119 04416 1858 2 0-4026 o 8051 04313 I 722 2 2 0-4043 08895 04221 1-025 24 o 4027 o 9665 04137 1-553 2-6 03991 10375 o 4061 1-497 28 o 3942 1 1037 03991 1451 3-0 03884 I 1652 o 3926 1414 4-0 03559 14237 0-3662 i 298 50 03247 I 6237 03465 i 239 60 0-2982 I 7892 03310 I 202 II 02115 2 3269 o 2832 I I 34 The corresponding results for any other dimensions and con- ditions of loading may be derived from the following dimensional formulas, viz If the length of the surface, velocity, resultant load, and viscosity, instead of being each unity in the units employed, are respectively Length, L centimetres, Velocity, V centimetres per second, Load, P kilogiams per unit width, Viscosity, M C.G.S. units, then, for any given value of , h t and h z aie to be multiplied by LV*M* a a-, , , l 1 _ . , ,.,.,, ^TITM-H-TI .p, is unchanged, and F is to be multiplied by V*P*M*, * a z a i m ]y[iyi while c and/ are to be multiplied by -^ . It will be seen from Table III, combined with these dimensional formulae, that the thicknesses of the films of viscous fluid concerned VISCOSITY AND LUBRICATION 137 i lubrication are small, and comparable to the smallest linear mea- irernents which the mechanical engineer is accustomed to make. t is therelbie necessary in order to effect lubrication in the manner itended, and to secure the low frictional resistances which the icory indicates as attainable, that the workmanship of the bearings lall be of a relatively high order of accuracy. The fact, otherwise inexplicable, that the conditions and laws F viscous lubrication were not discovered until the end of the ineteenth century, is doubtless due to the circumstance that it was tily at about that epoch that mechanical workmanship became merally of such a quality that the necessary conditions were often )mphed with. With rougher workmanship the necessary con- nuous films cannot be formed, but the two members of the bearing )me into actual or viitual contact, at least at some points, and thus ting about mixed conditions of solid and viscous friction incapable * being referred to any simple or consistent laws Even with woikmanship which may be legarded as perfect the Itimate stage of failure initiated by any cause is contact of the solid irfaces, either directly or through the small particles of solid im- imties which aie always to some extent present in the lubricant 'here is thus suggested as a cnterion of the safety of any bearing om such failure, the thickness of the lubricating film, at its thinnest nt under the working conditions which reduce this thickness to minimum It will be seen from Table III, p 136, that for a bearing surface of iven length, with given velocity, load, and lubricant, the thickness of ie lubricant at the point of closest approach to the other surface, is a laximum when -2 = 2-2 . . . It is usual to adopt this ratio as that . 1 be preferred in designing beatings The table shows that with ds ratio the coefficient of friction, though higher than is attainable ith greater values of the ratio , is nevertheless already so small i lat its further reduction may usually be considered of little moment. must, however, be remembered that the optimum ratio, = 2-2 j iften taken as z o as a sufficiently close approximation) has, strictly )eakmg, been derived only from the special case of a bearing surface " infinite length and for the condition h = ex. It is hardly necessary to remark that if the velocity HI in the above ilculations be reversed, the equations for the pressures will be still (D312) 6* 138 THE MECHANICAL PROPERTIES OF FLUIDS valid, with merely a change of sign for both u and p It must, howevei, be lemembered that whereas in the case of positive values of p the intensity of pressure has no necessary limit, negative values of p, that is to say tensions, are not in general sustainable in fluids such as ordinary oils, and indeed in most forms of bearings positive values of/> less than H, the atmospheric pressure, aie usually incon- sistent with the assumptions made in the calculations, since, under those conditions, air will be drawn into the spaces assumed to be occupied by oil. The volume of fluid flowing between the surfaces per unit time may be calculated as follows: From (30), p. 132, the rate of change of the piessuie with x at the tear end of the bearing, i e. at x = a^ is dp _ _6/x dx c z Theiefore from (17), p. 122, and (25), p 130, the volume rate at which the fluid passes through unit width of the noimal plane at the reai end of the moving surface is - -^ .. (36) The same result would be obtained by calculating the inflow at the front edge, and it may also be seen at once from the considciation that at the point of maximum piessure x = x lt theic being no flow due to late of change of pressure, the volume rate at which fluid 11 passes the normal plane is entiiely due to the mean velocity 1 , 2- acting ovei the film thickness, which is Under the same assumptions as in Table III, p 136,1110 value of Q for the condition = 2-2 is 2-78 X io~ 4 c. c. per second <h. per centimetre of the transverse dimension of the bearing. VISCOSITY AND LUBRICATION 139 Self-adjustment of the Positions of Bearing Surfaces The question naturally arises how it is possible to secure in actual :arings the exact locations of the bearing parts shown to be necessary r the preceding calculations, and as illustrated in figs. 15-17, and >w it is that so delicate an adjustment is not liable to be destroyed r inevitable wearing of the parts The explanation is that in suc- ssful types of beaiings the paits are self-adjusting, their correct utual location being automatically brought about by their relative otion and continually corrected for any slight wear which may ke place. Take for instance the case of the infinite plane slipper illustrated fig. 15, of which fig. 19 is a sec- m on any plane parallel to XZ. It has been seen from Table I that if the latio of 2 to a { is z the resultant piessuie of the lid acts at the point x = a, 'iere a a = 0-4221 X ( 2 ~ a i)> T id that with the value of h t Flg I0 ven in the table and unit values ju. and n the total resultant piessurc is i Kgm per unit uisverse width. Conveisely, if a load of I Kgm per unit dth be applied to the slippei at the point x a as indi- ted by the anow in fig 19, and the slippei be moved with lit velocity and supplied with fluid of viscosity I C.G S , it will ke up the same position. Expeuence, moi cover, shows that such [iiilibnum is stable for the displacements which are liable to occur the operation of the bearings. In the case of plane slippers the id must in practice be applied as shown in fig 19, that is to say, lough an actual or viitual pivot ol some kind with which the slippei provided at the correct point. Actual examples \vill be illustrated the descriptions of thmst bearings given in the later parts of this apter. In the case of cylindrical journal bearings, however, there is other mode of self-adjustment possible, which, though not so icient as the pivot method, is even simpler, and which undesignedly ok place in bearings of this class long befoie Reynolds' principle is discovered, and rendered them superior in efficiency to all other isses of bearings known at that time. MO THE MECHANICAL PROPERTIES OF FLUIDS Self-adjustment in Journal Bearings This action, is illustrated for the ordinary foim of fixed journal bearings in figs. 2oa, b, c. We will assume that the bearing is one of a pair of journal bearings, as for the shaft of an electuc motor, consisting of a cylindrical brass, or pair of semi-cylmdiical half- brasses, of which only the lower half cylinder is noimally effective. The radius of the bearing is, necessarily, gi eater than that of the journal. The load W is assumed to be the weight of the shaft and parts attached to it, acting vertically downwards When the journal is at rest its position m the bearing is that shown in fig zoa. The journal and bearing are then in contact along the lowest generating lines of their cylindrical surfaces When, however, the shaft begins to rotate, foi example, in the clockwise direction as indicated m the figures, the oil at the light-hand side ot the journal is subjected to a traction directed fiom the wider to the narrower part of the interspace between the journal and the bearing. On the principles which have been explained, the oil in this space will consequently exert a fluid pressure. On the opposite, or left-hand side of the journal, on the contraiy, the mtci space increases m thickness in the direction of motion, and consequently, as explained on p. 137, the pressure m the oil film will fall, becoming negative unless, as is sometimes the case, air is fice to enter, when atmospheric pressure will tend to be established. The journal will consequently tend to move towards this left-hand side, the point of contact between journal and bearing shifting from the lowest gener- ating lines to some higher line towards the left hand. Oil tinder piessure will thus be admitted between the paits of journal and bearing, and this action will be progressive until the resultant up- waid piessure becomes equal to the load W on the bearing. At VISCOSITY AND LUBRICATION 141 onslant speed a stable condition will be reached as shown in fig. 2ob. 'he point of closest approach, C, will be somewhere on the left-hand de of the vertical, with a portion of the interspace above and to ic left of C still diverging in the direction of motion The oil in us latter space will in consequence exert a negative pressure on ic journal, as indicated by the arrow P 2 The resultant of this force ad the positive resultant pressure P ls exerted by the oil in the ght-hand converging portion of the interspace, will be equal and pposite to W, the load on the journal. If the speed of the journal is increased, the amount of con- ergencc and divergence of the respective parts f the journal, for a given load W, will auto- mtically dimmish, the limiting condition with ifimte speed (or zero load) being that illus- ated in fig 2oc, the journal becoming then Dncentric with the bearing It may be noticed that in all cases the ivergmg portion of the film, and the neaily Flg 3I arallel portions in the immediate neighbourhood f C, though of respectively negative and zero value for the support f load, are subject to sheai of equal or greater intensity than the ffective pressure-producing film on the right hand and lower 11 faces of the journal For this icason such journal bearings, with ic biasses embracing a semicircle or other relatively large arc, are ecidedly inefficient compared to a pivoted bearing of small arc such > that illustrated in fig 16, in which self-adjustment takes place i the same mode as that described in connection with fig 19 It is also readily seen that, in all cases, the interspace between ic journal and a segmcntal cylindrical bearing surface can only be mvergent throughout its length if the arc of the bearing surface less than 90 It is indeed desirable, in order to secure a fairly ipid rate of convergence thioughout, that the arc should be limited i 45 at most It will be seen that in such a case as that illustrated in fig 21, is possible, without pivoting the brass, for the icsultant fluid ressme to be vertical and thus in equilibrium with the load W, ithout the formation of any diverging interspace, and this even hen the radius of cuivature of the brass is the same as that the journal The latter is a convenient condition, as it Imits of the simplest and most accurate method of accurately rmmg the bearing surface, namely by scraping or lapping it. 142 THE MECHANICAL PROPERTIES OF FLUIDS It is to be lemarked, however, that automatic self-adjustment in journal bearings with rigid (i.e. non-pivoted) brasses, as m fig. 2ob or 21, is only possible when there are not more than two bearings on a shaft, or if the shaft is flexible, as otheiwise, since a thud beaimg will be invariably out of alignment to an extent comparable with or greater than the thickness of effective oil-films, it is not possible for each of the journals to adjust itself to its correct position. With any number of pivoted bearings, however, if the pivots are appioxi- mately in the vertical plane through the axis of the shaft, each bearing will exert a vertical resultant piessure, and by providing adjustments for the pivots in the vertical direction only it is possible to divide the total load carried by the shaft equally between the beanngs. Exact Calculation of Cylindrical Journal and Bearing The mathematical solution of the viscous motion foi the case illustrated in figs, 2oa, 21 was given by Reynolds (5, p 158) The solution was simplified by Sommerfeld (8, p. 158), of whose process a brief lesume will now be given. In fig. 22, O and O' are the centies and r and r -f- 8 the radii of the cylindrical journal and semi-cylindrical bearing, both of infinite extension m the dnec- tion of their axes. <> Let OO' = e, and - = a, a having therefore different values in difleient cases, varying fiom i, when the journal Fig 33 and bearing are in contact, to oo when they ate concentric. Let i/r be the angle between OO' and the vertical and (j> an angulai co-ordinate measured from the direction OO', the co-ordinates for the ends of the bearing-biass being ~, and iA -f - as indicated in 2 2 the figure. Thus the linear co-ordinate, x, in the direction of motion of the brass relatively to the journal is now constant rfi. Then from (27), p. 131, rd$ h* (37) VISCOSITY AND LUBRICATION 143 md since p II when < = i/i - , and when $ = ^ + - , r J 2 a , if p is the fluid pressure and q the circumferential traction per unit width at <f t and P the total load on the bearing per unit width, /> + - P cosi/r * (p II) cos^rJ^ J z J ~z md /> + E A|, + I P sin^ w (p IT) sm<l>rd(l> + I 2 # coafod(f> = o. But since . . jr fa r (p H) COS(bd<f) = \ (p , 7T L ind r5 r + r"/> (/> - H) smfW = - (p - H) cos./, + -77 ^ -E "- -",-1 / -E ^ md since p = II, both when $ = ift -\- - and when cf> = ijj -, 2 2 30 that the terms not under integral signs vanish, r-H . P r COS! ^' (38) /-V "1* md 2 Now from (33), p. 133, and (37), p. 142, in which the second term on the right can be neglected on account of the smallness of h compared to r. 144 THE MECHANICAL PROPERTIES OF FLUIDS Thus the equations can be written /l/f +7T/2 {(h hj)lh?}sm<f*ty = (P/r) cosi/, *::", /\li + tr[Z {(h hj)/h 9 }cos(f>d(f> = (P/r) smift, V-rr/2 or, since h = e(a + cos</>), 7/j = e(a -j- these equations become J (a J (a (1 JL\2 I ( (X "1 COSC&I G * \ Cff - i - vwoyy i \jp*t\j(t i ifc-i . v / /* COS(P o w i S] + -// , . -,^^ = (a + cos<ji) 2 (a H- cos0) 3 6r X 7T the integrals as before being from -- to tft 2 2 These integrations can be effected by usual methods,* and from the results Sommerfeld calculated the following numerical table, Table IV, in which r] -, and the " coefficient of fnction ",/= - -, r Pt where M is the moment of the frictional tractions about O. TABLE IV a a A co^ (1 z / I 90 I'D f\ X i oo I '02 1 20 o 998 PYI 2 f J?o. x o 012 X o 94 I-I3 129 - 098 X 0-04 X 091 1'5 135 - 093 X 008 X o 92 2-4 133 -088 X 0-14 X i oo 63 128 o 72 X 0-29 X 134 33'9 120 o 50 X 0-62 X 2 17 00 90 o X oo X oo If the coefficient of friction / be plotted with z^ as the variable, *T cos^ . f(a + cos<) a f d<f> I .am I drp = ff> a / - . - 7 J a -f cobf/> ' J a + cosf/p J a H- cosip = ^ _ a ? =tan -x / tan Va a -i \ Differentiate this with icspcct to to get intcgials in second line of (40). In- tegials in first line come at once, since d(a + cos^) = suvfxty. VISCOSITY AND LUBRICATION 145 s P being constant, then for various values of v) Q = - we have a series of cuives, m which as % increases from zero, the coefficient of friction falls at first to a minimum value about 8 per cent lower than its initial value, then with further increase of % the coefficient gradually rises and finally increases to an asymptotic approximation to the straight line/ = i. The value of % for which the coefficient of friction is a minimum is approximately = P82 12-5 X ftf* 2 " In actual bearings the initial value of the coefficient of friction will be much higher than that calculated, since with very low velo- cities, and values of a only slightly greater than i, the journal and bearing will be, owing to minute roughnesses of their surfaces, in metallic contact instead of being separated by a very thin continuous film, as assumed in the theory It is to be observed that in these calculations of Sommerfeld's the portion of the film between the point of closest approach, <f> = < , and <}> = i/f -f- is subject to a negative pressure. The possibility 2 ot such a condition may icasonably be postulated in very wide beanngs, but can hatdly be assumed in bearings of usual proportions unless special means are employed for preventing the entry of air at the sides Approximate Calculation of Cylindrical Bearings The method and results of Sommerfeld's investigation given above apply to the case of a cylindrical bearing whose angulai length in the circumfeiential dnection is 180. A similar process may be applied to beanngs of smaller angulai length, as in fig 21, such beanngs, as explained on p 141, being pieferable in practice In these cases, howevei, there is little value in the assumption that the suiface of the brass is a circular cylinder, and, especially in the pivoted type, it is usually sufficient to assume that the thickness of the mtei space is a Imeai function of </>, that is to say to apply in the case of a veiy wide bearing the method and results of pp. 131-133. If a closer approximation is desired, the form of the bearing may 146 THE MECHANICAL PROPERTIES OF FLUIDS be approximately represented by the equation h cx m = C(n/>)"' with an appropriate value of m differing from unity The solution of this case has been given by Rayleigh (20, p. 159), who, however, found that in the numerical applications which he made of it, the results did not differ very materially from those derived from the simpler formula, h = ex. Plane Bearings of Finite Width A more important modification of the Reynolds' theory of beanngs with uniformly varying interspaces, is that which it requires for its application to bearings which are of limited width, and in which, consequently, there is a transverse flow of the fluid under pressure i to the sides, i e m the direction of Y, as well as flow m the direction of the relative motion X. The solution, as given A i a by Michell (9, p 158), m- */"' volves rather lengthy cal- f a s culations, and we can give * 2 only an indication of the method and a few working formulas and constants In fig 23 (which corresponds to fig 15 for the case of infinite width), ABDC is a rectangular plate m the plane z = ex (the length of the plate being # 2 a^ and its width b) sliding m the dnection of X with velocity u^ The pressure is assumed to be uniform eveiy where except in the inteispace between the plate ABDC and the infinite fixed plate in the plane % = o, i e the boundary conditions of the plate are p = IT, when x = 15 or x = fl 2 , for all values of y, and also when y = QJ or y = b, for all values of x Between the two plates p must satisfy the differential equation (26), p 130, i.e 9 /;<M , 9 fi$P\ ^ f ( dh . dh\ /r*-f -f -( h + M is- + v i^~ } J dx\ dx dy\ dyJ ^\ *dx l dy 7 j or, since h = ex, and = o, dy a<t , j xdx d z cV VISCOSITY AND LUBRICATION 147 This equation may be written in the form ox x ox oy c x TT { . TTV . I . i^ry . . i miry . \ ( sin- + - sini- +...+- sin -^ + . . . ) = o, \ b 3 b m b / .ince the sum of the series in brackets, for all the values of y with vhich we are concerned, viz. y = o to y = b, is -. 4 To solve this differential equation so as to give p as a function >f x and y, it is assumed that there is a solution of the form p = n + pj_ + p 3 -f . . . + p m + . . . ad inf. (42) .. . mrry f oirt ' S m *^ J "** * in which p m == > ITITTX m being a function of x only. The integer m can have only odd values, because p H must be ymmetrical on both sides of y = - . ^ miry Thus p n = S 00 '" b , /here ? is odd. If for brevity we write 24^^ = A, and = , oc 2 6 . miry sin =-, o Thus the coefficient of sin ^ in equation (41), p. 146, is 4 - i -i- 2 r 4 8 THE MECHANICAL PROPERTIES OF FLUIDS Every such coefficient must vanish, and consequently the factoi within the brackets may be equated to zero, of which equation the particular integrals are the BessePs Functions, I x (^) and K^), and the complete integral may be written in either of the foims &, = AA() + BJ^O - k(i + 2 -f -^ + ...),... .(44) or = A' w I 1 a)H-B' M K 1 (0-^( The second form, useful when is veiy large, being " asymptotic " The coefficients A^, A' OT , B OT , W m are to be deteirmned so as to make g m vanish for x == a 1} and x = 2 , and hence p m vanish for all values of y on these two lines These coefficients can only be determined aiithmetically, numeii- cal values being given to the quantities <z 1} a Z) and b. The steps of the calculation, with tables, aie given in the paper (9, p 158) The coefficients A m , &c , having been calculated, the values of p for as many points x, y, as may be desired aie also calculated aiith- metically, and when p is known the total fluid pressure supporting the block is determined by aiithmetical or graphical summation, from the relation P = f a *f b pdxdy ...... (46) J a^J The fnctional traction by (33), p 133, is F = log e , pei unit width, and a i M = Hog? = Alog .(47- for the whole of the square shoe, of area A = b(a z a). The point of action of the resultant pressuie is found by an arithmetical summation of moments By way of examples, a lew numerical formulae will be given. The total pressuie on a square bearing in which flg flj - - d^ o is Jr . c being, by comparison with formula (31), p. 132, only 0-421 of the total piessure on a portion of equal area, equal length and inclina- tion of a plane of infinite width, thus showing the effect of the escape of oil fiom the sides of the bearing. VISCOSITY AND LUBRICATION 149 The position of the centre of pressure for the finite square block is at a distance 0-42% from the rear edge, as compared with o 431%, in the infinite beaiing. (See Table III, p 136.) The coefficient of friction is 10-3^. A further calculation serves to show that of the total quantity of oil which enters the inter- space at the leading edge of the square shoe approximately one-sixth passes out at each of the sides and the remaining two-thirds at the rear edge. Similarly, in the case of a bearing whose width transverse to the motion is only one-thud of its length, so that a z ~" a i a \ ~ 3^ the total pressure is ind the centre of piessure is 0-39%, from the rear end. These results, as already explained, are equally applicable to ournal bearings as to plane slide bearings provided that the foim of he beaiing surface and the position of the pivot are such that h = ex, ind <2 a a a^ Arithmetical evaluations of the pressures and frictional co- 'ffiuents given by the above theory have been calculated for an >xtensive series of beaiing blocks of varying proportions by Torao Cobayashi (30) Cylindrical Bearings of Finite Width Mathematical treatment of cylindncal bearings of finite width, 'onespoiidmg to the theory given above for plane bearings, does tot yet exist This is unfortunate, since the limitation of width ias an even gi eater cflect in a cylmdiical than in a plane bearing n reducing the pressures generated, and particularly so if the arc ubtended by the cylindrical bearing approaches a semicircle as it isually does in the conventional type of journal bearing. By way of illustration of this statement, we may take a journal 'earing in which the bearing-shell subtends an arc of 120, nd in which the thickness of the film at the inlet is double its hickness at the outlet (as in the plane bearing previously discussed). Such a bearing is illustrated in fig 24,* in which r is the radius * Fiom (28), p 159. 150 THE MECHANICAL PROPERTIES OF FLUIDS of the journal, r ~\- 8 that of the bearing-shell, the distance between their centres being e, while 26 is the width of the bearing. It is readily seen that in such a bearing the areas through which oil may escape at the sides of the bearing are much greater relatively to the areas at the front and rear (at which the oil would enter and leave in a two-dimensional bearing) than is the case in a plane bearing of similar length and width. In the particular case in which the width of the bearing is equal to the radius of the journal, and in which the radii of journal and bearing are equal, so that 8==o 2,b = r, the respective areas of the leading, trailing and side openings are in the proportions 2:1: 10-3. In other words, the oil which enters the interspace at the leading edge has more than 10 times greater area by which to leave at the sides than at the rear Pressures are consequently determined almost entirely by the conditions at the sides, and a two-dimensional solution would convey a very false idea of the actual conditions A more useful approximation in such a case can be obtained by treating the bearing as infinitely long in comparison with its width. On this assumption the pressures over the portion of the bearing near its trailing end (which is the only portion m which effective pressures will be generated) are given by P VISCOSITY AND LUBRICATION 151 being the axial co-ordinate measured from the middle circum- irence, and h the varying thickness, determined by h = 8 -j- cos#. Experimental Results The curves given in the right-hand half of fig 24 are derived rom an extensive series of tests of a pivoted journal bearing of which he circumferential length was 6 98 cm. and the width was 6-35 m , the block being thus not quite square From examination of his diagram it will be found that for a given load the coefficient of action varies approximately as V/^'u while for a given value of fiu lt varies nearly inversely as the square root of the load, both these esults being m accordance \\ith the formulas above The facts tated are brought out more explicitly in the following table, which hows that the values of F, P and pu^ as read off the right-hand art of fig 24, make Fy'(P//xM 1 ) approximately constant. The ift-hand part of fig 24^ will be explained on p. 157. TABLE V F P/ju/! V(P//ii) io 3 FV(P//<i) o 0008 o 12 o 35 o 28 OOI2 067 26 31 0016 037 19 30 OO20 023 15 30 0024 017 -13 31 Types of Pivoted Bearings The chief practical field of application of plane pivoted bearings , to thrust bearings These usually take the form of an annular incs of " shoes " or " blocks " pivoted upon fixed points in the ationary casing and presenting their plane working surfaces to a lane-surfaced annular collar fixed on the rotating shaft. Such a N E3 01 am u S "s 152 I'K, _>() I IHU SI SllOl S I I(r 28 FlIKl'Sl Hi V.KIM, [OK IIOKI/OMVI SlIUlS, I'ating piigf i ^, VISCOSITY AND LUBRICATION 153 rust bearing arranged for a vertical shaft is shown in fig. 25. In is bearing the thrust shoes, t, are fixed in the lower part of the sing of the bearing which also serves as the casing of a journal taring for the thrust shaft. (The journal bearing is of the flexibly- voted type described on p. 155.) In order that the casing may form a reservoir for oil to Fig 23 ibiicate both bearings, the journal surface is formed not diiectly pon the shaft, but on the outer surface of a collar T, attached to ic sleeve S, and forming also the thrust collar which revolves upon le annulus of thrust shoes t. This annulus is shown separately in o; 26 (The flexible journal ring is shown in fig. 31, facing p. 154). In figs 27 and 28 is shown another type of thiust bearing, con- snient for application to horizontal shafts. In this form, which . adapted to take thrusts in either axial direction, two pivoted thrust- loes R only are employed for each direction of thrust, each pair eing mounted in a common housing H, which is itself pivoted on the \n\\soi Hi \KINC, Snoi I' ic, 31 LMU.L JOURNAL BTAKING Facing page 134 >/ Q s? (\A ^V * X \ v - ? VISCOSITY AND LUBRICATION 155 yvver part of the fixed casing on an edge e at light angles to the ivoting nbs, rr, of the individual shoes. Fig 28 is a photograph of the parts of the bearing which is shown y longitudinal and cross sections in fig. 27. In fig. 29 aie shown two views of a pivoted journal bearing shoe, icing one of an annular series of four arranged for the journal bearing f a vertical shaft The pivoting edge e is clearly seen on the back f the shoe Flexible Bearings The comparatively small clearances and slight relative inclinations ictween coacting bearing parts, requisite to produce effective lubri- ating films, allow of a modified type of con- traction for achieving the same purposes as re attained by pivoted bearings It is evident hat in these a spring, or other continuous but leformable connection, may be substituted for he rolling or rotating contact of a pivot. Such spring may be either a separate pait attached Fl , 30 joth to the shoe and its supporting member, or nay be an integial part of one or both of these provided such part s made with the necessaiy degiee of flexibility to allow of the shoe leflectmg under the load. Alternatively, as in a type of construction )roposed by Ferranti,* a pair of springs may be used to connect the hoe and its suppoit, viz a comparatively stiff spiing at the rear and i lighter spring at the front of the shoe This construction, which is llustiated in fig 30, will evidently have the effect of applying he resultant load at a point P, behind the middle point of the ,hoe, much as if it were applied to a ngid pivot at that point. The chief advantage of a flexible construction is that it enables .mail or relatively unimpoitant beanngs to be simplified, by con- itmctmg a number of bearing shoes integral with, but flexibly con- iccted to, a common suppoitmg member. A serious disadvantage s that the flexibility involves more or less lisk of fracture of the lexible part, a danger which is to some extent oveicome by giving lexibility to a portion of the shoe itself. The large journal bearing, ilready mentioned on p. 153, is constructed in thk way, and is llustrated in fig. 31, and in fig. 25, p. 153. In the former the individual shoes, S, may be seen attached to the * British patent No. 5035/1910. 156 THE MECHANICAL PROPERTIES OF FLUIDS supporting ring R, by flexible necks N, and having also their leading portions, L, reduced in thickness for some distance from the leading edges. Limitations of the Theory As the shoes of such thrust bearings as are illustrated in figs 24*2: to 28 are usually of small radial width compared to their mean radii, the formute given for rectangular bearing slippers may usually be applied to them with sufficient accuracy for practical purposes in spite of their sectonal form. A more exact calculation can be made when required by a process which refers the co-ordinates of the sectonal shoe to those of the rectangular shoe.* Of greater practical importance are the departures from the results of the calculations which in some cases arise from the insufficiency of the physical assumptions which have been made, especially as to the constancy of the coefficient of viscosity An experimental method of solution, imagined and applied by Kingsbury (29), is free from most, if not all, of these limitations This method utilizes the identity which exists between the equations connecting pressure and volume-flow in viscous liquids, and potential and current in an electrical conductor The conductor used is a conducting liquid contained within solid, non-conducting boundaries shaped to represent in correct proportion (though on an exaggerated scale as legards thickness of the conductor) the lubricating film to be investigated The results obtained by Professor Kingsbuiy agiee closely with those of the mathematical investigations, e g those of the plane bearing of finite width given on pp. 146 of the piesent chapter. The method has been applied to both plane and cylindrical bearings of various ratios of length to width It was shown in Table I, p. 118, that the viscosity of lubricating oils diminishes rapidly as the temperature rises. In a well-loaded pivoted bearing, carrying for instance a mean pressure of 70 Kgrn per square centimetre, and with the product \JM-^ amounting to 2000 C G S., and with usual dimensions, it can easily be deduced from calculations of the energy expended in overcoming the viscous friction, and of the heat capacity of the quantity of oil flowing through the lubricating film, that apart from conduction of heat through the metal, the oil would rise in temperature some 50 C. in passing * See (16, p. 158) Correspondence. VISCOSITY AND LUBRICATION 157 uough the bearing. Conduction will diminish this rise of tempera- ire, but in most cases of heavily loaded bearings it is still sufficient i make the viscosity of the oil in the rear portion of the film much wer than in the leading portion. Thus, other conditions remaining lalteied, the outflow of oil at the rear will take place with a less pid fall of pressure m that direction, and the point of maximum essure will be shifted towards the front of the bearing. In fig. 18 LC dotted curve pi, p. 135, Revue B.B.C. (19, p. 159), is figured i the assumption that the rise of temperature of the oil is such at its viscosity at exit is reduced to one-half of its value at entry, ie conditions being otherwise the same as those for the full-line irve as already explained on p. 135. The lower values of the fluid pressure throughout the film and e shift of the point of maximum pressure towards the leading edge e clearly seen. The point of action of the resultant pressure is so moved forward relatively to its position with constant oil tem- TAture, and it may even happen that the centre of pressure is at, in front of, the middle point of the bearing block If, for example, c direction of motion of a pivoted bearing is reversed, so that the vot is before instead of behind the centre of the bearing, it is 11 possible in many cases for a lubricating film to be formed and essures generated m it in equilibrium with the load Such an "ect is shown in the left-hand half of fig 24*2, which shows the suits of revusing the bearing In such a case the oil film is neces- rily thinner, and the coefficient of friction higher than for the rrect direction of motion, but neveitheless the capability of being versed in this manner, and of even then working with coefficients of ction lowci than those of non-pivoted bearings, is a valuable opcrty of the pivoted type When, however, pivoted bearings are iploycd in this manner, it has always to be remembered that their ccess when running reversed depends upon the lubricant having a nsideiable rate of diminution of viscosity with rising temperature >r example, an experimental thiust bearing which ran very suc- ssfully in both dnections with water and with a mineral oil of low icosity as lubucants, 01 with carbon bisulphide when running in 2 noimal direction, completely failed to run in the reversed direc- >n with the last-named fluid, doubtless on account of the peculiarity its viscosity-temperature relation, which has already been men- ned on p 117 Effects of the same nature, which arise in the use of air as a meant in pivoted thiust bearings, have been pointed out and 158 THE MECHANICAL PROPERTIES OF FLUIDS experimentally investigated by Stone * (24, p. 159). With air, owing to the viscosity of gases increasing with rising temperatures instead of diminishing as in liquids, pivoted bearings tend to be much less stable as to the inclination of the pivoted shoe than with liquid lubricants. On the other hand, as the same author has also remarked, the increase of the viscosity of the air film with temperature tends to increase the thickness of the film when a rise of temperature takes place owing to excessive load or undue resistance. The risk of direct contact of the bearing elements thus tends to become less as the bearing heats up, instead of greater as with liquid lubricants. Calculation and experiment agree in showing that the successful use of air as a lubricant demands the highest refinements of work- manship, with modeiate loads and relatively high speeds. BIBLIOGRAPHY OF ORIGINAL WORKS ON VISCOSITY OF FLUIDS AND VISCOUS THEORY OF LUBRICATION 1 POISEUILLE. " Recheiches expeiimentales sur le mouvement des liquides " Memoir es de VAcademie des Sciences, 9, 1846 2. STOKES, G G "Theories of the Internal Friction of Fluids in Motion, etc " Collected Papers, Vol. I, p. 75 3. HIRN, A " Eludes sur les prmcipaux phenomenes que presentent les frottements medials, etc " Bulletin de la Soadti Industnelle de Mulhouse, 1855 4 BEAUCHAMP TOWER Proc. Inst Mech Eng , 1883 and 1884. 5 OSBORNE REYNOLDS " On the Theory of Lubrication " Phil Trans Roy Soc London, 1886, p. 157, also Collected Papers, Vol II, p 228 6 GOODMAN Manchester Association of Engineers, 1890 7. LASCHE " Die Reibimgsverhaltmsse in Lagern " Zeit thrift deutschet Ingcmeure, 1902 8. SOMMERFELD. " Hydi odynamische Theone der Schmieimittel- reibung." Zeitschrift fur Math u Phys., 1901, 50, p 97. 9 MICIIELL, A. G M. " The Lubncation of Plane Surfaces " Zeit- schnft fwr Math u. Phys , 1905, 52, p 123 10 BRILLOUIN La Viscositd (Gautliiei-Villars, Pans, 1907) 11 HOSKING " Viscosity of Water " Phil Mag , April, 1909, p 502 * These expeurnents were made by means of a thrust beating consisting of quaitz-crystal thtust shoes and a glass thiust collar, the beanng suifaces being worked to true planes by optical methods Monochromatic diffraction bands produced by the closely adjacent pair of beaung surfaces at a slight mutual in- clination gave an immediate and very accurate measure of the thickness of the lubricating film VISCOSITY AND LUBRICATION 159 2 ARCIIBUTT and DEELEY. Lubrication and Lubricants, 2nd ed. (London, 1912) 3. CAROTHERS, S P. " Portland Experiments on the Flow of Oil in Tubes." Proc. Roy Soc , A, 87, No. A. 594, Aug , 1912 4. FAUST, O. " Internal Friction of Liquids under High Pressure." Gottingen Institute of Phys. Chem., 21st June, 1913. (Quoted in Report of British Lubricants and Lubrication Inquiry Committee, 1920) 5 GUMBEL " Das Problem der Lagerreibung " Berliner Bezirks- verem deutscher Ingemeure, 1st April, 1914 6 NEWBIGIN, H T. " The Problem of the Thrust Bearing " Mm. Proc Imt. C E , 1914 7 MARTIN, H M " Theoiy of Lubrication " Engineering, July, 1915, p 101. 8 STONE, W " Viscometer " Engineering, 26th Nov , 1915 9. " De F " " Pahers de butee modernes " Revue BBC, Jan -April, 1917 RAYLEIGH, LORD " On the Theory of Lubrication " Collected Papers , Vol VI, p 523 1 HYDE, J H. " Viscosities of Liquids (Oils) at High Pressuies ' Proc Roy Soc A , 97, No A 684, May, 1920. 2 MARIIN, H M "The Theoiy of the Michell Thrust Bearing" Engineering, 20th Feb , 1920 3 LANCIIESTER, F W " Spin Geai Erosion " Engincenng, 17th June, 1921, p 733 1 Si ON I , W "A Proposed Method foi Solving Problems in Lubn- cdtion " The Commonwealth Engmect (Melbouine), Nov , 1921 3 STONILY, G " Journal Beatings " Engineering, 3id Maich, 1922 3 HERSEY, M. D and SIIORF, II "Viscosity of Lubucants under Pressure " Amer Soc of Meek Engineers, Dec , 1927 7 BOSWALL, R O "The Theoiy of Film Lubrication", pp xi, 280 (Longmans, London & New York, 1928) 3. MICIIELL, A G M " Piogiess of Fluid-film Lubrication " Trans of Amer Soc of Mech Engineers, M S P 51/21, Sep -Dec , 1929 ) KINGSBURY, A " On Pioblems in the Theory of Fluid-film Lubn- cation, with an expenmental method of solution " Amer Soc of Mech Engineers, Dec , 1930 ) KOBAYASIII, TORAO " A Development of Michell's Theoiy of Lubri- cation " Report of the Aeronautical Research Institute, Tokyo University, No 107, June, 1934 A comprehensive bibliography of the whole subject is contained in Notes on the Histoiy of Lubncation ", Parts I and II, by M D Hersey, wrnal Amer. Soc of Naval Engineers, Nov , 1933 and Aug., 1934. CHAPTER IV Stream-line and Turbulent Flow Stream -line Motion The motion of a fluid may be conveniently studied by con- sidering the distiibution and history of the stream lines, i.e the actual paths of the particles. If these paths or stream lines preserve their configuiation unchanged, the motion is called steady or stream-line motion. (See Chapter II, p. 57.) If the stream line be imagined to foim the axis of a tube of finite sectional aiea having imagmaiy V V \y J J I boundaries, and such that its aiea \J\v\ /^s at different points in its length V- >A f ^ ^ i s inversely proportional to the velocities at these points, this is termed a " stream tube ".* Such stieam lines must always have a continuous cuivature, since, to cause a sudden change in direction, an infinite foice acting at light angles to the dnection of flow would be necessary. It follows that in steady motion a fluid will always move in a curve around any sharp coiner, and that the stieam lines will always be tangential to such boundaiies, as indicated in fig. i, which shows the geneial foirn of the stream lines of flow from a sharp-edged orifice. With a very viscous fluid, the effect of cohesion may introduce compara- tively large forces, and the radius or curvature may then become very small. * See an alternative way of putting this idea, Chaptei II, p. 57. Fig i 100 STREAM-LINE AND TURBULENT FLOW 161 Stability of Stream -line Motion Several conditions combine to determine whether, in any parti- alar case of flow, the motion of a fluid shall be stream-line or tur- ulent. Osborne Reynolds, who first investigated the two manners [ motion by the method of colour bands,* came to the conclusion lat the conditions tending to the maintenance of stream-line motion e (1) an increase in the viscosity of the fluid; (2) converging solid boundaries, (3) free (exposed to ail) surfaces; (4) curvature of the path with the greatest velocity at the outside the curve; (5) a reduced density of the fluid. he reverse of these conditions tends to give rise to tuibulence, does a state of affairs in which a stream of fluid is projected into body of fluid at rest. The eflect of solid boundaiies in producing turbulence would >pear to be due rather to their tangential than to then lateral flness One remaikable instance of this effect of a boundary Assessing tangential stiffness is shown by the effect of a film of on the suiface of water exposed to the wind The oil film eits a very small but appicciablc tangential constiamt, with the 5ult that the motion of the water below the film tends to become stable This results m the formation of eddies below the surface, d the energy, which is otherwise imparted by the action of the wind form and maintain stable wave motion, is now absoibed m the stitution of eddy motion, with the well-known eflect as to the lling of the waves. Where two streams of fluid are moving with different velocities 5 common suiface of sepaiation is in a very unstable condition ynolds showed this by allowing the two liquids, carbon bisul- ide and water, to foim a horizontal surface of separation in a ig horizontal tube. The tube was then slightly tilted so as to produce elative axial motion of the fluids, when it was found that the don was unstable for extremely small values of the relative ocity. This also explains why diverging boundaries are such a cause turbulence. Experiment shows that in such a case as shown in *Phil Trans, Roy. Soc., 1883. (D812) 7 i6a THE MECHANICAL PROPERTIES OF FLUIDS fig. 2 the high- velocity fluid leaving the pipe of small section is projected as a core into the surrounding mass of dead water, thereby giving rise to the conditions necessary for eddy formation. More recent experiments* tend to show that the foregoing conclusions as to the effect of the curvature of the paths in affecting the manner of motion, are only true where the outer boundary of Fig z the fluid is formed by a solid surface, and that in some cases as shown at the impact of a steady jet on a plane surface, at the efflux of a jet from a sharp-edged orifice, and in motion in a fiec vortex curved motion, with the velocity gieatest at the inside and not at the outside of the curve, tends to stream-line motion. Generally speaking, wherever the velocity of flow is increasing and the pressuie diminishing, as where lines of flow are converging, there is an ovei- whelmmg tendency to stability of flow In a tube with conveiging boundaries it is this which leads to stability, and it is because this effect is sufficiently great to oveicome the tendency to turbulent motion to which all solid boundaries, of whatever form, give rise, that the motion in such tubes is stable for very high velocities. Hele Shaw's Experiments The fact that stream-line motion is possible at fairly high velo- cities between parallel boundaries if the fluid is viscous, and if the distance between the boundaries is small, has been taken advantage of by Dr. Hele Shaw,f who produced stream-line motion in the flow of glycerine between two parallel glass plates, and showed the form of the stream lines by introducing coloured dye solution at a number of points. By inserting obstacles between the glass plates the form of the stream lines corresponding to flow * Memoirs, Manchester Lit and Phil Soc., 55, 1911, No. 13. f Trans. lust. Naval Architects, 1898, p. 37. STREAM-LINE AND TURBULENT FLOW 163 ough a passage or around a body of any required shape can is be obtained (figs. 3 and 4). Fig 3 The form of the lines to be expected in the case of two-dimensional v of a perfect non-viscous fluid around bodies of simple and Fifi 4 164 THE MECHANICAL PROPERTIES OF FLUIDS symmetrical stream lines are identical spite of the T shape, may be calculated,* and an examination of the obtained in the Hele Shaw apparatus shows that they in form with those thus obtained by calculation, in fact that in one case the forces operating are entirely due to ineitia, and in the othei to viscosity. It has been shown by Sir Geoige Stokesf that this is to be expected, for if PQ and P'Q'(fig. 5) be two boundaries of a stream tube, and if PP' and QQ' be normals to one of the boundaiies, ultimately these will become elements of two con- secutive equipotential lines, and if produced will meet on the centre of ciuvatuie of the tube, so that if v and v -|- Sv be the velocities at P' and P, and it r be the curvatme and t the thickness, PQ _ r + t. P'Q' Sv Again consideiing the equilibrium of the clement P'Q'QP, now imagined as part of a perfect non- viscous fluid, the centrifugal force will be balanced by the difference of noimal piessmcs (Sp) on the inner and outer faces, and by the resolved part of the diflei- ence of pressure due to the difference of level (S^) between the two faces If directions towards the centre of curvature be called positive, on resolving normally, 1) t On substituting for from (i) this becomes r -f- or -j- p * = o constant, * Hydrodynamics, Lamb, p 61, also Trans. Inst N. A., 1898. t British Association Reports, 1898, pp. 143-4. STREAM-LINE ANt> TURBULENT FLOW 165 hich is Bernoulli's equation of energy for a perfect non-viscous nid. It follows that the velocity relationship indicated in (i), p. 164, hich obtains when viscosity is the dominating factor, is also msistent with the stream-line flow of a non-viscous fluid. Critical Velocity The nature of the two modes of fluid motion was first demon- rated by Osborne Reynolds* in a series of experiments on parallel ass tubes of various diameters up to 2 in. These were fitted with ell-mouthed entrances and were immersed horizontally in a tank Fig 6 [ water having glass sides (fig. 6) In these expenments the water i the tank was allowed to stand until motionless. The outlet valve was then opened, allowing water to flow slowly through the tube, little water coloured with aniline dye was introduced at the itrance to the tube through a fine tube supplied from the vessel B At low velocities this fluid is diawn out into a single colour and extending through the length of the tube. This appears to e motionless unless a slight movement of oscillation is given to le water in the supply tank, when the colour band sways fiom side > side, but without losing its definition. As the velocity of flow is tadually increased, by opening the outlet valve, the colour band ecomes more attenuated, still, however, retaining its definition, ntil at a certain velocity eddies begin to be formed, at first inter- dttcntly, near the outlet end of the tube (fig. 7). As the velocity still further increased the point of eddy initiation approaches the *PhiL Trans. Roy, Soc , 1883. i66 THE MECHANICAL PROPERTIES OF FLUIDS mouthpiece, and finally the motion becomes sinuous throughout. The appaient lesser tendency to eddy foimation near the inlet end of the tube is due to the stabilizing influence of the convergent mouthpiece. The velocity at which eddy formation is first noted in a long tube in such experiments is termed the " higher critical velocity ". There is also a *' lower critical velocity ", at which the eddies in originally turbulent flow die out, and this is, strictly speaking, the true critical velocity. It has a much more definite value than the higher critical velocity, which is extremely sensitive to any distui- bance, either of the fluid before entering the tube, or at the entrance. Over the range of velocities between the two ciitical values, the HEAD VELOCITY Fis.8 fluid, if moving with stream-line flow, is in an essentially unstable state, and the slightest disturbance may cause it to break down into turbulent motion. The determination of the lower critical velocity is not possible by the colour-band method, and Reynolds took advantage of the STREAM-LINE AND TURBULENT FLOW 167 act that the law of resistance changes at the critical velocity, to letermine the values by measuring the loss of head accompanying lifferent velocities of flow in pipes of different diameters. On plot- ing a curve showing velocities and losses of head (fig. 8) it is found hat up to a certain velocity, A, for any pipe, the points lie on a .traight line passing through the origin of co-ordinates. From A o B there is a range of velocities over which the plotted points are /ery irregular, indicating general instability, while for greater velo- nties the points lie on a smooth curve, indicating that the loss of lead is possibly proportional to v n . To test this, and if so to determine the value of , the logarithms LOG H )f the loss of head h and of the velocity were plotted (fig. 9) Then if , 7 h = k v ", log h = log k -{- n log v, he equation to a straight line inclined at an angle tan" 1 n to the ixis of log v, and cutting off an intercept log k on the axis of log h. On doing this it is found that if the velocity is initially turbulent the plotted points lie on a straight line up to a certain point A, the value of n for this portion of the range being unity. A.t A, which marks the lower critical velocity, the law suddenly changes and h increases rapidly. There is, however, no definite relationship between h and v until the point B is reached. Above i68 THE MECHANICAL PROPERTIES OF FLUIDS this point the relationship again becomes definite, and within th( limits of experimental eiroi, over a moderate range of velocities the plotted points he on a straight line whose inclination varies witt the roughness of the pipe walls. The values of n determined in this way by Reynolds are* Mateual of Pipe. n Lead . . . i 79 Varnished . . . . i 82 Glass ... . i 79 New Cast lion . . i 88 Old Cast lion . . 2-0 those for cast iron being deduced from experiments by Darcy When tested over a wide range of velocities, it is found that the value of n in the case of a smooth-walled pipe is not constant but increases somewhat as the velocity is increased Between A and B the value of n is greater than between B and C, and the inci eased resistance accompanying a given change in velocity is gieatei even than when the motion is entnely tuibulcnt This is due to the fact that within this range of velocities eddies aic being initiated in the tube, and the loss of head is due not only to the maintenance of a moie or less uniform eddy regime, but also to the initiation of eddy motion. Messis Barnes and Coker* have determined the cutical velocity in pipe flow by allowing water to flow through the given pipe which was jacketed with water at a higher temperatuie The tcmpeiatuie of the watei dischaiging from the pipe was measured by a delicate thermometer. So long as the motion is non-sinuous, tiansmission of heat through the water is entiiely due to conduction and is extremely slow, so that the theimorneter gives a steady reacting sensibly the same as that in the supply tank. Immediately the critical velocity is attained, the rate of heat tiansmission is increased due to convection, and the change from stream-line to tuibulent motion is marked by a sudden increase in the temperature of the discharge, * Proc. Roy, Soc. A, 74, STREAM-LINE AND TURBULENT FLOW 169 The law governing the relationship between the critical velocity I the factors involved was deduced by Reynolds from a considera- i of the equations of motion: for if the state of motion be supposed depend on the mean velocity in the tube and on the diameter, acceleration may be expressed as the difference of two terms, : of which is of the nature pvfd, and the other of the nature pv z . was then inferred that since the relative value of these terms bably determines the critical velocity, the lattei will depend on le paiticular value of the ratio p/pvd To test the accuracy of s conclusion experiments were made on pipes of diffeient dia- ters, and with different values of //- obtained by varying the iperature of the water between 5 C. and 22 C. The lesults of the expeiiments fully justified the foregoing elusions, and showed that the cutical velocity in a straight allel pipe is given by the formula P iie b is a numerical constant, and where P oc /j,fp. If the unit of *th is the foot, b equals 25-8 for the lower critical velocity, and ) for the higher velocity, while if t tempeiaturc in degrees itigrade, T P = I -f- O 033681? + O OOO22lt Z Moie iccent expeiiments by Coker, Clement, and Barnes* and *rs earned out by Ekmanf on the ongmal apparatus of Reynolds, iv that by taking the greatest care to eliminate all disturbance ntiy to the tube, values of the higher cutical velocity considerably itei than (up to 3 66 times as great as) those given by the above nula may be obtained. The probability is, in fact, that there is definite higher ciitical velocity, but that this always inci eases i decreasing disturbances. A general expression for the lower critical velocity in a parallel ;, applicable to any fluid and any system of units, is * dp _ 2300^ _. * Trans Roy Soc , 1903, Proc. Roy. Soc A, 74. f "Arluv for Matematic " Ast. Och, Fys , 1910, 6, No 13. (D312) i 7 o THE MECHANICAL PROPERTIES OF FLUIDS Thus foi water at o C., p,/p v~i92Xio~ 5 m foot-pou second units, so that >v k = -_ zL ft.-sec , wheie d is in feet. While for air at o C., v = 14-15 X io~ 5 ; /. v k 3 ft.-sec. where < is in feet. In this connection fig. 10* is of interest, as showing the icsul of experiments on a number of pipes of diffeient diameteis, wi air and watei flow, in which values of Rfpv z aie plotted as oidmat against the corresponding values of vd/v or of log(vd/v) Ilcie R the suiface friction per unit area of the pipe wall. The tuive consis of two parts connected by a narrow veitical band correspondii to a value of vd/v of approximately 2300, over which the points ft the various pipes are somewhat irregulaily disposed This ban indicates the range of instability between stream line and tiue tin bulent flow. The left-hand curve, coi responding to speeds belo the critical value, is calculated from the foimula , d theoietically conesponding to stieam-hnc flow f It will be seen tin the points for both air and water flow he closely on this cuive, an that the break-down of the stream-line motion takes place in a cases at appioximately the same value of vd/v As may be shown by an application of the pimciplc of dynamic* siimlaiity,J formula (3) is a paiticular case of the geneial foimul kv 71, _ * I ' which is applicable to all cases of fluid motion. Heie / is the lengtl of some one definite dimension of the body The value of th constant k now depends only on the ioim of the surfaces over whicl flow is taking place. Thus in flow past similar plates immeised n water and in air, Eden has shown by visual observation that th< * Stanton and Pannell, Phil Trans Rov Soc A, 21*4- j Chap V, p 200 1 Chap V, p 193 Advisory Committee for Aeronautics, T.R , 1910-11, p 48. AM M-LIf VD V 4-6 STREAM-LINE AND TURBULENT FLOW 171 >e of flow, especially in the rear of the plate, is identical for identical ues of vl/v, where / is the length of any particular side of the ite. Critical Velocity in Converging Tubes In a converging tube the angle of convergence of the sides has arge effect on the critical velocity. At all ordinary velocities the tion in tubes or nozzles having moie than a few degrees of con- gence may be consideied as non-sinuous. Experiments on the v of water through circular pipes having sides converging uni- mly at an angle 6 gave the following approximate values for the r er critical velocity, at 14 C.* 5 Deg 7 5 Deg 10 Deg. 15 Deg 'At large section (3 in 1 diametei) .. ..} '* r 94 * 44 3*5 \ ltlc f l At throat (i| in dia-^ f f locit ^ meter) . 6o 77* 977 12 9 -sec ' ^dSSSET 1 ' 21 ".} 27 34S +34 " 3 ; lower critical velocity in a i-^-in. parallel pipe at this temperature 20 ft. per second. Should the ratio of higher to lower critical cities have the same value in a conical pipe as in a paiallel pipe, would mean that in the case oi a i|-m jet discharging from a merging nozzle with steady flow m the supply pipe, the critical city would have the following values 5 Degrees. 7 5 Degrees. 10 Degrees 15 Degrees tical velocity, ft -sec 39 50 63 84 n flow through a pipe bend, the velocity at which the resistance es to obey the laws of laminar flow is less than in a straight pipe, re is now a considerable range of velocity over which the resis- e is proportional to a power of the velocity higher than unity, n which turbulence is not developed. This is due to the develop- * Gibson, Proc, Roy. Soc. A, 83, 1910, p. 376. 172 THE MECHANICAL PROPERTIES OF FLUIDS ment of a cross circulatory current superposed on the laminar flow. The critical velocity is not well defined but experiments * indicate that full turbulence is developed at a somewhat higher velocity than in straight pipes. The Measurement of the Velocity of Flow in Fluids Several methods are available for measming the flow of fluids in pipes. Of these, the use of the Ventun meter or of the Pitot tube are the most common Recent investigations into the possi- bilities of the hot-wire anemometer have shown that this is capable of giving excellent results, and that it is likely to be especially valuable for the measurement of pulsating flow. The Venturi Meter The Ventun meter, invented by Clemens Heischel in iSSi , affords perhaps the simplest means of measuring the flow of a liquid When fitted to a pipe line of diameter greater than about 2 in its indications are, under normal conditions, thoroughly reliable so long as the - 2% d i\- Fit,', ii Venturi Meter velocity in the pipe line exceeds i ft. per second, and the discharge may then be predicted, even without calibration, to within i or 2 per cent. The meter is usually constructed of approximately the propor- tions shown in fig. u, and consists essentially of an upstream cone usually having an angle of convergence of about 20, connected to * C. M. White, Proc, Roy. Soc. A, 123, 1929, p. 645. STREAM-LIKE ANt> TURBULENT PLOW downstream cone whose angle of divergence is about 5 30', by asy curves. One annular chamber surrounds the entrance to the tieter, and a second surrounds the throat, the mean pressures at hese points being transmitted to these chambers through a seiies >f small holes in the wall of the pipe. The two chambers are con- tected to the two limbs of a differential pressure 'gauge which records heir difference ot pressure h in feet of water. For this purpose a J-tube containing mercury may ie used as in fig. n. In this ase if the connecting pipes are all of water it may readily be hown that the difference of iressure in feet of water is equal 3 12-59 times the difference of ;vel of the tops of the mercury counter olumns By using an inverted J-tube with compressed air applied to the highest poition t the tube, the difference of icssure may be directly re- orded in feet of water When n automatic recoid is desned, ie type of mechanism shown in g 12 may be used. If P, A, V and p, a, v re- resent the pressures in pounds er square feet, the areas in q[uaie feet, and the mean velo- Flg tties in feet per second respec- vely at the entrance and throat of a meter whose axis is hori- ontal, neglecting any loss of energy between entrance and throat, ieinoulli's equation of energy becomes Driving^ Mechanism u I 4o C UJ -Recording Mechanism for Venlun Metei X 2 W W or = .4. ( W Y!/(_ 2g\.\a A\ ft.-sec. (4) 174 THE MECHANICAL PROPERTIES OF FLUIDS Actually owing mainly to fiictional losses the velocity is slightly less than is indicated by formula (4), and is given by / 2gh v==c \// A \ 2 ; ft -- sec -> (s) \~a) wheie C varies from about 0-96 to 0-995,* usually increasing slightly with the size of meter. When used to measure pulsating flow, the value of C is reduced. The effect is, however, small for any such percentage fluctuations of velocity as are usual in practice, even with the discharge from a icciprocating pump. For accurate results the meter should be installed m a stiaight length of pipe lemoved from the i fluence of bends. Such bends set up whirling flow in the pipe, and this tends to increase the effective value of C. The Venturi meter may also be used to measuie the flow of gases.f In this case, for air, the discharge is given by Q = CAj3P,W lt . . ..(6) where P x is the pressure at entrance in pounds pel squat e foot, Wj is the weight per cubic foot at P and temperature T; and where, if p 1 is the pressure at entrance in pounds per square inch, j> 2 is the pressme at thioat in pounds per squaie inch, m is the ratio of areas at entiance and thioat, n is the index of expansion (1-408 foi dry air expanding adiabatically), If TJ be the absolute tempeiature at entrance on the Fahrenheit scale, on wiiting W x = ' ^ 1 (the value for dry air) (6) i educes to T i Q = i -ioC%/M~L Ib. per second, (7) VT X wheie #! is the aiea at entrance in square inches. *For a discussion of the vanability of the coefficient C, see "Abnormal Co- efficients of the Ventuu Meter ", Proc Inst C E,, 199, 1914-5, Pait I \ "Measurement of Aii Flow by Venturi Meter", Proc. Inst. Mech E., 1919, P 593, " Commeicial Metering of Air, Gas, and Steam ", Proc. Inst. (7. E., 1916-7, Part II, 204, p. 108. STREAM-LINE AND TURBULENT FLOW 175 Experiments * indicate that the value of the coefficient C is not nstant, but that it diminishes as the ratio p 2 /Pi is increased approxi- ately as indicated in the following table 06 098 07 0-97 0-8 0-96 0-9 0-94 I'O 0-91 Measurement of Flow by Diaphragm in Pipe Line The coefficients of discharge of standard sharp-edged orifices scharging freely are known with a fairly high degree of accuracy, d where such an onfice can be used for measuring the steady flow her of a liquid or of air, the results may be relied upon as being Fig 13 Curate within i or 2 per cent, if suitable precautions are taken- ymg to the convenience of the method and the simplicity of the paratus, much attention has recently been paid to the use of fices through diaphragms in a pipe line for measuring the flow. If D be the diameter of the pipe and d that of the orifice (fig. 13), idgsonf states that the coefficient of discharge C for sharp-edged *Proc. /Twit Mech E , Oct 1919, p. 593. tProe. IKS/. C.E., 1916-17, Part II, p. 108. 176 THE MECHANICAL PROPERTIES OF FLUIDS orifice in a plate of thickness o-ozd, and for ratios of d/D less than o 7 is 0-608 for water or air when p z /pi is greater than o 985 and is equal to 0-914 0-306 p z /Pi for air, steam, or gas when j& 2 /Pi is less than o 98, pressures being measured at the wall of the pipe immedi- ately on each side of the diaphragm. A rounded nozzle, if well designed, has a coefficient which vaiies fiom about 0-94 in small nozzles to 0-99 for large nozzles, either for water or air, if p^/Pi is greater than o 6. Fig. 14* shows a form of nozzle in which the coefficient lies between o 99 and o 997. The Pitot Tube For measurements of the flow in pipes or in unconfined streams where the velocity is fairly high, the Pitot tube is capable of giving excellent results This usually consists of a bent tube teimmating in a small orifice pointing upstream, which is surrounded by a second tube whose direction is paiallel to that of flow. A series of small holes in the wall of the outer tube admit water, at the mean pressuie Tubes 03 Jftnn Won Two holes, one on each side thick r80D ^ OD through outer wall diam 02" Fig is in their vicinity, to its mtcnor, which is connected to one leg of a manometer. The other leg is connected to the cential tube carrying the impact onfice. If v is the velocity of flow immediately upstieam from this onfice, the pressuie inside the onfice, wheie the velocity is zero, is equal to the sum of the statical picssuie at the point, plus kv z /2g ft of water, where k is a constant whose value approxi- mates closely to unity in a well-designed tube. It follows that the difference of level of the two legs of the manometer equals ktP/2g, Figs. 15, 16, and 17 show modern types of this instiument. Fig. 15 shows the type used for measuring the air speed of aeroplanes * Engineering, ist Dec , 1933, p. 690, STREAM-LINE AND TURBULENT FLOW 177 and for wind tunnel investigations. A tube of this type having the dimensions shown gave K = i-oo within i per cent.* The tube illustrated in fig. 16 gave a value of C = 0-926 when cali- brated by towing through still water, and 0-895 when calibrated in a 2-in. pipe. The low value of C in still water is probably due to the fact that the pressure orifices are too near the shoulder of the pressure pipe. If this - . weie lengthened, with the orifices faither back, the coefficient would probably be higher. The difference between the calibration in still water md in the small pipe is o be expected, since he velocity at the sec- ion of the pipe contam- ng the pressure orifices s of necessity increased >y the presence of the ube, and the pi assure s lecorded by the static >ressure column will onsequently be less than ti the plane of the im- >act orifice. This effect /ill increase with the atio of the diameter of ube to that of the pipe, nd unless this latio is mall, the tube should be calibrated in a pipe of approximately the ime dimensions as that in which it is to be used. It is prefer- ble, foi pipe woik, to use a simple Pitot tube having only an npact onfice, and to obtain the static pressure from an orifice in ic pipe walls in the plane of the impact orifice. Using the tube i this way, the coefficient C may usually be taken as o 99, within per cent. In the type of Pitot tube shown in fig 17, and known as the Pitometer ", the pressure at the downstream orifice is less than the # " The Theory and Development of the Pitot Tube ", Gibson, The Engineer, 7th July, 1914, p. 59- v///// JOIA Fig 1 6 Details of Pitot Tube 178 THE MECHANICAL PROPERTIES OF FLUIDS statical pressure in the pipe and the coefficient is less than in the normal type. For the tube shown, calibrations in flowing water in pipes give a mean value of C 0-916. Owing to eddy formation at the downstieam orifice the coefficient of such a tube fluctuates within fairly wide limits. The Pitot tube may be calibrated either in still water or in a current. In the lattei case the mean velocity is computed from readings taken at a large number of points in a cross section, and TO DIFFERENTIAL PRESSURE GAUGE the coefficient of the mstiument is adjusted so as to make this mean velocity agree with that obtained fiom weir flow, current meter, -or gravimeter measurements on the same stieam. Without exception, obseivers have found that a still- water rating gives a somewhat higher value of C. The explanation would appear to be twofold. In the first place, the velocity of flow in a moving current is never quite steady, but suffers a series of periodic fluc- tuations, and since the Pitot tube is an instrument which essentially measures the mean momentum, or the mean (z> 2 ) of the flow and not its mean velocity, any such fluctuation superposed on a given mean velocity will give an increased head reading. In the second place, when metering a flowing current the average tube cannot be used at STREAM-LINE AND TURBULENT FLOW 179 Doints very near the boundary where the velocity is least, and for this eason also the mean recorded velocity tends to be too high. It follows that although a still-water or still-air rating represents he true coefficient of the instrument, this requires to be reduced omewhat for use in a current, the effect increasing with the un- teadiness of the current. Where a high degree of accuracy is equired, the rating should be carried out under conditions as nearly s possible resembling those under which measurements have after- wards to be made. For measurements of the flow in pipes, the instrument should e used if possible at a section remote from any bend or source f disturbance. For approximate work the velocity of the central lament may be measured. This when multiplied by a coefficient fcich varies from o 79 in small pipes to 0-86 in large pipes gives ic mean velocity Alternatively the velocity may be measured at le radius of mean velocity, which vanes from o ja in small pipes > 075^ in large pipes, where a is the radius of the pipe. These dues, however, only apply to a straight stretch of the pipe, and it is neccssaiy to make measuiements near a bend, and in any case >r accurate results, the pipe should be traversed along two dia- eters at right angles, and the velocities measured at a series of dii If <$r is the width of an elemental y annulus containing one ncs of such measuiements whose mean value is v, the discharge then given by ( a Q = I 2-nrvdr Jo Scvcuil methods are available for determining the mean velocity flow of a liquid in an open channel. This may be obtained: (a) by the use of current meteis giving the velocity at a seiies points over a cross section of the channel; (b) by the use of a standard weir; (c) by the use of floats; (d) by chemical methods. This method is best adapted to rapid d megular sti earns, although it may be applied to the measure- atit of pipe flow. It consists in adding a solution of known strength some chemical for which sensitive reagents are available, at a iform and measured rate into a stieam,* and by collecting and * For a description of the method of introducing the solution uniformly, refer - <e may be made to Mechanical Engineering, 44, April, 1922, p. 253, or to dro-electric Engineering > Gibson, Vol. I, p. 29. i8o THE MECHANICAL PROPERTIES OF FLUIDS analysing a sample taken fiom the stream at some lower point wheic admixture is complete. The solution should be added and the sample taken at a number of points distiibuted over the cross section. Various chemicals may be used Unless the water is distinctly biackish, common salt is suitable. If blackish, sulphuric acid or caustic soda may be used. With a solution consisting of 16 Ib. of salt per cubic foot of water, a dilution of i in 700,000 will give, on titration with silver nitrate, a precipitate weighing i mgm. pei litre of the sample, and the gravimetiic analysis of such a sample will enable an accuracy of i per cent to be attained. If Q = discharge of stieam to be measuied in cubic feet per second, q = quantity of solution intioduced in cubic feet per second, Cj = concentration of salt in the natural stream watei in pounds pei cubic foot, C 2 = concentration of salt in the sample taken downstieam in pounds per cubic foot, C = concentration of salt in the dosing solution in pounds per cubic foot, then Q = " and if V it V 2 , and V are the volumes oi silver nitiate solution lespec- tively necessary to titrate unit volume of noimal stream water, of the downstieam sample, and of the dosing solution, V-V 2 \ - v ) q ' Vj/ (e] By injecting colour and by noting the time icquiied for this to cover a measured distance. A close approximation to the tiue discharge may be obtained either by the use of a weir, of current meters, or by chemical methods if suitable precautions are taken.* The colour method would not, m general, appear to be so reliable, and float measurements cannot be relied upon for any close degree of accuracy. The Effect of Fluid Motion on Heat Transmission Apart from the effect of radiation, the heat transmission between a solid surface and a fluid in motion over it will, for a given differ- ence in temperature, be proportional to the rate at which the fluid *See Hydraulics, Gibson, 1912 (Constable & Co ),p 346. STREAM-LINE AND TURBULENT FLOW 181 articles are carried to and from the surface, and therefore to the iffusion of the fluid m the vicinity of the surface. Such diffusion epends on the natural internal diffusion of the fluid at rest, and n the eddies produced in turbulent motion which continually bring -esh particles of fluid up to the surface In stream-line flow the scond source of diffusion is absent; the heat transmission can only ike place in virtue of the thermal conductivity of the fluid; and le rate of heat transmission is very small. Assuming that H, the eat transmitted per unit time per unit of surface, is proportional ) 6, the difference of temperature between surface and fluid, the Dmbmed effect of the two causes may be written H = A0 + BM ........... (8) here p is the density of the fluid, A and B are constants depending a its nature, and v is its mean velocity As pointed out by Reynolds,* the resistance to the flow of a uid through a tube may be expressed as R = Afv + B'/*; 2 , ......... (9) id vanous consideiations lead to the supposition that A and B i (8) are proportional to A' and B' in (9) For assuming, as is now 2neially accepted, that even in turbulent flow theie is a thin layer f fluid at the suifacc which is in stream-line motion, the heat trans- ussion thiough this layer will be by conduction, and fiom the oundaiy of this layer to the mam body of fluid by eddy convec- on In stieam-lme flow the tiansfei of momentum which gives se to the phenomena of viscosity is due to internal diffusion, while i turbulent motion the tiansference of momentum is due to eddy Dnvection, so that it would appear that the mechanism giving rise ) icsistance to flow is essentially the same as that giving rise to eat transmission, both m stieam-lme and turbulent motion The following geneial explanation of the Reynolds law of heat ansmission is due to Stanton *]* Neglecting the effect of conduc- vity compaied with that of viscosity, the ratio ot the momentum >st by skm fiiction between two sections Bx in apart, to the total lomentum of the fluid, will be the same as the ratio of the heat :tually supplied by the surface, to that which would have been ipplied if the whole of the fluid had been carried up to the surface. * Reynolds, Manchester Lit and Phil Soc., 1834 f " Note on the Relation between Skin Friction and Suiface Cooling ", Tech, eport, Advisory Committee for Aeronautics, 1913-3. 182 THE MECHANICAL PROPERTIES OF FLUIDS Thus in pipe flow: if 8p is the difference in pressure at the two sections; ST is the rise in temperature between the two sections; W is the weight of fluid passing per second; v m is the mean velocity of flow; T m is the mean temperature of the fluid; T s is the temperature of the surface; a is the radius of the pipe; the above relationship becomes QJ./ 9.\ T7frC' r n op(Tra*) Wol , v W ~ W(T,-TJ (I > T'" The heat gained per unit area of the pipe per second is wheie a is the specific heat. If R is the lesistance per unit area, T> __ 7m2 dp so thai if H is the heat transmitted per unit area per second, H = Ro < T .- T J (II) *'j Since, as pointed out by Reynolds, the heat ultimately passes horn the walls of the pipe to the fluid by conductivity, a correct expression for heat transmission should involve some function of the conduc- tivity, and for this reason expression (n) can only be expected to give approximate results. In spite of this it enables some lesults of extieme practical value to be deduced. Thus if R = kv n , and if a be assumed constant, Hoc ^-'(T.-TJ, (12) so that if the lesistance be pioportional to a 2 , and if T s T m be maintained constant, the heat transmitted per unit area will be proportional to v, and since the mass flow is also proportional to v , the change in temperature of the fluid during its passage through STREAM-LINE AND TURBULENT FLOW 183 the pipe will be independent of the velocity. Otherwise, the heat transmitted will be directly proportional to the mass flow. The general truth of this was demonstrated experimentally by Reynolds,* who showed that when air was forced through a hot tube, the temperature of the issuing air was sensibly independent of the speed of flow. In the case of the flow of hot gases through the tubes of a boiler, or of the water through the tubes of a condenser, n is usually less than 2 and has a value of about 1-85. Moreover, in the former case any increase in the velocity of flow will be accompanied by an increase in the temperature of the metal surface, so that for both reasons the heat transmitted is not quite proportional to the mass flow, and the issuing gases are slightly hotter with a high velocity than with a low velocity of flow. The difference is, however, not great, and it appears that by increasing the velocity of flow of the fluid, the output of a steam boiler, or of a surface condenser, may be considerably increased without seriously affecting the efficiency. This is in general accordance with Nicholson's f investigations on boilers working under forced draught. These showed that by increas- ing the mass flow, the heat transmitted to the water was increased in almost the same proportion, while the tempeiature of the flue gases was only slightly increased. Numerous other observers J have verified the general truth of the relationship expiessed in equation (12), p 182, for the flow of liquids and gases through pipes. Its more general application to other cases of heat dissipation in a current still awaits experimental proof. Experiments on the heat dissipation from hot wires of small diameter in an air current, show that this is proportional to v s , which, if this relationship is correct, would indicate that the resistance should be proportional to v l 3 . This is not in agreement with the generally accepted result that the resistance is proportional to v z . An examination of the experimental data shows, however, that the product of vd in the wires on which the heat measurements were made, was small, and an examination of the curve showing R/pv z d 2 plotted against vd/v (fig. 4, Chap. V), shows that in this part of the * Memoirs, Manchester Lit and Phil. Society, 1872 f " Boiler Economics and the Use of High Gas Speeds ", Trans. Inst of Engi- neers and Shipbuilders in Scotland, 54; " The Laws of Heat Transmission in Steam Boilers ", J T Nicholson, D.Sc , Junior Institute of Engineers, 1909. JStanton, Phil. Trans, Roy Soc, A, 190, 1897, Jordan, Proc. Inst. Mech. Engmeeis, 1909, p 1317, Nusselt, Zeitschnft des Vereines deutscher Ingemeure, Z3rd and soth Oct , 1909. 184 THE MECHANICAL PROPERTIES OF FLUIDS range the curve is very steep, indicating that the icsislance is pro- portional to a value of n much less than 2. Although the data are insufficient to indicate the exact value of n they do not, at all events, disprove the foregoing hypothesis The difficulty in foiming any definite decision as to the general validity of the hypothesis arises fiom the fact that in most icsistance experiments on cooling systems, it has been tacitly assumed that the resistance is propoitional to v 2 , and the published data usually give the average value of the coefficient of resistance based on this assumption. Thus the resistance of honeycomb radiatois is known to be neaily proportional to a 2 , while the heat tiansmission pci dogiee difference of temperature is approximately proportional to v &s . Experiments by Stanton and by Pannell* show that while equation (u), p. 182, gives modeiate icsults for air flow thiotigh pipes, the calculated results obtained with water as the fluid aie very different from those deduced expenmentally, as appeal s itom Table I. TABLE I Pipe Dia, Cm 0736 0736 139 488 488 488 488 Mean Tern- Flictlon HeatTi. ins- Mean peiatuie AOCLIUU, mittc( j Vel.Cm D ^ es Calones Surface, Fluid, ^rL,^ pei Sq Cm pci Sec 443 5-08 3-28 o 0162 o 0205 o 0300 o 0369 per Sec Surface, Deg C Fluid, Deg C per a 296 282 1593 298 296 5*65 3965 26 o 123 2 47-2 20 96 50 6 69 o 473 21 21 171 940 362 227 3 *' 1180 374 225 51. 1480 435 235 8 i. 2188 43 26-2 149 RffCn-T,,,) Fluid V,W 1235 10 5 10-8 6-5 o 0109 00155 0-0266 0-0267 Water Air. It will be seen that in the case of air flow the heat transmission calculated from equation (n) is about 76 per cent of that observed, while for water the calculated value is twice as great as that observed. *Phil Trans. Roy Soc A, 190, Tech Report, Advisory Committee for Aero- nautics, 1912-3 STREAM-LINE AND TURBULENT FLOW 185 Mr. G. I. Taylor* suggests that equation (u), p. 183, may be modified to take into account the effect of conductivity by assuming that there is a surface layer of thickness t, having laminar motion, through which heat is conveyed by conduction; that the velocity it the inner boundary of this layer is U, and the temperature Tjj md that between the centre and this layer heat transmission is due L o eddy convection. The temperature drop in the laminar layer, of conductivity k, s given by rri r " f R is the resistance per unit area at the surface, this will be equal o the resistance to shear of the lamina. , R - so that T,-T, = (13) ly analogy with (u) the rate at which heat is transmitted from the lyer by eddies is H = R ^~P. .. ..(14) (v m - u ) ubstituting for in (14) from (13) gives Tf U If = r, *> .-U i-r rp rp * 1 * m T.-T, U U = i A OjU T x T * *i H = 7 r\k (is) his equation is identical with (n) if the quantity in brackets is * Advisory Committee for Aeronautics, Reports and Memoranda, No. 272, 1916. i86 THE MECHANICAL PROPERTIES OF FLUIDS unity, i.e if fco- = k. For air this is very nearly the case, since k i-6ju,C ft where C v is the specific heat at constant volume, and C f . = , so that ~ = 0-88. In the case of water, however, at i '4 k 20 C., ft. = o-oi, k = 0-0014, cr i-o, and ^r = 7*i. K Stanton * has shown that the value of r necessary to bdng the results as found from equation (15), p. 185, into line with the expen- mental results for water quoted in Table I, p. 184, is 0-29, and that similar experiments by Soennekei f require a mean value of 0-34. Taylor, from an examination of data by Lorentz,J concludes that the ratio is approximately 0-38. Some idea of its value in the case of air may be deduced from direct measurements by Stanton, Marshall, and Bryant, of the velocities in the immediate vicinity of the pipe wall. These measurements would appear to indicate that true laminar flow is instituted at a point where is appioxi- v m mately 0-14. They show, however, that it is erroneous to assume that at this point the change to true turbulent flow is abrupt, but that the change is gradual over an appreciable ladial depth of the fluid. It follows that equation (15) has not a strictly rational basis, but that by assuming r = 0-30 it gives results which are, for piactical pur- poses, not senously in error. The ratio drop in temp in surface layer __ T s T x __ / r N/xa drop in temp, in rest of tube Tj T m \i r' k' Thus, if the effective value of r = 0-30, the ratio is 3-0 for watei at 20 C., and 0-38 for air. Reynolds j| has shown by an application of the piinciple of dynamical similarity that in the case of pipe flow dp _ v z ~ n n B 71 8^ ~ dF* Vm A ' * Dictiona.} y of Applied Physics, Vol I, p 401. j- Komg Tech. Hochschule, Munich, 1910 I Abhandlung ilber theoretische Physic, Band, I, p. 343. In air flow R = o 002/3 Vm 2 approximately, and Stanton 's experiments (Proc. Roy Soc. A, 1920) indicate that t is approximately o 005 cm. when /" = 0-00018 and V m 1850 cm. per second This makes U O'l^Vm- |J Chapter V. STREAM-LINE AND TURBULENT FLOW 187 nd if this value of be used in (10), p. 182, on writing vX V = 7rr 2 pv, the expression becomes dl _ B" g V*-* 1 n- Z( _ -** (T *~" T)> ......... (I6) /7T* /here now T is the temperature of the fluid, and - is the tem- erature gradient along the pipe. ^ x If T 4 is sensibly constant along the pipe, integration of (16) ives , T, T! __ B" g v 2 ~ n n- 2 , g ~-T ~ A #="" Vm ' here / is the length of the pipe, and T x and T 2 are the temperatures ' the fluid at inlet and outlet. Stanton,* from experiments on heat transmission from water > a cold tube and vice versa, deduced the expression, for small dues of T! T 2 , log *ri-Tl = *&=- ^" a/ {(I + aTs)(l + ^ TJ} ' ' (l8) here, in C G S units, a = o 004 and ft = o-oi It will be noted at this expression is identical in form with (17), except for the st two factors, which were intioduced to take into account the Feet of the variation m conductivity, with temperatuie, of the suiface rn of water. In those experiments in which the heat flow was from etal to water, k had a mean value of 0-0104 With flow in the other tection k was, however, distinctly less, having a mean value of proximately 0-0075. 3plication of the Principle of Dimensional Homogeneity to Problems involving Heat Transmission The principle of dimensional homogeneity, Chap. V,f mayieadily extended to problems involving heat transmission. In this case, addition to the three fundamental mechanical units, a thermal it is needed to define all the quantities involved. Taking tempeia- e T as this unit, the new quantities, heat flow H, conductivity k, * Trans. Roy Soc. A, 1897 f See also a note by Lord Rayleigh, Nature, 95, 1915, p, 66, i88 THE MECHANICAL PROPERTIES OF FLUIDS and specific heat o- which are now involved, may be expressed dimensionally as _ JTl - /heat flow perl _ /energy pei) _ MT 2 3 \ , , / - \ I - 1VJLJU( 6 . \ unit time j ( unit time j k I ^ x k^* 1 \ _ H 1 sectional area X Tj LT I heat per unit mass } Jrlt r n^nm i O 1 = { / == == Ju t JL . Inse in temperature] MT If attention be confined to the large class of problems of practical importance, involving the transmission of heat between a fluid and a surface moving with relative velocity, where temperature differ- ences are so small not exceeding a few hundred degrees that radiation is only of secondary importance, the only quantities in- volved are H T & n- w 7 n // 11, J. , K, a, v, i, p, fj, We select /, v, p and a as the four independent quantities, and combine them with the other four H, T, k and p, in turn, so as to obtain 8 4 (=4) dimensionless quantities K, as explained at p 198; i e. we write PI = l x v v p z a n , and similarly with T, k and /x We thus find Kj = ; K 2 = ; K 3 = ' ~ r~~ --4 j > v* ivpa Ivp i ,/ H O-T k LC \ whence M -^-5-, -, - , ) = o ; \l 2 v 3 p v z Jvpa Ivp' * or H = 1V , Ivpa Ivp/ which, by combining the last two terms, becomes H = P /V^I, A, JL) \v z aii Ivp/ At such speeds as are usual in the case of air flow over air-cooled engine cylinders, of the flow of gases through boiler flues, or of heating or cooling liquids through pipes, experiment shows that the heat flow is sensibly proportional to v n , where n is between 0-5 and i-o, its value depending on the type of flow and the foirn of surface. Expeiiment, moreover, indicates that if radiation be neglected the heat flow is directly proportional to the difference of temperature STREAM-LINE AND TURBULENT FLOW 189 etween the fluid and the surface, in which case the function m (19) lust be of the form F( , -), and(io) becomes a 2 \fft Iv/' v w H _ opvaTp _?_ f* } ( 20 ) J. J. fJl' (s(J JL Jl I 5 ~ I * t** \^jZf\JJ \ (T/i /W/) / (, \ 1 W / k \ J } f( }, 10; Decomes ^ ^ - H = p n l l + n v n (j?- n aTf(~\ (21) \ CTL6/ / k \ ' the fluid to which k, a and p belong is a gas, /( J is a constant, > cr/Lt/ T / K \ f the kinetic theory of gases. In this case, we may take for /( ) 7 7 1 \OXl' /v /A \ ther A or B( ) , these being constants; thus obtaining the cr/x \ a/x / ternative forms for H, or Hoc / 1|n (yp(r) w A 1 - n T ............. (22) F is the total resistance to the steady motion of the fluid, then ice the heat loss per degree difference of temperature, per unit >ecific heat, is of dimensions _ o-T Vt-tT- 1 T bile F is of dimensions ML/- 2 , the ratio F/- is of dimensions ' crT It = v, so that the index n in H oc v n should be less by unity than e corresponding index m F oc v n ' From this it appears that with >w so turbulent as to give the n 2 law of resistance, n in equation 2) becomes i, and H oc PvpaT, ............... (23) ule with stream-line flow (n' = i and n = o), Hoc&T .................. (24) >r example, in the case of flow through similar pipes, where the erm may be taken to represent the diameter, equation (23) indi- tes that in such circumstances H is independent of the conduc- >ity of the fluid. Also since the weight of fluid W passing 190 THE MECHANICAL PROPERTIES OF FLUIDS a given section per second is propoitional to the product d z vp Hoc WaT It follows that in similar pipes the heat transmission per unit degree difference of temperature between wall and fluid is proportional to the weight of fluid passing, or in other woids, that with a given inlet temperature the outlet temperature is independent of the weight of fluid If, however, n is somewhat less than i, as is usually the case in practice, equation (22) shows that nin H " oc so that with a given pipe and fluid, the heat transmission does not increase quite so fast as the mass flow, and the outlet temperature will increase somewhat as the flow is increased. CHAPTER V Hydrodynamical Resistance A body in steady motion through any real fluid, or at rest in moving current, experiences a resistance whose magnitude depends Don the relative velocity, the physical properties of the fluid, the ze and form of the body, and, at velocities above the critical, also son its surface roughness. At velocities below the critical, where the flow is " stream line ", e resistance is due essentially to the viscous shear of adjacent layers the fluid. It is directly proportional to the velocity, to the vis- >sity, and, in bodies of similar form, to the length of corresponding mansions Thus the resistances to the motion of small spheres such velocities are proportional to their diameters * With stream-line motion there is no slip at the boundary of solid d fluid, and the physical characteristics of the surface do not "ect the resistance. At velocities above the critical, wheie the motion as a whole is finitely tuibulent, theie would still appear to be a layei of fluid contact with the surface in which the motion is non-turbulent f ic thickness of this layer is, however, very small, and any increase the roughness of the surface, by increasing the eddy formation, ;reases the resistance. At such velocities the resistance is due in rt to the viscous shear in this surface layer, but mainly to eddy mation in the main body of fluid. This latter component of the sistance depends solely on the rate at which kinetic eneigy is being 'en to the eddy system, and is proportional to the density of the id and to the square of the velocity. Although the viscosity of a fluid provides the mechanism by which dy formation becomes possible, and by which the energy of the dies, when formed, is dissipated in the form of heat, it has only rery small effect on the magnitude of the resistance in turbulent >tion, and, as will be shown later, it can have no direct effect in * See H. S Allen, Phil. Mag , September and November, 1900. t Stanton, Proc Roy. Soc. A, 97, 1930. 191 iga THE MECHANICAL PROPERTIES OF FLUIDS a system in which the resistance is wholly due to eddy formatioi and in which the resistance is, in consequence, pioportional to v'' Experiments carried out over a limited iange of velocities hav usually shown that with turbulent flow the resistance of any givei body is proportional to v n , where n is slightly less than 2, althougl experiments on flow in rough pipes, on the icsistance of cylindeis of inclined plane surfaces, and of air-ship bodies, show that in sucl cases the variation from the index 2 may be within the limits o experimental error. With smooth pipes, however, n may be as low as 1-75, and with ship-shaped bodies of lair form in water is usually about i 85. Moie recent experiments* indicate that no one constant value of n holds over a veiy wide range of velocities, but that n inci eases with the velocity, and that a formula of the type R = Av + Bv z where A and B and C are constants depending upon the form and roughness of the body and on the physical propeities of the medium, more nearly represents the actual results Over a moderate iange of velocities a single value of n can usually be obtained which gives the resistance, within the errois of observation, and in view of the convenience of such an exponential formula it is commonly adopted in practice. At velocities above the critical, the direct influence of viscosity increases with the depaiture of the index n fiom 2 When n = 2 the resistance is proportional to the density of the fluid, and, in similar bodies, to the squaie of corresponding linear dimensions. Between the low velocities at which the motion is stieam-lme, and the high velocities at which it is definitely tuibulent, there is a range over which it is extremely unstable, and in which the resistance may be affected considerably by small modifications in the form, piesentation, or surface condition of the body. Thus the resistance of a sphere, at a certain velocity whose magnitude depends on the diameter, is actually increased instead of being diminished by reducing its loughness. In problems occurring in practice, however, velocities are in geneial well above the critical point. One noteworthy exception is to be found in the flow of oil through pipe lines in which, owing to the high viscosity of the fluid, the motion is usually non-tuibulent. In hydrodynamical problems it is usual to assume that the *N.P.L., Collected Researches, 11, 1914, p. 307. HYDRODYNAMICAL RESISTANCE 193 esistance depends solely on the relative velocity of fluid and body, ad that it is immaterial whether the body is at rest in a current of uid, or is moving through fluid at rest. Although there is not much irect experimental evidence on this point, it is probable that while ith stream-line motion the resistance is identical in both cases, in irbulent motion it is not necessarily so, and that it may be sensibly reater when the fluid is in motion than when the body is in motion. This is to be expected when it is realized that in a fluid in motion ith a mean velocity v, many of its particles have a higher velocity, ) that the kinetic energy is greater than that given by the product f the mass and the square of the mean velocity. Any difference ising from this effect will in general only be small, but compara- vely large differences may be expected over the range of velocities i which the motion is unstable, owing to the fact that, with a sta- onary body, the interaction occurs in a medium which has an initial ndency to instability owing to its motion. Thus the system of eddy formation in the rear of any solid body Ivancing into still water may reasonably be expected to differ om that behind the same body in a current of the same mean 'locity, owing to the instability in the latter case of the medium which, and from which in part, it is being maintained. Except in the case of stream-line flow, the -laws of hydro- rnamical resistance can only be deduced experimentally Much formation can, however, be obtained regarding these laws from i application of the two allied principles of dynamical similarity id dimensional homogeneity. Dynamical Similarity Two systems, involving the motion of fluid relative to geometn- lly similar bodies, are said to be dynamically similar when the paths iced out by corresponding particles of the fluid are also geometrically "mlar and m the same scale ratio as is involved in the two bodies The densities of the fluids may be different, as also the velocities ith which corresponding particles describe their paths If the ;nsities in the two systems are in a constant ratio, and the velocities corresponding particles are also in a constant ratio, then the ratio corresponding forces can be determined. In fact, the scale ratio velocities and that of lengths being both given, the scale ratio of nes is determined, and therefore also the scale ratio of accelerations f means of the fundamental relation, force = mass X acceleration, e ratio of corresponding forces can then be found. (D312) 8 *c,4 THE MECHANICAL PROPERTIES OF FLUIDS In two systems, denoted by (i) and (2), if w be the weight of unit volume, p the density, v the velocity, / any definite lineai dimen- sion, and r the radius of curvature of the path, these forces are in the ratio F 2 T! El l l Pz 4 2 It follows that for dynamical similarity corresponding velocities must be such as to make the corresponding forces due to each physical factor proportional to pl z v z . The velocities so related are teimed " Coiresponding Speeds ". Where the only physical factor involved is the weight of the fluid, since the force due to this is proportional to p/ 3 , the required condition will evidently be satisfied if = v* Y 4 On the othei hand, if viscous forces are all important, since dv the force due to viscosity equals p, - pei unit area, wheie fi is the dl coefficient of viscosity, Fj ^ ^ /^ Vj / g F /w- 7 2 v M 2 4 v z p I a /v \ 2 and for this to equal ~ - 1 - ( ) it is necessaiy that Pz 4 Vcy 2 y or that - 1 where v is the " kinematic viscosity " ft /p. Generally speaking, whei ever the influence of gravity is involved in the interaction between a solid and a fluid, as is the case where surface waves or surface disturbances are produced, and where the direct influence of viscosity is negligible, corresponding speeds are proportional to the square roots of corresponding linear dimensions; while where gravitational forces are not involved and where the forces are due to viscous resistances, corresponding speeds are HYDRODYNAMICAL RESISTANCE 195 inversely proportional to corresponding linear dimensions, and directly proportional to the kinematic viscosities The flow of water from a sharp-edged orifice under the action of gravity is an example of the first type of interaction, while the resistance of an air-ship, or of a submarine submerged to such a depth that no surface waves are produced, is representative of the second type. The lesistance of a surface vessel is one of a series of typical jases, of importance in practice, in which both gravity and viscosity are involved, and in which therefore no two corresponding speeds will satisfy all requirements. In other words, the speeds which give ^eometucally similar wave formations around two similar ships will not give similar stream-lines in those parts of the systems subject to viscous flow If, however, the influence of one of these factors s much greater than that of the other, approximately similar results, iVhich may be of great value in practice, can be obtained by using corresponding speeds chosen with reference to the important factor. Thus in tank experiments on models of floating vessels the corre- sponding speeds are chosen with reference to the wave and eddy effects, and are proportional to the square root of coi responding mear dimensions. This involves a scale error for which a correction s made as explained on p 213. Dimensional Homogeneity In view of the value of the lesults which may be obtained by he use of the principle of dimensional homogeneity in problems nvolvmg fluid resistance, the method of its general application will low be outlined The principle of dimensional homogeneity states that all the eims of a correct physical equation must have the same dimensions That is, if the numerical value of any one term m the equation lepends on the size of one of the fundamental units, every other erm must depend upon it in the same way, so that if the size of he unit is changed, every term will be changed m the same ratio, ind the equation will still remain valid The quantities which occur m hydrodynamics may all be defined n teims of three fundamental units The most convenient units are isually those of mass (M), length (L) and time (t) Example i. Suppose some physical relationship to involve only our quantities, say a force R, a length /, a velocity v, and a density p. 196 THE MECHANICAL PROPERTIES OF FLUIDS Let it be assumed piovisionally that the relationship is of the fornr R oc tWp*. Expressed dimensionally, this gives ML*- 2 = L" . L"t- v . M"L- 3a , and, on equating the indices of like quantities, whence z= i, y= 2, x ~ 2 It follows that R oc I 2 v z p, provided the initial assumption as to the form of R is correct. It is possible, however, to obtain the result without making this assumption. What we have really proved, in fact, is that the quantity is dimensionless Also, since it is given that there is a relationship between R, /, v, p, the quantity R/l 2 v z p is certainly some function of/, v, p, say/(/, v, p). Now, since the units of /, v, p are independent we can, by changing the unit of /, say, change the numerical value of / without changing the numerical values of v or p This change does not change the value of /(/, v, p), since it does not change the value of the dimeiisionless number R/l 2 v z p, to which /(/, v, p) is equal. Hence the function / does not involve /, and similarly it does not involve v or p; it is therefore a mere numeiical constant, so that R oc Pv*p Example 2 If more than four quantities of diflerent kinds are involved, for example R, /, >, p, //,, wheie //, is a viscosity (dimensions ML,- 1 *" 1 , p. 28), the assumption R oc l a> v ja p z p? would not allow us to determine the values of x, y, z, p by considerations of dimensions, since there would be only three equations in four unknowns. It is possible, however, to obtain a new dimensionless number, not in- volving R, but of the form Equating the dimensions of //. to those of l a v b p, we find i = c, i = a + b 3^, i = 1)\ and c i, b = i, a = i. Thus nflvp is dimensionless. HYDRODYNAM1CAL RESISTANCE 197 For brevity, we shall denote the two dimensionless numbers now found by Kj, K 2 ; i.e. R j^. fjt, -f,. , == "!> "; == -"-2 \I,) fo/) It will now be proved that if there is a relationship between R, /, v, p, //. it can be expressed in the form Ki=/(K.), (2) where the form of the function / remains undetermined. In fact, since R by assumption is some function of /, v, p, p, and since we :an substitute Ivp K 2 for p, it follows that R// 2 u 2 /? is some function )f 7, v, p, K 2 , or say Kj = <(/, 0, />, K 2 ). Then exactly the same argument as that given under Ex. i proves hat the function (j> does not involve I or v or p, but only K 2 , and his is what was to be proved The relation K t = /(K 2 ) may of ourse be written in other forms, as for example K 2 = F(Kj) or i(K,, K 2 ) = o. ^he general theorem of dimensionless numbers The geneial theorem, of which the two preceding results are 'articular cases, may be stated as follows * (1) Let it be assumed that n quantities Q,, Q 2 , , Q w which re involved in some physical phenomenon, are connected by a Nation, F(Q lt Q, . . . , Q n ) = o, (3) Dntaming these quantities and nothing else but pure numbers (2) Let k be the number of fundamental units (L, M, t, . . .) 'quired to specify the units of the Q's (3) Let QJ, Q 2 , . . , Q 7c be any k of the Q's that are of independent nds, no one being derivable from the others, so that these k might, we so desired, be taken as the fundamental units (4) Let Qa, be any one of the remaining n k quantities Q, and t , which we denote by K x , be the dimensionless n on & oX ^1 v ^2 ' ^/c lantity formed from Q a and powers of Q lf . . . , Q fc . *E Buckingham, Physical Review, IV, 1914, p 345, Phil Mag , Nov , 1931, 696. 198 THE MECHANICAL PROPERTIES OF FLUIDS (5) Then the theorem is that the equation F(Qi, Qa , Qn) = o is leduciblc to the foim Kx, K 2 , . , K n _ 7c ) = o, (4) or, alternatively, The pi oof follows exactly the same lines as m the two particulai examples already given The actual forms of the functions (f> and / can only be deduced fiom experiment Jn the most geneial case of a dynamical relationship between any numbei of quantities, say n, there will be n 3 quantities (K) of zeio dimensions, composed of poweis of the quantities, and deduced in the manner explained above The relationship between the n quantities originally considered reduces to a relationship be- tween the n 3 quantities K 1} K 2 , . , K n _ 3 Hence if all but one of the K's are known, the last one is deteimmed Some oi the K's may often be written down fiom inspection Suppose, ioi example, that a certain phenomenon involves a time i and an acceleration g, in addition to the R, /, v, p, p, of Ex 2 above. Then we see at once that the new K's are tv/l and gift* The re- lation between the seven quantities is therefore of the form ./ R //, to gl\ , </'( ,., , -,-, -7, .,) ^ o, (6) \ rvp Lvp I V" ' 01, say, T> 70 o ,/ V tV Pl t\ rr t **$}** r- "*' JIX *" ' b \s /-> T ) Tt is useful to remailc that any product of poweis of the K's is dimension less. Hence if we multiply the second and third of the aiguments of <}> in (7), we get a new dirnensionless number tgfv, which may peilectly well leplace one of the two, tv/l and gl/v z , in (6) and (7). If two 01 moic quantities of the same kind are involved, as, for example, in the case of the resistance of an anship body, where both the length and diameter of the body affect the result, these may be specified by the value of any one, and by the ratios of the others to this one. Thus, in the problem just considered, if besides HYDRODYNAMICAL RESISTANCE 199 t, v, p, p, g, t and the length / of a body, there are also involved the ireadth b and the depth d of the body, the solution is * , , , ........ (8) lvp v v 2 I // It is clear from the above examples that the actual arithmetic ivolved m working out an application of the principle of dimensions i of the simplest possible kind. The real difficulty is m making jre that all the essential quantities concerned in the phenomenon are emg taken into account. If this preliminary condition is not ilfilled, the result obtained will be quite erroneous Resistance to the Uniform Flow of a Fluid through a Pipe An examination of the factors involved m the non-accelerated totion of a fluid through a pipe would indicate that the pressure rop_/>// per unit length of the pipe may depend in some way on thcdia- teter d, the velocity of flow, and the density and viscosity of the fluid i a liquid where the eJlect of elasticity is negligible, it is difficult * imagine any other factoi likely to aikct the pressure drop, except ie roughness of the pipe walls , and, so long as we only considei pipes the same degree ol toughness, the geneial um educed lelationship of the form F(d, v, p, p, p/l) = o (9) ere the number of dirnensionlcss quantities K is 5 3 = 2 oiclly as m a former example (p 196) we find T<r _ P d K __ /* XXj , J-Vg . Ipv" avp icie is some advantage m working with v, the kinematic viscosity, nch is equal to /././/?, rather than with //, itself. Hence K 2 = v/dv The reduced relationship may therefore be written 10 d leie the form of the function c/> remains to be found from experi- mt. Note that the value of the function (f> for all values of its jument can be found by varying one only of v, d, v. 200 THE MECHANICAL PROPERTIES OF FLUIDS With str cam-line flow, experiment shows that - is proportions / v\ ki> to v. It follows that r/n y~ must equal ---,, and that \av/ 1 va P _ 7 ~ 'd 1 ' where k is a numeiical coefficient. This is Poiseuille's expressio for the resistance to viscous flow, the coefficient k having the vain 32.* If the flow is turbulent the piessure giadient is approximate! proportional to v n , where n is between 1-75 and 2-0 In this cas (V \ I V \ ( V \ 2 ~" - } must be such as to make <i(- -) = k'{ } , so that v' \Jv' -/ dv P __ , d P __ j,pv" /v\ ~~ or where h is the diflercnce of head at two points distant / apar expressed as a length of a column ot the fluid This is tl Reynolds | foimula for pipe flow If the lou'going assumptions aic coirect, on plotting observe values ol /., against simultaneous values of '' , in any series < IV" p dv experiments in which chflerent liquids 01 pipe s of different diameter but ecjually lough, are used, the points should he on a single cutv That this is the case foi fluids so widely different as ai water, and oil, has been shown by vaiious observers, notab by Stanton and Pannell \ (sec fig 10 at p, 170, where R = pdltf An example is given m iig. i, where expenmental points fc both aii and water He evenly about the two ciuves showi The agieement m the case of air is only close where tl drop in prcssuir is so small that the effect of the change of densii * Gibson, Hydraulic 1 * and its Applications (doubtable & Co., 1913), p. 69. 1 tfetfntifit. I'tipd 1 !, Osboinc Reynolds, Vol. 11, p. 97, y Phtl. Tians, Roy, floe. A, 214, 1914, p 199. HYDRODYNAMICAL RESISTANCE *>t is negligible. For large changes of pressure it may be shown* that formula (12) becomes cT Here T is the absolute temperature of the gas, C is the constant obtained from the relationship pV = CT,f and v m and p m are the mean velocity and pressure in the pipe. The results of experiments on the flow of air through pipes by several experimenters, with dia- r-a so ~ C 40 so t Z 3456789 Values of v" Scale t division - 2870 units Fig i meters ranging from 0-125 m to 9 1 **> anci at velocities from 10 to 40 ft -sec , confirm the accuiacy of this formula Below the critical velocity, n = i, and the formula becomes showing that the pressure drop is now independent of the absolute pressure in the pipe, a result confirmed by experiment. Equations (10), p. 191, (n), p. 200, show that, with an incompres- sible fluid, if the resistance to flow varies as v z , $(-=-) degenerates \dv/ into a numerical constant. Viscosity ceases to have any direct influence on the resistance,! and true similarity of flow should be obtained in all pipes at all velocities. If n is less than 2, the Gibson, Phil Mag , March, 1909, p 389. f In the case of air, if the mass is unity, and if p be measured in pounds weight per square foot, the value of C is 53-18 X 32 z = 1710, while if p be measured in pounds per square inch C becomes 1 1 -9. J See footnote on p. 209 (D812) 8 * 202 THE MECHANICAL PROPERTIES OF FLUIDS equations show that for similarity of flow in two pipes of diffeiei diameters, or conveying different fluids, it is necessaiy that shoul vd be the same in both cases. This has been shown to be tiue over a moderate range < diameters by Stanton.* who measured the distribution of velocit with turbulent flow across the diameters of two rough pip< of different diameters and repeated the measurements for tw smooth pipes. In the rough pipes (n = 2) the velocity distribt tion (CC', fig. 2) was the same in both pipes and at a AA Velocity distribution In btfo smooth pipes in which l^d is constant One pfpe t diareter 4 93cm, Velocity at axis 1525cm parsecond . . T40CM} 1017 ... 0-4 A B Smooth p!ps diameter 4 33 cm cc' Velocity Distribution In hvo rough pipes soscm Sii-sscm atoli velocities Radius, In terms ofrapius of pipe o*t 06 Fig a velocities. In the smooth pipes (n < 2) identical cuives wei obtained only when was the same in each pipe. Undei othe vd conditions the curves were sensibly identical from the centre u to a radius of about 0-8 times the radius of the pipe, but differs appieciably at larger radii (AA' and AB, fig. 2). In any type of pipe the " critical velocity " at which the type o motion changes from stream-line to turbulent is obtained with constant value of , and this is generally true for fluid motion unde vd other circumstances. As already indicated, experiment shows that the resistance ti * Proc. Roy. Soc. A, 85, 1911. HYDRODYNAMICAL RESISTANCE 203 flow in a smooth pipe where n is less than 2 is not strictly propor- tional to any one power of the velocity, and Dr C. H. Lees* has shown that Stanton and PannelPs results for smooth pipes are very closely represented by the empirical relationship where a = 0*35 and a and b are constants, so that In the rough pipe the ratio of friction to v z increases with velocity, and Stanton and Pannell suggest, for both rough and smooth pipes, an expression of the form T> 9 f A V I T7- P T F = pa a ]A- + K+B ( vd in which K depends only on the roughness of the pipe. It will be noted that this relationship is similar to the one obtained by Messrs Bairstow and Booth fiom expenments on the normal icsis- tance of flat plates (p 213) Skin Friction The lesistance to the endwise motion of a thin plane through a fluid is usually termed " skin friction " Expressing the resistance by fAv", where A is the wetted surface, the values of / for vanous sm faces in water weie detei mined by Mr Froude f In these expenments a series of flat boards was suspended vertically fiom a carnage diiven at a uniform speed and was towed through the still water m a large basin The boaids were -fV in thick, 19 in deep, and varied m length fiom i ft to 50 ft The top edge was submerged to a depth of if in., and the boards were fitted with a cut- water, whose resistance was detei mined separately A short resume" of Mr Fioude's icsults is given on the follow- ing page, these paiticular figures lefemng to a velocity of 10 ft -sec Here A refeis to varnished surfaces 01 to the painted surfaces of iron ships, B to surfaces coated with paraffin wax, C to surfaces * Proc Roy Soc A, 91, 1914, p 46 f Btitish Association Report, 1872. 204 THE MECHANICAL PROPERTIES OF FLUIDS 8 00 vO Q M O O OO V - CO CO N U tjL, H O O 1-M o 10 1 m o VO CO N OO N ol *^j H O O rj- 10 o o co 10 Q N o O N i ^ O\ M CJ O 1) <U M M CO CN N CO N m O O 00 OO M N <I H O O 10 OO O o N vO oo Q ft O o 00 CO p\ M vO N u 4> ^ O O <U fe "* O oo c CO vO N pq M O O to N VO- " " oo CO N ^rj M O O o o o\ CO f- Q rt o o ^ o CO ,ps, (*J\ Pt O <u f O fe o d ln 00 OA CO CO Q H o O o H rt- CO <tj N 8 <+$ " 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 4-ft board i water would conespond to a velocity of 325 ft -sec with the 6- it bojid in air Taking the lesistance m air as proportional to 1 8s , the lesistance per square foot of the i6-ft. board at this speed is R = o 000457 X (3 <3 5) J 8s = 0-00402 Ib , and R <z; a = o 00000380 'he value of R v z for the 4-ft. board in water is 0-00325, the * Phil Mag., 8, 1904, pp. 58-66. 206 THE MECHANICAL PROPERTIES OF FLUIDS ratio of the two being 855, a value only about 4 per cent greatei th, the relative density o water and air at 60 F. In view of the fact that one set of experiments was earned o in still water, and the other in an air current whose flow was n perfectly uniform, the agieement between the two sets of resul is very close. Resistance of Wholly Submerged Bodies Where a body is submerged in a current to such a depth th no surface waves are formed, gravity has no effect on the resistanc This will happen with a deeply cubmeiged submarine, 01 wil an air-ship. If the speed is constant,, so that there is no accelen tion, and if the liquid is incompressible, or if in a compressib fluid the speedc do not approach the acoustic speed so that piessui changes arc so small that the compressibility may be neglectei the resistance R may evidently depend upon the iclative velocil of fluid and body, on the density and viscosity of the fluid, and on tl size and shape of the body. In a series of geometiically simil; bodies each is defined by a single linear dimension /, and the icsii tance R will be given by the relationship F(R, /, v, P , p) = o As in the previous examples K 2 Inserting the dimensions of /, v, p, R, and /x, and detei mining th indices x y, s, a, b, c, so as to make K x and K 2 dimensionless, give the values of K.JL and K 2 These are R - lpv p or R - plW <l> jpv The value of the unknown function <j> might be found by plottin HYDRODYNAMICAL RESISTANCE 207 R 7?) bserved values of - against simultaneous values of , and 7 2 z> 2 /> v y finding an empirical equation to represent the curve joining the dotted points. Moreover, it should be noted that since the values f both terms in the function are dependent on v, the form of the unction, for any liquid, can be determined from experiments on single body at different speeds in the same medium. In the case of model experiments, if the medium is the same 3r model and original, and if the suffix m denotes the model, then, F the speeds be chosen so that v m = , we shall have o that </>( j becomes a constant, and R / 2 v* The speeds thus related are " corresponding speeds ", and at these peeds the model and its original are " dynamically similar " In his case the corresponding speeds are inversely pioportional to the meai dimensions, and at these speeds the icsistance of the model nd of the original are equal Unfortunately this relationship would involve such high speeds Q the case of the model as to be of no practical value. If, however, he model cxpenments can be carried out in a medium whose lune- natic viscosity is less than that of the original, the corresponding peeds aie reduced Thus by using compressed air m a wind tunnel he corresponding speed is reduced in the same ratio as the density 3 increased, since the kinematic viscosity of air varies inversely as ts density. Such a wind tunnel is in operation at Langley Field USA) Adopting a working pressure of 20 atmospheres ?!S = 1-JL V 20 7 OT ' o that with a i/io scale model, the corresponding speed in the vmd tunnel would be one-half that of the onginal, and at these peeds 2o8 THE MECHANICAL PROPERTIES OF FLUIDS __ R Pv z 20 With bodies whose resistance is sensibly proportional to th square of the velocity, the form of the function must be such t to make <f>(\ a constant whose value depends only on the shape c the body, and the resistance is given by R It is now immaterial at what speed the model experiments are carrie' out, so long as this is above the " critical speed ". This discussion applies equally well to any case of motion of totally immersed body in a medium whose compressibility may b neglected. Resistance of Partially Submerged Bodies When a body is paitially submeiged, or, although submerged is so near the surface that surface waves are produced, part of tin resistance to motion is due to this wave formation. The influenc* of gravity must now be taken into account, and we have the relation shl p Since there are now 6 quantities involved, 6 3 (= 3) K's are required. Taking /, v, and /> as convenient independent quan tities and proceeding as before, K! = R/l<Wp* t K Inserting the dimensions of R, /, v, p, p, and , and deteimining the indices #, a, a, &c , necessary to make K a , K 2 , K 3 dimensionless gives the values of K 1$ K 2 , and K 3 . These are R K - ^ K - gl - ~ l so that M =?-}. (~),(^}\ = o, Vwp/ or HYDRODYNAMICAL RESISTANCE 209 [n the case of model experiments, it is necessary for dynamical similarity that each of the arguments of $ shall have the same value for both model and original. But if, as is usual in practice, both are to operate in water, v is sensibly constant, while g is also constant, so that for exact similarity both Iv and v z /l would require to be constant In other words, neither v nor I can vary It follows that the lines of flow and the wave formation around a ship and its model in the same fluid cannot simultaneously be made dynamically similar It remains to be seen whether any of the arguments in- volved in the function <f> may reasonably be neglected so as to give an approximation which is likely to be of use in practice Experiment shows that the resistance of a ship-shaped body at the speeds usual in piactice is proportional to v", where n is approxi- mately 1-83, and the nearness of this index to 2-0 indicates that the direct effect of viscosity is small If it be assumed that this influence of viscosity is negligible,* the argument v/lv may be omitted from the equation, which now becomes, R = If now be made the same both for the model and the original, <7j2 R, Pm l m 4 Pn$J Pm D /n' where D is the displacement. In this case the " coiiesponding speeds " at which wave and eddy formation are the same for ship and model, are given by lm Model Experiments on Resistance of Ships In practice these corresponding speeds are used, but allow- ance is made for the diffeient effects of viscous resistances in the two cases by the well-known " Froude " method *This does not involve the assumption that skin friction is ummpoitant or that viscosity plays no pait in the phenomenon It is m effect assuming that skin friction, instead of being pioportional to v" where n is slightly less than 2, is pro- portional to w a . In this case the i esistance is due to the steady rate of formation of eddies at the surface of the body, and, once these have been formed and have left the immediate vicinity of the body, the rate at which they are damped out by viscosity has no effect on the drag. 2io THE MECHANICAL PROPERTIES OF FLUIDS In determining the icsistance of any proposed ship, a scale model is made, usually of paraffin wax, and is towed through still watei, the lesistance corresponding to a number of speeds being measuied 60 55 5 90- -00 30 i Iff Speed of Model in knots 4 o 50 2O 25 Corresponding speed of shfps In knots Fig. 3 by dynamometer. A curve AA, fig 3, is plotted showing lesistance against speed. The length and area of the wetted surface being known, the skin friction (f^A^^ ) is calculated, the coefficient being taken from Froude's results on the resistance of flat planes towed endwise,* * See pp 203, 204 HYDRODYNAMICAL RESISTANCE iti ri he curve BB of frictional resistance can now be drawn, and the itercept between AA and BB gives the eddy- and wave-making esistance of the model. If now the horizontal scale be increased i the ratio S : i, where S is the scale ratio of ship and model, and " the vertical scale be increased in the ratio D : i, this intercept ives the eddy- and wave-making resistance of the ship at the corre- ponding speed. If fresh water is used in the tank, the vertical cale is to be increased again in the ratio of the densities of salt and resh water The skin friction (fAv n ) of the ship is next calculated nd set down as an ordmate from BB to give the curve CC. The itercept between the curves AA and CC now gives, on the large cales, the total resistance of the ship The resistance at any speed v may be calculated from model bservations at the corresponding speed v m , as follows' Total resistance of modeh _ .. , , ,s \ = R OT Ib. (observed) . . . . J m Skin friction of modeh _ . . n (calculated) .. ../ '"*?> 1D ' .'. Wave- + eddy-iesist-j _ __ - - p Ib ance of model . J "~ '" J ^ V " - r 1D * .'. Wave |- eddy-iesist-^| , ance of ship in salt I = DF X -^~ Ib. 62 4 water . ' ^ Total resistance of 6 DF , fAv n lb _ ship .. ../ 62-4 Scale Effects Resistance of Plane Surfaces of Wires and Cylinders of Strut Sections From what has already been said, it will be evident that in lost model experiments some one of the factors involved tends 3 prevent exact similarity and introduces some scale effect, t is only when this effect is small, and when its general icsult i known, that the results of model experiments can be used dth confidence to predict the performance of a large-scale rototype. The resistance of square plates exposed normally to a current, THE MECHANICAL PROPERTIES OF FLUIDS affords a typical example of scale effects. Expressing the resistance of such plates as R = K'pa 2 in absolute units (British or C.G S ) = Kw a , where R is in pounds weight per squaie foot, and v is in feet per second, experiments show the following values of K' and of K. K' K Size of Plate. Dmes * Canovetti f Eiffel J Stanton 56 56 55 56 59 61 62 52 62 62 00135 00134 i ft. squaie 3 ft diametei (circular) 00133 00136 10 m. square 14 , 00142 20 , 00147 27 , 00150 00126 00148 00149 39 . 2 , 5 ft squai 10 ft squ e ire n Such experiments show that while is almost independent of v, it increases by about 18 per cent as the size of the plate is increased from 2 in. to 5 ft. square. As the compressibility of the air has been neglected in deducing expression (15), p. 206, it is impossible to say without further examination that this effect is not due to compressibility. Indeed if compressibility has any influence on R, a dimensional effect can occur which may be in accordance with a v 2 law of resistance, for, when this is taken into account, (15) becomes R = wheie C is the velocity of sound waves in the medium. This may be written * Proc. Roy /Soc , 48, p 252. f Socie'te' d 'Encouragement pour 1'Industne Nationale, Bulletin, 1903, 1, p. 189. I Eiffel, Resistance de I'Air N.P.L, Collected Researches, 1, p. 261. HYDRODYNAMICAL RESISTANCE 213 ind since by hypothesis R is proportional to v z , this becomes T> \xi investigation of the possible effect of compressibility shows, lowever, that this is less than i per cent for speeds up to oo miles per hour, so that an explanation of the observed limensional effect based on this factor is not admissible. It has been suggested* that since, as shown by Mr. Hunsaker's ibservations on circular discs, there is a critical range of speed letermined by the form of the edge, and not dependent on the size n t> >f plate, the apparent discrepancy between = constant and = <D t triable may be due to the results of vaiious observers having >een affected by such ciitical phenomena, which were not, however, ufficiently marked to attiact attention. To determine whether this xplanation is valid would require an experimental investigation of he forms of edge which have been used. The more piobable explanation would appear to be that while xpenments on any one plate have been taken as showing the i esist- nce to be pioportional to v z , this is only appioximately true. An examination by Messis. Bairstow and Booth ,f of all reliable xpenments on squaie plates, leads to the conclusion that an empirical ormula of the type R = ^ /)2 + ^ /)3 (iy) jives a close appioximation to the experimental results. For values >f vl ranging from i to 350, a and b have the values 0-00126 and 1-0000007 lespectivcly Here v is in feet per second and R m pounds Neglecting compicssibihty this indicates that (f>( } of equation 1 6), p 212, equals m + nvl, wheie m and n are constants for any miticular fluid under given piessure and tempeiature conditions. It may be noted that experiments by StantonJ show that the >ressmes on the windward side of a square plate are not subject to dimensional effect, but that the whole variation can be traced to hanges in the negative pressure behind the plate. * By Mr. E Buckingham, Smithsonian Miscellaneous Collections, 62, No 4, an., 1916. f Technical Report, Advisory Committee for Aeronautics, 1910-1, p 21. %Proc. Inst. C. E., 171; also Collected Researches of the National Physical ,aboratory, 5, p 192. 2i 4 THE MECHANICAL PROPERTIES OF FLUIDS Resistance of Smooth Wires and Cylinders A somewhat similar scale effect is obtained from experi- ments on the resistance of smooth wires and cylinders. A series of such tests on a tange of diameters from 0-002 in. to 1-25 in., with R v ranging from 10 to 50 ft.-sec ,* shows that, on plotting 3 J against or log , a narrow band of points is obtained which in- v v 10 20 30 40 50 Loq & -MO I/ Fig 4 eludes all the experimental results (fig. 4). This shows that for a given value of the value of - is the same for all values of v and of v d. From this it appears that whereas experiments on a single wire or cylinder indicate that R is nearly proportional to v z , true similarity of flow is only obtained when is constant. v It becomes very necessary to satisfy this condition with low vd values of , owing to the changes which may occur in the type * Reports and Memoranda, Advisory Com. for Aeronautics, No. 40, March, 1911; No. 74, March, 1913, No 102. HYDRODYNAMICAL RESISTANCE 215 of flow around such bodies at comparatively low velocities or with small diarneteis. As in all other cases of flow, as this factor is reduced a critical value is ultimately reached where the type of flow undergoes a definite and rapid change, so that the function < ceases to be even approximately constant. For a given body in a given medium, this critical value of corresponds to a critical speed which may be calculated from the 032 28~ -x ,,, ,. \ s*~S s Ss S^S 5rt - , 4' ^***^S * s J/ s r S s <- 024 -* I S.020 n S. y cc <Q \i ' ' 3 5 008 - 004 2-4 6 8 10 12 14 16 18 2o 21 2* Values of v d in foot second units Fig 5 Resistance of Strut Sections values of d and v, li its value has once been experimentally deter- mined for bodies oi the given form by varying any one of the vari- ables d, v, and v. In some such bodies as spheies and cylinders the law of resistance may change widely with compaiatively small alteration in the con- ditions; thus, for example, at ceitam speeds the resistance of a sphere may actually be reduced by toughening the surface. In carrying out any such experiments, theiefore, it is of the greatest importance that the geometiical similarity between a model and its prototype should be as exact as possible, and that where possible vd should be kept constant. The variation in the type of flow at a definite critical velocity has been well shown in the case of flow past an inclined plate by ai6 THE MECHANICAL PROPERTIES OF FLUIDS C. G. Eden*. By the aid of colour bands in the case of water, anc smoke in the case of air, photographs of the eddy formation in th< rear of the plates of different sizes were obtained. These shov BLERIOT BETA BF, 34- B.F 35 BABY Fig. 6 Strut Sections. that the types of flow were similar for both fluids and for all the plates so long as was maintained constant, and that the change v over from one type to the other took place at a critical velocity, defined by v ont oc - , in each case. / * Tech. Report of Advisory Committee for Aeronautics > 1910-1, p. 48; also R. and M., No. 31, March, 1911. HYDRODYNAMICAL RESISTANCE 217 ID The curve of fig. 5*, p. 315, shows the change in with a pdtr variation in vd in the case of a strut of fair stream-line form. Here R is the resistance per foot run of the strut. The curve shows that the resistance is very nearly proportional to v z for values of ^greater than 5, but that as vd is reduced below this value the law of resistance suffers a rapid change. Since, below the critical velocity, Roc v, the ordinates of the curve to the left of the critical point will be pro- portional to -, and this part of the curve will be hyperbolic. v The following tablef shows the resistance of typical strut sections of the types and sizes shown m fig. 6. Resistance of too Ft Type of Stiut of Strut m Pounds at 60 ft -sec Cucle, i in diametei 43 o Ellipse axes, i in X 2 in 22 2 Ellipse axes, i m X 5 m 15-2 De Havilland 25 5 Farman 22 9 Blcnot A 23 7 Blenot B 24 5 Baby . 7 9 Beta 6 9 B F 34 72 B F 35 .. 63 B F. 35, tail foiemost 10 9 * Applied Aerodynamics, Banstow (Longmans, Green, & Co , 1920), p. 392. f Tech Repot t of Advisory Commi tteefor Aeronautics, 1911-2, p 96 CHAPTER VI Phenomena due to the Elasticity of a Fluid Compressibility Compressibility is defined (Chapter I, p 16) as the reciprocal of the bulk modulus, i e. by - ( ) v \op/T The compressibility of water varies with the temperature and the pressure, the values of the bulk modulus, obtained by different observers, being as follows * These values are in pounds per square inch. Temperature, Degrees C Authority o. 10 ao" 30 40 318000 333000 Grassi f 293000 303000 319000 322000 TaitJ ,, ( At low 283000 301000 319000 334000 347000 352000^ piess ures 292000 311000 328000 340000 347000 347000 | , i At 2 tons 300000 321000 332000 346000 339000 339000-^ At i ton per sq in The bulk modulus K of sea water is about 9 per cent greater than that of fresh water. *See also Parsons and Cook, Proc. Roy Soc A, 85, 1911, p 343 At 4 C. Parsons finds K = 306,000 Ib. per square inch at 500 atmospheres, 346,000 Ib. at 1000 atmospheres, and 397,000 at 2000 atmospheres. Results of experiments by Hyde, Proc R S A, 97, 1920, are in close agreement with these. f Annales de Chimie et Physique, 1851, 31, p 437 %Math. and Phys Papers, Sir W. Thomson, Vol. Ill, 1890, p. 517. 218 PHENOMENA DUE TO ELASTICITY OF A FLUID 219 For purposes of calculation at temperatures usual in practice, e modulus for fresh water may be taken as 300,000 Ib. per square ch, or 43-2 X io 6 Ib per square foot. The compressibility is so small that in questions involving water rest or in a state of steady flow it may be assumed to be an incom- essible fluid. In certain important phenomena, however, where a idden initiation or stoppage of motion is involved the compressi- hty becomes an important, and often the predominating factor. In such cases the true criterion of the compressibility or elasticity a fluid is measured by the ratio of its bulk modulus to its density, nee it is this ratio which governs the wave propagation on which ich phenomena depend. In this respect air is ooly about eighteen rates as compressible as water For olive oil the value of K at 20 C is 236,000 Ib. per square ch (Quincke) and for petroleum at 16-5 C is 214,000 Ib per [uaie inch (Martini). The following are values of K for lubri- itmg oils at 40 C * Tons per Castor Oil Sperm Oil 1X ?,H 1 P l1 square inch 1 291000 242000 287000 2 302000 252000 291000 5 330000 285000 315000 Sudden Stoppage of Motion Ideal Case If a column of liquid, flowing with velocity v along a rigid pipe F unifoim diameter and of length / ft., has its motion checked by the LStantaneous closure of a ngid valve, the phenomena experienced e due to the elasticity of the column, and are analogous to those btaining in the case of the longitudinal impact of an elastic bar gainst a rigid wall. At the instant of closure the motion of the layer of water in mtact with the valve becomes zero, and its kinetic energy is con- srted into resilience or energy of strain, with a consequent sudden se in pressure. This checks the adjacent layer, with the result lat a state of zero velocity and of high pressure (this at any point * Hyde, Proc, R. S. A, 97, 1920. 220 THE MECHANICAL PROPERTIES OF FLUIDS being p above the pressure obtaining at that point with stea* flow) is propagated as a wave along the pipe with velocity V # .* This wave reaches the open end of the pipe after t sec., whe Static. L a Aitruos ' Instant of Valve. Closure, Pressure Pressure ,1-1. V P Static V P V Pressure Attnaf Tressure Fig i Ideal Case of Water-hammer in Pipe Line t = l-rV p . At this instant the column of fluid is at rest and in a state of uniform compression. This is not a state of equilibrium, since the pressure immediately * Vj) * s the velocit y of propagation of sound waves m the medium, and is equal t0 A/~of ' where is the weight in pounds per cubic foot, and K is in pounds per square foot. PHENOMENA DUE TO ELASTICITY OF A FLUID 221 aside the open end of the pipe is p greater than that in the surround- tig medium. In consequence the strain energy of the end layer 3 reconverted into kinetic energy, its pressure falls to that of the urrounding medium, and it rebounds with its original velocity v owards the open end of the pipe. This relieves the pressure on he adjacent layer, with the result that a state of normal pressure nd of velocity ( v) is propagated as a wave towards the valve, caching it after a second inteival / ~ V p sec. At this instant tru vhole of the column is at normal pressure, and is moving towards he open end with velocity v. The end of the column tends to leave he valve, but cannot do so unless the pressure drops to zero, or so iear to zero that any air in solution is liberated. Its motion is con- equently checked, and- its kinetic energy goes to reduce the strain nergy to a value below that corresponding to normal pressure. The pressure d ops suddenly by an amount equal to that through yhich it originally rose, and a wave of zero velocity and of pressure below normal is transmitted along the pipe, to be reflected from he open end as a wave of normal pressure and velocity towards the r alve When this wave reaches the valve 4/ V p sec aftei the QStant of closure, the conditions are the same as at the beginning >f the cycle, and the whole is repeated. Under such ideal conditions the state of affairs at the valve would >e represented by fig i a At any other point at a distance / x from he open end the pressure-time diagram would appear as m fig. i b If the velocity were such as to make p greater than the normal bsolute pressure in the pipe, the first reflux wave would tend to educe the pressure below zero. Since this is impossible, the pressure ould only fall to zero, and the subsequent rise in pressure would >e correspondingly reduced. Actually, at such low pressures any lissolved air is liberated and the motion rapidly breaks down. Effect of Friction in the Pipe Line The effect of friction in the pipe line modifies the phenomena n a complex manner. In the first place the pressure, with steady low, falls uniformly from the open end towards the valve, and the >ressure at the valve will be represented by such a line as AB (fig. 2). )n closure the pressure here will rise by an amount p as before. When the adjacent layer is checked its lise in pressure will also >e p, but since its original pressure was higher than that at the r alve, its new pressure will also be higher. It will therefore tend to J22 THE MECHANICAL PROPERTIES OF FLUIDS compress that portion of the column ahead of it, and will lose some of its strain energy in so doing. This will result in the pressure at the valve increasing as layer after layer is checked, but since this secondary effect travels back fiom each layer in turn with a velocity Vp, the full effect at the valve will not be felt until a time 2/ -r V^ after closure. At this instant the pressure will have risen by an amount which to a first approximation may be taken as dp Vz,* where dp is the pressure- difference at the ends of the pipe under steady flow. When reflux takes place the end layer, having transmitted part Static al 1- f f Pressure A Pressure Fig 2 Water-hammer as Modified by Friction of its energy along the pipe, can no longer rebound with the original velocity i). Moreover, since at the instant when the distuibance again reaches the valve, and the column is moving towards the open end, frictional losses necessitate the pressure at the valve being dp greater than at the open end, the pressure drop will be less than in the ideal case. * With steady flow the pressure distant x from the valve will be greater than that at the valve by an amount dpj. Therefoie excess strain energy of a layer of length Sx at this point, due to this pressure c dx(P. x .* j Assuming this excess energy to be distributed over the length x between this //}/! V V\ % layer and the valve, it will cause a use in pressuie p', where (p'Yx = <W -jr ) . Integrating to obtain the effect of all such layers from o to x gives \ * / ^ fo1)\^x 11 i 3t> dv ~ I -f- 1 , and when x I, p = -*= This is only a first approximation, since the equalization of piessures will be accompanied by surges which will introduce additional frictional losses. PHENOMENA DUE TO ELASTICITY OF A FLUID 223 If the velocity of the first reflux be kv t where k is somewhat less tan unity, the pressure diagram will be modified sensibly as in =' 2 '. Friction thus causes the pressure wave to die out rapidly, with- Jt affecting the periodicity appreciably. Magnitude of Rise in Pressure, following Sudden Closure Assuming a rigid pipe, on equating the loss of kinetic energy pound of fluid to the increase in its strain energy or resilience: zg p = /r V g 'g' here p is the rise in pressure, and v the velocity of flow in feet per 'cond. Putting K = 43-2 X io 6 Ib -sq ft., ,, w 62-4 Ib -c ft , g =- 32 2 ft.-sec. 2 , this becomes p = gi6ov Ib.-sq ft. = 63-72; Ib.-sq in Effect of Elasticity of Pipe Line Owing to the elasticity of the pipe walls, pait of the kinetic Lergy of the moving column is expended in stretching these, with resultant increase in the complexity of the phenomena, a reduction the maximum pressure attained, and an increase in the rate at bich the pressure waves die out. The state of affairs is then dicated in figs. 3 a and b, which are reproduced from pressure- ne diagrams taken from a cast-iron pipe line 3-75 in. diameter and ;o ft. long f Fig. 3 a was obtained from behind the valve and fig. 3 b at a point ; ft. from the open end of the pipe. The elasticity of the pipe line may modify the results in two * If a cube of unit side be subject to a pressure increasing from o to p, the ange in volume will be p K, and since the mean pressure during compres- in is p 2, the work done is p z aK f Gibson, Water Hammer in Hydraulic Pipe Lines (Constable & Co., 1908). 224 THE MECHANICAL PROPERTIES OF FLUIDS ways. If the pipe is free to stretch longitudinally, at the insta of closure the valve end of the pipe and the water column w move together with a common velocity ,* less than v, and a wa' of longitudinal extension will travel along the pipe wall. Tl instantaneous rise in pressure at the valve will now be equal to (v S Since the velocity of propagation is much greater in metal tha Static Status Pressure Fig 3 Water-hammer in Experimental Pipe Line in water, this wave will be reflected from the open end of the pipe and will reach the closed end again before the reflected wave in the water column. At this instant the closed end of the pipe will rebound towards the open end with velocity u, and will produce an auxiliary wave of pressure equal to /Kw M-W v s * It may readily be shown that u = v\ w m a m V m \> where Wm, am, and Vm < JM/t V p ' f, refer to the weight per cubic foot, the cross-sectional area, and^ the velocity ot propagation in the metal, and w, a, Vp, to the corresponding quantities for water, PHENOMENA DUE TO ELASTICITY OF A FLUID 225 in the water column. This will increase the pressure to the value 8 which would obtain if the pipe were anchored. So that the effect Dn the maximum pressure attained during this first period zl ~ "V p is zero. The effect on the subsequent history of the phenomenon s complex The net effect, however, is to superpose on the normal Dressure wave a subsidiary wave of high frequency (V OT 4/, tfhere V, M is the velocity of propagation of waves in metal) and of nagnitude /Kro 8 If, as is usual in practice, the pipe is anchored so that no appreci- ible movement of the end is possible, this effect will be small. The second effect of the elasticity of the pipe line is due to the act that, since the walls extend both longitudinally and circum- erentially under piessure, the appaient diminution of volume of he fluid under a given inclement of pressure is gi eater than in a igid pipe The effect of this is to reduce the virtual value of K to a value ', where * i __ i , r I _ 4 K 7 ~" K 2tE\ 5 a vhere r is the ladius of the pipe, t is the thickness of the pipe walls, i) is the modulus of elasticity of the material, i \a is Poisson's latio for he material (approximately 0-28 for iron or steel) If the pipe is so anchored that all longitudinal extension is pre- ented, but that circumferential extension is free, this becomes i _ i , zr K 7 ~~ K *E" ""he use in pressure due to sudden stoppage of motion is now equal g * Hydraulics and its Applications, Gibson (Constable & Co , 1912), p. 235. CD 812) 9 22 6 THE MECHANICAL PROPERTIES OF FLUIDS Valve Shut Suddenly but not Instantaneously If the time of closure t, while being finite, is so small thi t< - - * V ~ V~' the disturbance initiated at the valve has tiavelled a distance s and has not arrived at the open end when the valve i caches it seat. In this case, if each part of the column is subject to th same retardation (a), the relationship force = mass X acceleration wax gives p = , v vVs, and since a = - =2 ?, t x 2/V 'w iK.zo this makes p = * = v\l Ib. per square foot, S V g the value obtained with instantaneous stoppage Whatever the la^ of valve closure then, if this is completed in a time less than / V^ the pressure rise will be the same as if closure were instantaneous Sudden Stoppage of Motion in a Pipe Line of non- Uniform Section In such a case the phenomena become veiy complicatec Let 4> / 2 , 4> &c., be the lengths of successive sections of a rigi pipe, of areas a lt a z , 3 . Following sudden closure of a valve a the extremity of the length 4> a wave of zero velocity and of pressur 63-7 >! Ib. per square inch above normal is transmitted to the June tion of pipes i and 2. Here the pressure changes suddenly to 63-72 above normal. This is maintained in the second pipe during th passage of the wave, and is followed by a change of pressmc to 63-72 at the junction of 2 and 3, and so on to the end of the line. BL immediately the pressure at the junction of i and 2 attains its valu 63'7# 2 > the wave in pipe i is leflected back to the valve as a wave c pressure 63-7^2 an d of velocity ^ v z , to be reflected from the valve a wave of zero velocity and pressure 63-7 {v z (2^ v z ) } above norma This wave then travels to and fro along pipe i , making a complel journey in time 4 -f- V # sec., until such time as the wave in pipe : PHENOMENA DUE TO ELASTICITY OF A FLUID 227 lected from the junction of 2 and 3 with pressuie 63-7^3 above rmal and with velocity v 2 v &) again reaches the junction of nd 2 At this instant it takes up a velocity and pressure depending that at the junction end of pipe i, and as this depends on the latio the lengths of the branches I and 2, it is evident that after the it passage of the wave the phenomenon becomes very involved. Where a pipe is short the period of the oscillations of pressure my point becomes so small that the pencil of an ordinaiy indicator unable to record them, and simply records the mean pressure the pipe. Thus where a short branch of small diameter is used the outlet from a long pipe of larger bore, the pressure as recorded an indicator at the valve will be sensibly the same at any instant in the large pipe at the point of attachment of the outlet branch. Moreover, where a non-uniform pipe contains one section of >reciably greater length than the remainder, this will tend to Dose its own pressure- change on an indicator placed anywhere the pipe. These points are illustrated by the following results of expen- ats by S B Weston * In each case the outlet valve was on the i. length. Details of Pipe Line ii ft of 6-m pipe 58 ,, 2 99 4 4 J " ii ft of 6-in. pipe 58 2 48 il 3 3 48 ij 4 i 82 ft of 6-m pipe 66 , 4 4 I , 2 7 Calc 154 322 Piessures, Pounds per Squaie Inch. Obs Calc Obs Calc 73 129 ist i j-in pipe ij-in pipe 6 9 143 71 127 3-in pipe 2j-m pipe 8 9 and Obs H'5 pipe 715 75 180 65 71 5 61 142 126 355 121 H3 114 1 80 150 45 150 1 80 139 268 203 67 207 268 196 6-m pipe 1 20 49 52 22 90 4-8 H9 62 645 36 II 2 6-6 223 82 97 52 16 7 158 466 122 2OI 99 35 36-8 * Am. Soc. C. E , 1 9th Nov, 1884 228 THE MECHANICAL PROPERTIES OF FLUIDS Sudden Initiation of Motion If the valve at the lower end of a pipe line be suddenly opened, the pressure behind the valve falls by an amount p Ib. per square inch, and a wave of velocity v towards the valve and of pressuie p below statical pressure is propagated towards the pipe inlet. The magnitude of p depends on the speed and amount of opening of the valve, and if the latter could be thrown wide open instan- taneously the piessure would fall to that obtaining on the discharge side. In experiments by the writer* with a 2-|-m. globe valve on a 3f-in. main 450 ft. long, with the valve thiown open through o 5 of a complete turn, the diop in pressure was 40 Ib per square inch, the statical pressure in the pipe being 45 Ib. per squaie inch, and on the discharge side zero With the valve opened through o-io of a turn the drop was 20 Ib. per squaie inch, while with 0-05 of a turn it was n Ib. per squaie inch In each case the time oi opening was less than 0-13 sec (/ V p ) With a pipe so situated that the original statical picssmc is everywhere greater than p, this pressuie wave i caches the pipe inlei with approximately its original amplitude, and at this instant the column is moving towards the valve with velocity v and picssmcy below normal. The pressure surrounding the inlet is howcvci maintained noimal so that the wave letuins from this end with normal picssuie ant with velocity 2v relative to the pipe. At the valve the wave i reflected, wholly or in part, with a velocity which is the difieienc between 2v and the velocity of efflux at that instant, and since th velocity of efflux will now be greater than v, the wave velocity wil be less than v, and the use in pressure less than p above noimal This wave is reflected from the inlet to the valve and heie the cycl is repeated, the amplitude of the pressure wave diminishing lapidl until steady flow ensues. Fig. 4 shows a diagiam obtained undc these circumstances. Where the gradient of the pipe is such that beyond a ccitai point in its length the absolute statical piessuic is less than tl^ drop in pressure at the valve, the motion becomes partly discontinuoi * Gibson, Water Hammer in Hydraulic Pipe Lines (Constable & Co., 1908). PHENOMENA DUE TO ELASTICITY OF A FLUID 229 this point on the passage of the first wave, which travels on to the let with gradually diminishing amplitude. The amplitude with hich it reaches the inlet determines the velocity of the icflected Valve , i. A yr=r?Y Closed Valve open, f \ / v ' Steady f lota Fig 4. Diagram of Ptesauie (per squaie inch) obtained on Sudden Opening of a Valve ave, which will be less than in the preceding case, and under such rcumstances the wave motion dies out rapidly. As the valve opening becomes greater, the efficiency of the valve a reflecting surface becomes less, so that with a moderate opening e pressure may never even attain that due to the statical head Valve opened Valve open. - -L . Steady flout 4olhf-yj"u^ Fig 5 Sudden Opening of Valve his is shown in fig. 5, which is a diagram obtained from the experi ental pipe line when the valve was opened suddenly (time < -i- rough half a turn. /> Wave Transmission of Energy In the systems in common use for the hydraulic transmission energy, water under a pressure of about 1000 Ib. per square inch supplied from a pumping-station and is transmitted through pipe ics to the motor. This method involves a continuous flow of the orking fluid, which in effect serves the purpose of a flexible coupling ;tween the pump and the motor. 2 3 o THE MECHANICAL PROPERTIES OF FLUIDS It is, however, possible to supply energy to a column of flui enclosed in a pipe line, to transmit this in the form of longitudim vibrations through the column, and to utilize it to perform mechanics work at some lemote point. Such transmission is possible in virtu of the elasticity of the column.* If one end of a closed pipe line full of water under a mean pres sure p m be fitted with a reciprocating plunger, a wave of alternat compiession and rarefaction is produced, which is piopagated alon the pipe with velocity V r If the plunger has simple harmoni motion, the state of affairs in a pipe line so long that, at the give instant, the disturbance has not had time to be i effected from il further end, is represented in fig 6 The pressure at each poir Fig 6 will oscillate between the values p m p, and the velocity betwee the values v, where v is the maximum velocity oi the pistoj At the instant in question, particles at A, C, E, and G aie oscillatir to and fro along the axis of the pipe through a distance r on eac side of their mean position, while paiticlcs at B, D, F, and H ai at test. If n be the number of revolutions of the ciank pci secom the wave-length X = V p n ft In a pipe closed at both ends such a state of vibiation is icilectc from end to end, forming a seiies of waves of pressure and veloci whose distribution, at any instant, depends on the latio of the lengi I of the pipe to A. In the cases where / is icspectivcly equal to A/4, A/2, and stationary waves are produced as indicated in fig y."j The exce * A number of applications of this method have been patented by Mi . < Constantmesco. "j. The pressure and velocity oscillate in a "statiomuy " manner, i e there t definite points called " nodes " wheie theic is no change m piessuie and hkewi points where the water does not move See any textbook on Sound, e.g. Datt Sound (Blackte), p 63, "Watson's Physics, Poyntmg and Thomson's Sound, & wheie the subject is fully explained for sound waves. This distribution, where I = A/4, is only possible wheie oscillation at the e A is possible, as where the pipe is fitted with a free plunger, PHENOMENA DUE TO ELASTICITY OF A FLUID 23 1 ^assure at a given point oscillates between equal positive and egative values, the range of pressure being given by the intercept etween the two curves. The velocity at the points of maximum nd minimum pressure, as at A, D, and B in fig. jc, is zero, while t the points C and E, where the variation of pressure is zero, the elocity varies from -\-v to v. In the case where 7 = A/4, the plunger, if free, would continue B (a,) A B (I) Fig 7 Stationary Waves in a Closed Pipe ) oscillate in contact with the end of the column without the appli- ition of any exteinal force. In a pipe fitted with a reciprocating plunger at one end and [osed at the other, the wave system initiated by the plunger will be iperposed on this reflected system. Thus if / = A/2 or A, the r ave initiated by the plunger will be reflected, and will reach the lunger as a zone of maximum pressuie at the instant the latter is Dmpletmg its m-stroke and is producing a new state of maximum ressure. The pressure due to the reflected wave is superposed n. that due to the direct compression, with the result that the pres- ire is doubled. The next revolution will again increase the pres- j.re, and so on until the pipe either bursts or until the rate of dis- pation of energy due to friction, and to the imperfect elasticity of ic pipe walls, becomes equal to the rate of input of energy by the lunger. 232 THE MECHANICAL PROPERTIES OF FLUIDS On the other hand, if the length of the pipe be any odd multiple of A/4, the pressure at the plunger at any instant, due to the reflected wave, will be equal in magnitude but opposite in sign to that primarily due to the displacement of the plunger, and the pressure on the latter will be constant and equal to the mean pressure m the pipe Except for the effect of losses in the pipe walls and in the fluic column, reciprocation may now be maintained indefinitely withou the expenditure of any further energy. For any intermediat< lengths of pipe, the conditions will also be inteimediate and th< wave distribution complex. Instead of closing the end of the pipe at B (fig. 8), suppose 5 piston to be fitted to a crank rotating at the same angular velocity in the same direction, and in the same phase as the ciank at A If the column were continued beyond B, the movement of thi Fig 8 piston would evidently produce in the column a sciics of wave forming an exact continuation of the wave system between A am B. There will now be no reflection from the surface of the piston and if the latter drives its crank and if the resistance is, at ever instant, equal to the force exerted on the piston by the wave system it will take up the whole energy of the waves pioduced by piston A It is to be noted that the piston B may be placed at any point in th pipe so long as its phase is the same as that of the liquid at the poin of connection. If more energy is put in by the piston A than is absoibed by F reflected waves will be formed, and the continuation oi the motio will accumulate energy in the system, increasing the maximur pressure until, as in the case of the closed pipe, the pipe will buisi This may be avoided by fitting a closed vessel, filled with liquic having a volume large in comparison with the displacement of tb piston, in communication with the pipe near to the piston. Altei natively this may be replaced by a spring-loaded plunger. In eithe case the contrivance acts as a reservoir of energy. If the piston is not absorbing the whole of the energy supplied from A, the liqui in this chamber is compressed on each instroke of the piston t PHENOMENA DUE TO ELASTICITY OF A FLUID 233 re-expand on the outstroke, and by giving to it a suitable volume, e maximum pressures, even when the piston B is stationary, ay be reduced to any requned limits If perfectly elastic, the servoir will return as much energy during expansion as it absorbed iring compression, so that the net input to the driving piston is ly equivalent to that absorbed by piston B. In the case of a pipe (fig. 7 c) whose length is one wave-length, d which is provided with branches at C, D, E, and B, respec- ely one-quarter, one-half, thiee-quarteis, and one wave-length >m A, if all the branches are closed, stationary waves will be oduced in the pipe as pieviously described. If now a motor running at the synchronous speed be coupled the branch at D, this will be able to take up all the energy given the column The stationary half-wave between A and D will tiish, being icplaced by a wave of motion, while the stationary ve will still persist between D and B Since there is no pressure vanation at C and E, motors coupled these points, with the remaining branches closed, can develop energy. If a motor be connected at any inteimediate point, part of the >ut of energy can be taken up by the motor The stationary ve will then persist, but be of reduced amplitude between A and motor, the wave motion ovci this region being compounded of => stationary wave and of a travelling wave conveying eneigy With a motor at the end B of the line, not absoibmg all the energy en by the generator A, there is, in the pipe, a system of stationary vcs superposed on a system of waves travelling along the pipe, that there will be no point in the pipe at which the variation of ssuie is always zero It follows that under these conditions a tor connected at any point of the pipe will be able to take some rgy and to do useful work In practice a three-phase system is usually employed, as giving re uniform torque and ease of starting A three-cyhndei gener- % having cranks at 120, gives vibrations to the fluid in thiee es, which are received by the pistons of a three-cylinder hydraulic tor having the same crank angles The mean pressure within system is maintained by a pump, which returns any fluid leaking t the pistons. (D312) 234 THE MECHANICAL PROPERTIES OF FLUIDS Theory of Wave Transmission of Energy The simple theory of the process is outlined below, on th< assumption that the friction loss due to the oscillation of the columr in the pipe is dnectly proportional to the velocity. Where such 5 viscous fluid as oil is used this is true, but where water is used i may or may not be true, depending upon the velocities involved If the resistance is equal to kv z per unit length as with tuibulen motion, an approximation to the true result may be attained b choosing such a frictional coefficient k' as will make k'v = kv z a the mean velocity. At velocities below the critical, k' = ~- ii pounds per squaie foot of the cioss section (Poiseuille) per urn length of the pipe. Consider the fluid normally in a plane at x } displaced fiom tha f\ plane through a small distance u, so that its velocity v = - Th at difference of pressure on the ends of an element of length S#, du to the variation in compression along the axis oi the pipe, is equal t K! dx and the equation of motion becomes dt 2 4 9# 2 4 <tf a 4 9^ ' ^ 0,0 == ^ o v "~ , > (i) or 9r oiC where a = A/ an< ^ ^ - ^ > v p pd z If 6 is small compared with spm, wheie ?z is the ficquency of t] vibration, a solution of equation (i) is u = u Q e * sm2-rrn(t -Y (2) which represents an axial vibration throughout the column, of maj mum amplitude % at the end where x = o. -J 6 A At any other point the maximum amplitude is e ? , grad PHENOMENA DUE TO ELASTICITY OF A FLUID 235 ally diminishing along the pipe owing to the friction term repre- sented by the term b. The excess pressure p, at any instant and at any point, is equal j-rdu to K -, i.e. ax b T ~^x _ 2777Z / X\ p = Kw e X cos27r#( t - - ) approx., a \ a/ ind the maximum excess pressure, p max , at any point, 27rwKw ~i- f N = - * a ................. (3) a The velocity of a particle at x is equal to du -t-* /_. si = 2rrmi e a cos27rw( -- 9i \ a ind the raaximum velocity, The eneigy transmitted by the excess pressure the mean >ressure conveys no energy on the average across a given section f the pipe in time Bt is equal to -fr ~ -~~ K -- 4 4 dx at The mean rate of transmission of eneigy per second over each stroke tf the plungei is thus given by rf K 4 There r is the duration of a stroke, i e. of a half-cycle. da K r du 8u ^ 1 __ __ -dt t TJ ox ot / x\ Wilting 2irn(t ) = a, \ a] dt 27m 2,7m - . , Kd z r du 'du, this becomes / - da 4 J ox ot TrK * (midu^fe * f <;) 236 THE MECHANICAL PROPERTIES OF FLUIDS and the horse-power transmitted, if the foot be the unit of length, is obtained by dividing expression (5) by 550. The loss of energy per unit length of the pipe, due to friction s and converted into heat, is 9E b ~5~ ~*^> ox a and the efficiency of transmission, through a pipe line of length / is bi E z ~ EO = i - . It should be noted that in any application of these results, if th< calculations are in English units, w _ 62-4 P g ~ 3^2 for water, while the value of p, is to be taken in pounds per squar foot, and the pipe diameter in feet. For a more detailed investigation of the theory, which become complex when a complicated pipe system is used, Mi Constant nesco's original papers should be consulted*. There is an exceedingly close analogy between wave it am mission by Constantmesco's system and alternating-cm rent electr power transmission; in fact, in the " three-pipe system " tl known facts of three-phase electrical cngmcciing can be appln with scarcely any except verbal changes. * The Theory of Somes (The Piopnetois of Patents Conti oiling Wave Tuir mission, 132 Salisbury Square, E.C , 1930) NOTE The foiegomg theory of wave tiansraission is due to H Moss, D.J See also Proc Insf Mech Eng , 1923. CHAPTER VII The Determination of Stresses by Means of Soap Films When a straight bar of uniform cross section is twisted by the application of equal and opposite couples applied at its two ends, it twists in such a way that any two sections which are separated by the same distance are lotated relative to one another through the same angle. The angle through which sections separated by a unit length of the bar are twisted relatively to one another is called the " twist ", and it will be denoted by the symbol 5 throughout this chapter. If the section is circular, particles of the bar which originally lay in a plane perpendicular to the axis continue to do so aitei the couple has been applied The couple is tiansmitted thiough the bar by means of the shearing force exeited by each plane section on its neighbour. The shearing stress at any point is, in elastic materials, propoitional to the shear strain, or shear In the case where a bar of circular cross section is given a twist 5, the shear evidently increases from zero at the axis to a maximum at the outer surface of the bar; at a distance r from the axis it is in fact r% If two series of lines had been drawn on the surface of the untwisted bar so as to be parallel and peipen- dicular to the axis, these lines would have cut one another at light angles. After the twist, however, these lines cut at an angle which difteis from a right angle by the angle r5, which measures the shear at the point in question. The shearing strain at the surface of any twisted bar can in fact be conceived as the difference between a right angle and the angle between lines of particles which were ongm- ally parallel and peipendicular to the axis. In the case of bars whose sections are not circular, the particles which originally lay in a plane perpendicular to the axis do not continue to do so after the twisting couple has been applied; the cross sections are warped in such a way that the shear is increased 837 238 THE MECHANICAL PROPERTIES OF FLUIDS in some parts and decreased in others. In the case of a bar of elliptic section, for instance, the point on the suiface of the bar where the shear is a maximum is at the end of the minor axis, while the point where it is a minimum is at the end of the major axis If the sections had remained plane, so that the shear at any point was proportional to the distance of that point from the axis of twist, the leverse would have been the case. The warping of sections which were originally plane is of funda- mental importance m discussing the distribution of stiess in bent 01 twisted bars. It may give rise to very large increases in stress In the case where the section has a sharp internal corner, for instance, it gives rise to a stress there which is, theoretically, infinitely great. The method which has been used to discuss mathematically the effect of this warping is due to St. Venant.* If co-ordinate axes Ox, Oy be chosen in a plane perpendicular to the axis of the bai , and if <j> represents the displacement of a paiticle from this plane owing to the warping which occuis when the bar is twisted, then St. Venant showed that </> satisfies the equation and that it must also satisfy the boundaiy condition _i = y cos(xn] x cos(yti), . (2) on fit wheie represents the rate of change of ^ m a dnection perpendiculai 9/2 to the boundary of the section, and (xn), (yn) icpiescnt the angles between the axes of x and y respectively and the normal to the boundary at the point (x, y). Functions which satisfy equation (i) always occur in paiis. II ift is the function conjugate to </>, i e. the other membei ol the pan ijj is related to </> by the equations d A = ?, ^ = -, ........ (3) dx 9y' "dy fix and ijj also satisfies (i). In the case under consideiation it turns oui that it is simpler to determine t/r and then to deduce <j> than to attempl to determine c/> directly. From (i), (2), and (3) it will be seen thai to determine ifj it is necessary to find a function ift which satisfies * See Love, Mathematical Theory o/ Elasticity , second edition, Chap. XIV. DETERMINATION OF STRESS BY SOAP FILMS 239 i) at all points of the cross section, and satisfies the equation ~ y cos(xn) x cos(yn) ........... (4) C/u t points on the boundary, where -^- represents the rate of variation OS f ?/( lound the boundary. (j*\} fj *V Now cos(xn) = ~- and cos(yn) = , so that (4) reduces to a ir / \ . - ~ Ij-f^ 2 H~ y z ), that is to say the boundary condition reduces i t/O \ / _j_ constant ........... (5) ""he advantage in using i/j instead of < is that the boundary condition 5) is more easy to satisfy than (2). The problem of the torsion of the bar of any section is therefore educed to that of finding a function ^ which satisfies -f- ^~ = o coc*"* ^/y*" 1 nd (5) There is an alternative, however If a function W be de- ned by the relation W ip %(x 2 -f- y 2 ), then ^F evidently must atisfy the equation t all points of the section, and 1 P = constant .......... (7) t the boundary This function 1 P, besides having a very simple boundary condition, ias also the advantage that it is simply related to the shear, in fact he shearing strain at any point is proportional to the rate of change i W at the point in question in the direction in which it is a maximum. Prandtl's Analogy It has only been possible to obtain mathematical expressions for >, i/, and W in very few cases The stresses in bars whose sections ic rectangles, ellipses, equilateral triangles, and a few other special hapes have been lound, but these special shapes are of little inteiest o engineers. There is no general way in which the stresses in wisted bars of any section can be reduced to mathematical terms The usefulness of equations (6) and (7) does not cease, however vhen W cannot be represented by a mathematical expression. It has 240 THE MECHANICAL PROPERTIES OF FLUIDS been pointed out by various writers that certain other physica phenomena can be represented by the same equations In some cases these phenomena can be measured experimentally fai moic easily than direct measurements of the stresses and strains in i twisted bar can be made. Under these ciicumstances it may b< useful to devise experiments in which these phenomena are mcasuiec in such a way that 1 F is evaluated at all points of the section Th< values thus found for W can then be used to dcteimine the stiessei in a twisted bai . Probably the most useful of these " analogies " is that of Piandtl Considei the equations which icpicscnt the smiace of a soa] film stretched over a hole in a flat plate of the same size and shap as the cioss section of the twisted bar, the film being slightly dis placed from the plane of the plate by a small picssme p. If y be the surface tension of the soap solution, the equation o the surface of the film is P52,v )2 W A 1 _L ~ JL P. o (K\ o o n ~ r, r - v^, ., i o i cx l fly 2y wheie % is the displacement oi the film horn the plane ot ,vy and and 3; are the same co-oidmates as beioie. Round the bouncUn i e the edge of the hole, % = o. It will be seen that if z is measiued on such a scale that 1 F ~ ^yz/j then equations (6) and (8) aic identical The boundaiy conihtioi are also the same. It appeals thcreloie that the value oi 1 F, coin spending with any values of x and y, can be lound by mcasuiing ll quantities pjy and % on the soap film. In other woids the soap film is a guiphical icpicscntation of tl function *Ffor the given cioss section. Actual values ol 1 F can I obtained from it by multiplying the oidimites by ^y//> To complete the analogy it is necessary to bung out the due connection between the measurable quantities connected with tl film and the elastic properties of the twisted bar If N is the modulus of rigidity of the matei ml and % the twi of the bai, the shear stress at any point of the cross secti( can be found by multiplying the slope of the W surface at tl point by N, so that, if a is the inclination oi the bubble to tl plane of the plate, the stress is (9) DETERMINATION OF STRESS BY SOAP FILMS 241 The torque T q on the bar is given by , dy, or T, = N5CV, ................... (10) where V is the volume enclosed between the film surface and the plane of the plate. The contour lines of the soap film in planes paiallel to the plate correspond to the " lines of shearing stress " in the twisted bar, that is, they run parallel to the dnection of the maximum shear stress at every point of the section. It is evident that the torque and stresses m a twisted bar of any section whatever may be obtained by measuring soap films in these respects. In order to obtain quantitative results, it is necessary to find the value of in each expenment. This might be done by measuring P y and p directly, but a much simpler plan is to determine the curva- ture of a film, made with the same soap solution, stretched over a circular hole and subjected to the same piessure difference, p, be- tween its two surfaces as the test film. The curvature of the circular film may be measured by obseivmg the maximum inclination of the film to the plane of its boundary If this angle be called /? then _ TT T ~~ -- 5' .......... ' p snip where h is the radius of the circular boundary The most convenient way of ensuring that the two films shall be under the same pressure, is to make the circular hole m the same plate as the expeiimental hole. It is evident that, since the value of ^.y/p for two films is the same, we may, by comparing inclinations at any desired points, find the ratio of the stresses at the corresponding points of the cross section of the bar under investigation to the stresses m a circular shaft of radius h under the same twist. Equally, we can find the ratio of the torques on the two bars by comparing the displaced volumes of the soap films. This is, in fact, the form which the investigations usually take. 24 3 THE MECHANICAL PROPERTIES OF FLUIDS As a matter of fact, the value of can be found fiom the test- * film itself by integrating round the boundary, a, its inclination to the plane of the plate If A be the area of the cross section, then the equilibrium of the film requires that 1 2y smack = pA (12) This equation may be written in the foim 4.v area of cross section TV ___ 2 V ~ p (perimeter of cioss section) X (mean value of sma) By measuring a all round the boundaiy the mean value of sma can be found, and hence may be deteimined This is, however, P more laborious in practice than the use of the cnculai stanclaid It is evident that if the radius of the cncular hole be made equal zA. to the value of , wheie A is the area and P the peumeter of the test hole, then sinjS = mean value of sino, It is convenient to choose the radius of the ciicular hole so that it satisfies this condition, in Older that the quantities measuied on the two films may be of the same oidei of magnitude. Experimental Methods It is seen fiom the mathematical discussion given above that, in ordei that full advantage may be taken of the mfoi matron on toision which soap films are capable of furnishing, apparatus is ic- quired with which three kinds of measuiements can be made, namely: (a) Measurements of the inclination of the film to the plane of the plate at any point, for the determination of stresses. (b) Determination of the contour lines of the film. (c) Comparison of the displaced volumes of the test film and circular standard for finding the corresponding torque latio. The earliest appaiatus designed by Dr. A. A Griffith and G. I. Taylor for making these measurements is shown in fig. i (see Plate).* The films are formed on holes cut in flat aluminium plates * From Proc. Inst. Mcch, Eng, t I4th December, 1917. o I'ai ing p<.if>e 2(3 DETERMINATION OF STRESS BY SOAP FILMS 243 the icquired shape. These plates are clamped between two Ives of the cast-iron box A (fig. i). The lower part of the box :es the form of a shallow tray J in. deep blackened inside and pported on levelling screws, while the upper portion is simply iquare frame, the upper and lower surfaces of which are machined rallel. Arrangements are made so that air can be blown into the ver part of the box in order to establish a difference in pressure tween the two sides of the film. In older to map out the contour lines of the film, i.e. lines of ual z, or lines of equal }F in the twisted bar, a steel point wetted th soap solution is moved parallel to the plane of the hole till it 5t touches the film. The point being at a known distance from the me of the hole marks a point on the film where z has the known lue. The requiied motion is attained by fixing the point (shown C in fig. i) to a piece of plate glass which slides on top of the 3t-non box. The height of the point C above the plate is adjusted fixing it to a micrometer screw B In older to record the position of the point C when contact with e film is made, the micrometer carries a iccordmg point D, which mts upwards and is placed exactly over C The record is made a sheet of paper fixed to the boaid E, which can swing about a nzontal axis at the same height as D To maik any position ot c scicw it is meiely necessary to puck a dot on the paper by mgmg it down on the lecoidmg point The piocess is repeated a gc number of times, moving the point to difleient positions on the m but keeping the setting of the rmciometer B constant In this ly a contour is pucked out on the papci. To puck out another ntour the setting of B is altered. The photogiaph shows an actual 2ord in which foui contouis traced in this way have been filled with a pencil. The section shown is that of an aeroplane pro- ller blade. To measure the inclination of the film to the plane of the plate e " auto-collimatoi " shown in figs ia and 2b was devised. Light )m a small electiic bulb A is reflected fiom the surface of the tn thiough a V-neck B and a pin-hole eyepiece C placed close the bulb. Dnect light fiom the bulb was kept away from the eye by small screen. The inclinometer D, which measuies the angle nch the line of sight makes with the vertical, consists of a nit level fixed to an aim which moves over a quadrant graduated degiees. In using the auto-collimator the soap-film box is 244 THE MECHANICAL PROPERTIES OP FLUIDS adjusted till the plane of the hole acioss which the film is sti etched is horizontal. The volume contained between the film and the plane of the hole can be measured m a variety of ways. One of the most simple is to lay a flat glass plate wetted with soap solution over the test hole in such a way that all the air is expelled from it. The volume contained between the spherical film and plane of the circular hole is then in- creased by an amount equal to the volume lequired. This increase in volume can. be determined in a variety of ways, one of the simplest being to make measuiements with the auto-collimator of the inclina- tion of the spherical film at a point on its edge. Accuracy of the Method Strictly speaking, the soap-film surface can only be taken to represent the torsion function if its inclination a is everywhere so small that sina = tana to the required older of accuiacy This would mean, however, that the quantities measuied would be so small as to render excessive experimental eriois unavoidable A compromise must therefore be effected. In point of fact, it has been found from expeiiments on sections for which the toision function can be calculated, that the ratio of the stiess at a point m any section to the stress at a point in a circular shaft, whose ladius 2A equals the value of for the section, is given quite satistactonly by the value of where a and B are the respective inclinations of the smjS ^ F corresponding films, even when a is as much as 35 Similaily, the volume ratio of the films has been found to be a sufficiently good approximation to the corresponding torque ratio, foi a like amount of displacement. In contour mapping, the greatest accuracy is obtained, with the apparatus shown in fig. i, when J3 is about 20. That is to say, the displacement should be rather less than for the other two methods of experiment In all soap-film measurements the experimental eiror is natuially aA greater the smaller the value of . Reliable results cannot be zA. obtained, in general, if is less than about half an inch, so that a shape such as a rolled I beam section could not be treated satis- DETERMINATION OF STRESS BY SOAP FILMS 245 ;tonly in an appaiatus of convenient size. As a matter of fact, wever, the shape of a symmetrical soap film is unaltered if it be ?ided by a septum or flat plate which passes through an axis of nmetry and is normal to the plane of the boundary. It is there- e only necessary to cut half the section in the test-plate and to ice a normal septum of sheet metal at the line of division. An beam, for instance, might be treated by dividing the web at a stance from the flange equal to two or three times the thickness of 3 web It has been found advisable to carry the septum down rough the hole so that it projects about J in. below the under side the plate, as otherwise solution collects in the corners and spoils 5 shape of the film. TABLE I SHOWING EXPERIMENTAL ERROR IN SOLVING STRESS EQUATIONS BY MEANS OF SOAP FILMS Radius Sin Error ' Error > & Square side, 3 in I" D* Deg I0 33 5S 2119 IS36 I4 QO : 500 + 24 -07 15 29 u at 34 i 364 r 337 i -550 -|-i o i o I2 ? 6 37t 2432 1263 1240 i 234 +24 +05 i 410 31 10 24 oo i 296 i 270 i 276 +i 6 o 5 , Ellipse 4 X o 8 in i 196 35 35 26 58 i 331 i 293 i 286 +3 5 +o 5 i Rectangle 4 X 2 in i 333 31 70 22 36 i 418 i 380 i 395 +i 6 i i ' Rectangle 8 X 2 in i 60 34 83 27 23 i 279 i 247 i 245 +27 +02 ! Infinitely long rec-j 6 6 6 OQO +o6 + tangle i in wide J j ^ o v o Ellipse 3 X i ni The values set down in Table I indicate the degree of accuracy 3tamable with the auto-colhmator in the determination of the aximum stresses in sections for which the toision function is known hey also give an idea of the sizes of holes which have been found tost convenient in practice. The angles given are a, the maximum iclination at the edge of the test film, and /3, the inclination at the * On 4-m. length. 246 THE MECHANICAL PROPERTIES OF FLUIDS ''A edge of the circular film of radius - . They are usually the means of about five observations and are expressed in decimals of a degree. The last two columns show the errors due to taking the ratio of angles and the ratio of sines respectively as giving the stress ratio. The error is always positive for a//?, and its mean value is 1-98 per cent. In the case of the average error is only 0-62 per cent. smj8 In only two instances does the error reach i per cent, and m both it is negative The presence of sharp corners seems to introduce a negative error which is naturally gieatest when the corneis aic neaiest to the observation point Otheiwise, theie is no evidence that the error depends to any great extent on the shape Nos 4, 5, 7, and 8 in the table are examples of the application of the method of noimal septa described above in which the film is bounded by a plane per- pendicular to the hole at a plane of symmetiy. Table II shows the results of volume determinations made on each of the sections i to 8 given in the picvious table. TABLE II SHOWING EXPERIMENTAL ERROR IN DETERMINING TORQUES BY MEANS OF SOAP FILMS No. *8. Section Equilateral triangle height, 3 in. Square side, 3 m Ellipse semi - axes\ 2 in X i m / Ellipse- 3 m X i in Ellipse' 4 in X o 8m Rectangle- sides, 4") m. X 2 m . . / Reactangle 8m. X \ 2 in. . . / Infinitely long rec-\ tangle . . . / Maximum Observed Calculated Inclma- Volume Toiquc Knot tion Deg Ratio Ratio I'ci cent 32 06 1953 I 985 -i 6 3039 I 416 I 43 2 i i 3050 I 143 I 133 -1 09 31 oi 36 12 2147 3 041 2 147 3 020 -j-o 7 3133 1456 M75 -13 3528 1-749 174-1 +03 36 oo 0858 0-848 + 12 * On 4-m length. DETERMINATION OF STRESS BV SOAP FILMS 24? The average error is o 89 per cent. In four of the eight cases con- idered the error is greater than i per cent and in three of these it i negative. One may conclude that the probable error is somewhat reater than it is for the stress measurements, and that it tends to e negative. Its upper limit is probably not much in excess of 2 er cent. The remarks already made regarding the dependence of ccuracy on the shape of the section apply equally to torque measure- lents When contour lines have been mapped, the torque may be found Lorn them by integration. If the graphical work is carefully done, tie value found in this way is rather more accurate than the one btamed by the volumetric method. Contours may also be used 3 find stresses by differentiation, that is, by measuring the distance part of the neighbouring contour lines; but here the comparison 5 decidedly m favour of the direct process, owing to the difficulties iseparable from graphical differentiation. The contour map is, eveitheless, a very useful means of showing the general nature of tie stress distribution throughout the section in a clear and com- >act manner. The highly stressed parts show many lines bunched Dgether, while few traverse the regions of low stress, and the direc- Lon of the maximum stress is shown by that of the contouis at eveiy iomt of the section. Furtheimore, the map solves the torsion pro- ilem, not only for the boundary, but also for every section having tie same shape as a contour line Example of the Uses of the Method The example which follows serves to illustrate the use of the oap-film apparatus in solving typical problems in engineeiing tesign. It is well known that the stiess at a sharp internal corner of a wisted bar is infinite or, rather, would be infinite if the elastic equa- tons did not cease to hold when the stress becomes veiy high. If he internal corner is rounded off the stress is reduced; but so far LO method has been devised by which the amount of reduction in tram due to a given amount of rounding can be estimated. This iroblem has been solved by the use of soap films. An L-shaped hole was cut in a plate. Its arms were 5 in. long >y i m wide, and small pieces of sheet metal were fixed at each end, lerpendicular to the shape of the hole, so as to form normal septa. The section was then practically equivalent to an angle with arms of 248 THE MECHANICAL PROPERTIES OF FLUIDS infinite length. The radius in the internal corner was enlarged step by step, observations of the maximum inclination of the film at the internal corner being taken on each occasion. The inclination of the film at a point 3-5 in. from the corner was also observed, and was taken to represent the mean boundary stress in the arm, which is the same as the boundary stress at a point far fiom the corner The ratio of the maximum stress at the internal corner to the mean stress in the arm was tabulated for each radius on the internal corner. The results are given in Table III. TABLE III SHOWING THE EFFECT OF ROUNDING THE INTERNAL CORNER ON THE STRENGTH OF A TWISTED L-SHAPED ANGLE BEAM Radius of Internal R atlo Maximum Stress Cornei Stress in Arm Inches o 10 I 890 o 20 I 540 o 30 I 480 040 1-445 0-50 I 430 0-60 I 420 070 t'^S 80 i 416 r oo i 422 1 50 I 500 2 OO I 660 It will be seen that the maximum stiess in the internal corner does not begin to increase to any great extent till the ladms of the corner becomes less than one-fifth of the thickness of the aims A curious point which will be noticed in connection with the table is the minimum value of the ratio of the maximum stress to the stiess in the arm, which occurs when the radius of the corner is about o 7 of the thickness of the aim. In fig. 3 is shown a diagram representing the appearance of these sections of angle-irons. No. i is the angle-iron for which the radius of the corner is one- tenth of the thickness of the arm. This angle is distinctly weak at the corner. DETERMINATION OF STRESS BY SOAP FILMS 249 In No. 2 the radius is one-fifth of the thickness. This angle-iron is nearly as strong as it can ? be. Very little increase in strength is effected by round- ing off the corner more than i this. No. 3 is the angle with minimum ratio of stress in corner to stress in arm. A further experiment was 2 made to determine the extent of the region of high stress in /" ' angle-iron No. i For this 3 purpose contour lines were mapped, and from these the slope of the bubble was found at a number of points on the F IS 3 line of symmetry of the angle-iron. Hence the sti esses at these points were deduced, The results are given in Table IV. TABLE IV SHOWING THE RATE OF FALLING-OFF OF THE STRESS IN THE INIERNAL CORNER OF THE ANGLE-IRON Distance fiom Ratio Stt ess at Point Boundary * Boundary Sti ess m Arm" Inches 000 1-89 O 05 i 36 O 10 I 12 O'2O 077 0-30 o 49 o 40 0-24 o 50 o oo It will be seen that the stress falls off so rapidly that its maximum value is to all intents and purposes a matter of no importance, if the material is capable of yielding. If the material is brittle and not ductile a crack would, of course, start at the point of maximum stress and penetrate the section. 250 Comparison of Soap -film Results with those obtained in Direct Torsion Experiments As an example of the order of accuracy with which the soap- film method can predict the toisional stiffness of bais and girders of types used in engineering, a comparison has been made with the experimental results of Mr. E. G. Ritchie.* The toisional stiffness of any section can be represented by a quantity C such that torque = CN, wheie N is the modulus of rigidity. C has dimen- sions (length) 4 . In Table V column 2 is given the value of C found by soap-film methods, while m column 3 is given the conesponding experimental results taken from Mr. Ritchie's papei . Section Angle: Angle. 1-175 X 1-175 in. i-oo X i oo in. Tee- 1-58 X 1-58 m. _ I-beam: 5-01 X 8-02 in. I-beam. 3-01 X 3-00 in. I-beam: 1-75 X 4-78 in Channel 1 o 97 X 2 oo m. TABLE V C (Soap Film) 0-01234111 4 o 0044 m 4 01451 m 4 1 160 in 4 o 1179 m 4 0-0702 m. 4 0-0175 m 4 C ('Dnect Toision Experiments) o 01284 m 4 0-00455 m 4 o 01481 in l 1-140 in 4 o 1082 in a o 0635 m 4 o 0139 m 4 Torsion of Hollow Shafts The method descubed above must be modified when it is desned to find the torsion function loi a hollow shaft In this case the lunc- tion satisfies the equation (6) and the boundaiy conditions aic W = constant on each boundary, but the constant is not necessanly the same for each boundary. In order to make use of the soap-film analogy it is therefore necessaiy to cut a hole m a flat sheet of metal to lepiesent the outer boundary, and to cut a metal plate to leprc- sent the inner boundary. These are placed in the conect relative positions in the appaiatus shown in fig. i, and they arc set so that they lie in parallel planes. The soap film is then stretched acioss the gap between them. The planes containing the two boundaries must be parallel, *A Study of the Circular Arc Bow Girder, by Gibson and Ritchie (Constable & Company, 1914). it they may be at any given distance apart and yet satisfy the con- tion that IP" = constant round each boundary. On the other hand ie contour lines of the film, and hence the value of IP, will vary eatly according to what particular distance apart is chosen. The ilution of the torsion problem must be quite definite, so that ust be possible to fix on the particular distance apart at which the anes of the boundaries must be set in order that the soap film retched on them may represent the requned torsion function. o do this it is necessary to consider again the function <, which presents the displacement of a particle from its original position ving to the warping of plane cross sections of the twisted material. his function < is evidently a single-valued function of x and y, ;. it can have only one value at every point of the material. In neral the values of W found by means of the soap-film apparatus ) not correspond with single-valued functions <j). On the other hand, ere is one particular distance apart at which the planes of the boun- ties can be placed so that the W function does correspond with single-valued function 0. To solve the torsion problem we must id this distance. If <j> is single- valued, I ds --= o when the integral is taken lound her boundary; and since = --, this condition reduces to 9* 8 * ~ds ~ o. Substituting W = ift ~l(x z + y z ) and remem- 7tl ring that 3 t n . 9X dy 3 /r 2 -f- y\ . dx 3 /# 2 -f y z \ dx dy ;. ftf -j- y*\ = JL ( ' J \ -j --- I .. - !_^_ \ = y x~, dn 8/2 dy \ 2 ] dn dx\ 2 / 9i 9^ will be seen that dtft, fds= ~- on J on lere A represents the area of the boundary. The condition that shall be single-valued is therefore f&Fj , . . ~-ds-2&, ......... (14) J 05) if erring again to the soap-film analogy, and putting W will be seen that (15) is equivalent to '/' = A.p (16) 552 THE MECHANICAL PROPERTIES OF FLUIDS Equation (16) applies to either boundary; it may be compared with equation (12), which there applies only to a solid shaft. Taking the case of the inner boundary, it will be noticed that Ap is the total pressure exerted by the air on the flat plate which constitutes the inner boundary, zylsmads* on the other hand is the vertical component of the force exerted by the tension of the film on the inner boundary. Hence the condition that < shall be single-valued gives rise to the following possible method of determining the posi- tion of the inner boundary. The plate representing it might be attached to one arm of a balance. The film would then be stretched across the space between the boundaries, and if the outer boundary was at a lower level than the inner one the tension in the film would drag the balance down. The pressure of the air under the film would then be laised till the balance was again in equilibimm. The film so produced would satisfy condition (16). As a matter of fact this method is inconvenient, and another method based on the same theoretical principles is used in practice, but for this and further developments of the method to such questions as the flexure of solid and hollow bars the reader is referred to Mr. Griffith's and Mr. Taylor's papers published in 1916, 1917, and 1918 in the Reports of the Advisoiy Committee for Aeronautics. Example of the Application of the Soap -film Method to Hollow Shafts As an example of the type of research to which the soap-film method can conveniently be applied, a brief description will be given of some work undertaken to determine how to cut a keyway in the hollow propeller shaft of an aeroplane engine, so that its strength may be reduced as little as possible. These shafts used to be cut with sharp re-entrant angles at the bottom of the keyway, and they frequently failed owing to cracks due to torsion which started at the re-entrant corners. It was proposed to mitigate this evil by putting radii or fillets at these corners, and it was requited to know what amount of rounding would make the shafts safe. The shafts investigated were 10 in external and 5-8 in. internal diameter. This was not the size of the actual shafts used in aero- * The factor 2 comes in owing to the fact that y is the surface tension of one surface md the film has two surfaces. DETERMINATION OF STRESS BY SOAP FILMS 253 )lanes, but it was found to be the size which gave most accurate esults with the soap-film apparatus. Some of the results of the experiments are shown graphically in he curve in fig. 4.* In this curve the ordinates represent the maximum SO I Outside diam of shaft to", 5 v Viam of hol&(concentHc) 5 $ Depth of keyway / o., VMft/i afkeunau z 5 3 O 20 The maximum stress is given as a multiple cf the maximum stress ma similar shaft without hey way (A) When the two shafts tti e twisted through the same angle (B) When they are subjected to the same torque The radius of the filkt isgtvenas a fraction of the depth oftKe key way The dotted curves show the respective stresses in the middle of the key way 01 02 03 04 05 O6 O 7 RADIUS OF FILLET IN CORNER OF KEYWAY Fig 4 Toraional Strength of Hollow Shaft with Keyway shear stress, on an arbitrary scale, while the abscissse lepresent the adms of the fillet, in which the internal corners of the keyway were -ounded off It will be seen that the shaft begins to weaken rapidly .vhen the radius is less than about 0-3 m . * This diagram and also that shown in fig 5 aie taken from Messrs. Griffith md Taylor's report to the Advisory Committee for Aeronautics, 1918. 254 THE MECHANICAL PROPERTIES OF FLUIDS The lines of shearing stress, i.e. the contour lines of the soap film, are shown in fig. 5 for the case when the radius of the fillet is 0.2 in. S Lines of Shearing Stress m the Torsion of a Hollow Shaft with Keyway It will be seen that the lines of shearing stress ai c crowded together near the rounded corner of the keyway. CHAPTER VIII Wind Structure During the present century great advances have been made in 5 field of aviation, and problems, some of them entirely new, others der a new guise, have presented themselves. Among the latter iy be included the problem of wind structure Slight changes of 5 wind, both in direction and in magnitude, are of little account some problems where only the average effect of the wind is of f moment On the other hand, for the aviator these small changes often of far greater moment than the general drift, especially en his machine is either leaving or approaching the ground. Now s just at this point that the irregularities are often greatest. Before proceeding to deal with the cause of these vanous irregu- ities, let us consider what are the governing factors in the move- nt of a mass of air over the surface of the globe. Apart from the difficulties of dynamics, the general problem is 2 of much complexity. In the first place, the surface of the earth not at all uniform. It consists of land and water surfaces, and a d suiface and a water surface behave quiet differently towards ar radiation, so that air over one area becomes more warmed up n that over another. Further, the land areas are divided into icrts and regions rich in vegetation, flat plains, and mountain ranges, am, water vapour, to whose presence in the atmosphere nearly all teorological phenomena are due, while being added at one place, not subtracted simultaneously at another, so that the amount sent in the atmosphere varies veiy inegulaily. These and other tors tend to render an exact mathematical solution of the problem ctically impossible. An approximate determination, however, of the effect of the th's rotation on the horizontal distribution of pressure when the moves over the surface of the globe in a simple specified manner, i be found. To obtain this approximate solution of the problem, we shall 255 256 THE MECHANICAL PROPERTIES OF FLUIDS assume that the air is moving horizontally* with constant linear velocity v, i.e. that a steady state has been reached. The forces acting on a particle of air, in consequence of its motion, under these conditions at a place P in latitude < arise from two causes, (i) the rotation of the earth, and (2) the curvature of the path in which the particle is moving at the instant, relative to the earth. The problem before us therefore is (i) to find the magnitude of the accelerations arising from these causes and (2) to show how the forces required for these accelerations in the steady state are provided by the pressure gradient. Consider first the effect of the rotation of the earth on great-circle motion. We shall suppose the particle is constrained, by properly adjusted pressure gradients, to move in a great circle through P with uniform velocity. The particle is therefore supposed to move in a path which is rotating in space about an axis passing through its centre. The rotation of the earth takes places about its axis NS, fig. i. The great circle Q'PQ is the specified path of the paiticle. The earth's rotation may be resolved by the parallelogram of rotations into two rotations about any two directions in a plane containing NS Let these two directions be the two perpendicular lines OP and OW, where O is the centre of the earth and P the point in latitude ^ referred to above. If the angular velocity of the earth about SN be co, then the component angular velocities are co cos</> about OW and CD sm< about OP. As the two axes are mutually perpendicular, it follows that any particle in the neighbourhood of P is in the same relation to OW as a particle on the equator is to ON. But a particle on the equator moving with uniform horizontal velocity has an acceleration directed only perpendicular to the axis ON, and therefore its horizontal velocity is not affected by the rotation about ON. Similarly the horizontal velocity of a paiticle near P is affected only by the component co sin< about OP, and not by the perpendicular component co cos< about OW. We need consider therefoie only the effect of the component co sm<ji. When the particle crosses the point P, it will travel a distance PA = vdt (see fig. id) in time dt, as the velocity is v. In the same interval of time, the line along which the particle started will have moved into the position PA', so that the element of arc ds = AA = PAco s'w.(j)dt. *Ie.m a plane perpendicular to the direction of the force compounded of the force of gravity and the centrifugal force. WIND STRUCTURE 257 Also ds or AA', which is described in a direction perpendicular to PA, may, by the ordinary formula, be expressed in the form North t Horizontal Plane at- P tacit' e t-Eaaf &/W 2 > where /is an acceleration in the direction perpendicular to the direction of motion Hence \f(dt} z = PAw sin^d? = since PA = vdt; i.e. / = 258 THE MECHANICAL PROPERTIES OF FLUIDS Hence the transverse force (F) necessary to keep a mass (m] of ai moving along a great circle, in spite of the rotation of the eaith, i given by F = mf = zmvai sin^, (i) acting, in the northein hemisphere, towards the left, in the southerr towards the right, when looking along the direction of th( wind. This expression, which is very nearly correct, shows that th deflective force due to the rotation of the eaith* on a mass of moving air is (i) directly proportional to the mass, to the horizontal velocity to the earth's angular velocity, and to the sine of the latitude of the place; (2) independent of the direction of the great circle, i.e. of 6 (fig. i); (3) always perpendicular to the instantaneous direction oi motion of the air and therefore without influence on the velocity with reference to the surface; (4) opposite to the direction of the earth's rotation. When the air moves, as specified, in a great circle, the acceleiation zvo) sin^ is the only transverse acceleration in the hoiizontal plane, for the acceleration arising from the curvature of the path relative to the earth (which exists even if co were zero) is radial and theiefoie has no appreciable component in the horizontal plane. Now suppose that the path is not a great circle but a small one, R'P (fig. i) In addition to the term 2,va> sm^ there will now be a term arising from the curvature of the path This term is inde- pendent of oj. Let fig. 2 be a section of the sphei e through a diameter of the small circle, PR' being the diameter. The path of the air at P is now curved, and if r is the radius of curvature of the path at P, 2,3 in the horizontal plane, is the acceleration in the horizontal plane r arising from the curvature of the path. But this acceleration is also the horizontal component of where / is PM, i.e. the radius of r curvature of the small circle. If a is the angular radius of the small circle it is also the inclination of the horizontal plane to the plane of the small circle (see fig. 2), hence = cosa: r r .*. r' = r cosa, * I.e the force F, reversed. WIND STRUCTURE 2 59 e. N (fig. 2) is the centre of curvature of the path in the horizontal lane. It is also clear from fig. 2 that R sina = r f , hence = cosa r r' . cosa = - cota. R sina R Fig 2 Relation between Radn of Curvature of the Path on the Earth and the Path in the Horizontal Plane PM = r' PN = r'/cosa = > PG the air is moving freely in space, i e if the barometric pressure umfoim, the resultant horizontal acceleiation is zero, i.e. 2VO) sm<j> -f- cota = o. [ence " free " motion is only possible when out of these accelera- ons has the opposite direction to the other, and = ~ cota 260 THE MECHANICAL PROPERTIES OF FLUIDS numerically, i.e. when the acceleration due to path curvature balances that due to the earth's rotation. When the barometric pressure is not uniform, we proceed thus: If the particle or element of air at P occupies the volume of a small cylinder, of length S in the direction of the outward drawn normal at P to the path of the air in the horizontal plane, and of unit cross-sectional area, the force on the air in the inward direction due to variation of pressure is ( ^n ) . The mass of air is p8?2 where \dn I p is the density at P, hence 8p s fi f . , . w 2 cotal -on = pt)n\ 2vaj sm< + dp ; i prf cota ., s " in. = p ZJa>sm ^+ R ............ (*) The formula is true for positive and negative values of v, lemem- 8i> bering that -j- is the gradient of pressure in the outward direction of the normal, and that the rotational term in the acceleration is towards the left hand when looking along the direction of the wind in the northern hemisphere. The two cases of cyclonic and anticyclomc wind (i.e. -f and ~v) are shown in fig. 3. The forces indicated in this figure are those required to keep the air in its assumed path, relative to the earth. These forces are provided by the pressure gradient. If we take the numerical value of the pressure gradient and the wind speed, then , , n^ + T5- cota on K for the cyclone, where ~ is the rate of rise of pressure outwards. For the anticyclone, " n , pv* n^ - - cota, ........... (30) t where ^- is the rate of rise of pressure inwards. Both cases are included in (2) without ambiguity. WIND STRUCTURE 261 These expressions give a value of the wind velocity called the f adient wind velocity. The direction of this gradient wind according > the previous reasoning is along the isobars, and is such that to one loving with it in the northern hemisphere, the lower pressure is i the left hand. It must be distinctly understood that in the above cpressions for the gradient wind a steady state has been reached; id further, it is assumed in arriving at these expressions that there Cyclone Anticyclone Fig 3 Pa = apvta sm</> = term ausing from rotation of the earth Pb = P."-" cota "~R~ = term arising from the angul it radius of the path no friction between the air and the surface of the earth ovei which s passing. Under actual conditions the relation cannot be satisfied exactly there is always a certain amount of momentum absorbed irom the earn of air by the friction at the surface. This absorption of ^rgy is manifested by the production of waves and similar effects water surfaces, on forests, and on deserts. Yet under the most favourable conditions this relation between wind and pressure can recognized, and therefore it must be an important principle in the ncture of the atmosphere. Also when we ascend into the at- sphere beyond the limits where the influence of surface friction ikely to be felt, we find very little difference in the velocity of the 262 THE MECHANICAL PROPERTIES OF FLUIDS wind for hours on end. According to Shaw,* " pressure distribu- tion seems to adjust itself to the motion of the air rather than to speed it or stop it. So it will be more profitable to consider the strophic balance between the flow of air and the distribution of pressure as an axiom or principle of atmospheric motion." This axiom he has enunciated as follows :f " In the upper layers of the atmosphere the steady horizontal motion of the air at any level is along the horizontal section of the isobaric surface at that level, and the velocity is in- versely proportional to the separation of the isobadc lines in the level of the section." Throughout this short study of wind structure we shall follow Shaw therefore, and regard the wind as balancing the pressure gradient. It may be argued that this assumption strikes at the root of the processes and changes in pressure distribution one may desire to study. The results of investigation appear to indicate, however, that in the free atmosphere, at all events, the balance is sufficiently good under ordinary conditions for us to take the risk and accept the assumption. Under special circumstances and in special localities there may occur singular points where the facts are not in agreement with the assumption, but the amount of light which can be thrown upon many hitherto hidden atmospheric processes, appears to justify our acceptance of it. In the expression for the calculation of the gradient wind the right-hand side consists of two terms The first term, 2pvw sm^, is due as we have seen to the rotation of the earth, and in consequence has been called the geostrophic component of the pressure gradient. 7) O The other part, - cota, arises from the circulation in the small R circle of angular radius a, and so has been termed the cydostrophic component. With decrease in <f>, i.e. the nearer we approach the equator, a remaining constant, the first component therefore be- comes less and less important, the balance being maintained by the second term alone practically. On the other hand, with increase in a, i.e. the nearer we approach to the condition of the air moving in a great circle, <j> remaining constant, the second term becomes less and less important, until finally with the air mov- ing on a great circle the gradient and the geostrophic wind are one and the same. Consequently in the pressure distributions in mean latitudes where the radius of curvature of the path is * Manual of Meteorology, Part IV, p. 90 t Proc. Roy Soc. Edin., 34, p. 78 (1913). WIND STRUCTURE 263 lerally very large, the geostrophic wind is commonly taken as 5 gradient wind. Having obtained expressions indicating the connection between j pressure gradient and the theoretical velocity of the wind, we ill now consider some of the reasons for the variations of the wind ocity from this theoretical value. In the equation for the gradient wind and in the statements made ;arding the effects of friction on the wind, there is nothing to licate that the flow of air is not steady But it is a perfectly well Scales Ihf-em JoyS- S7Jln. lOmi/kr 6in, IOft/j, . tOSln, Bwufort tinutb at 03, 21 IS 1 U (2 1^, 14 15 16 f 13 19 20 | E __ N ' N Lowor lnM/n - 2 in +J Fig <L Anemometer Record at Aberdeen Observatory, a6th September, 1923 >wn fact that the air, at all events near the surface, does not flow h a constant velocity even for a very short interval of time. This iteadiness in the wind velocity is exhibited very well by the records self-recording anemometers Fig. 4, which is part of the record 26th September, 1922, at Aberdeen, exhibits this moment- to- ment vanation. Not only does the velocity vary but the direction > shows a similar variation, as indicated by the lower trace in the ire. As a rule the greater the variation in velocity, the greater also variation in direction. The variations both m velocity and direction are very largely undent upon the nature of the surface over which the air is sing, i.e. the nature of the records is greatly affected by the osure of the anemometer. A comparison of figs. 4 and 4*2 reveals i very plainly The first, as stated above, is a record from the 264 THE MECHANICAL PROPERTIES OF FLUIDS anemometer at King's College Observatory, Aberdeen. The head of the instrument is 40 ft. above the ground, the instrument itself being housed in a small hut * in the middle of cultivated fields and placed about | mile from the sea. The second is a record from an ane- mometer situated about 5 miles inland from the first, at Parkhill Dyce, and belonging to Dr. J E. Crombie. The exposure in this case is over a plantation of trees, and though the head of the instru- ment is 75 ft. above the ground, it is only 15 ft above the level of Lower margin a Z //I, Fig 4 Anemometer Record at Parkhill, Dyce, aGth September, 1922 the tree-tops. The two records refer to the same day, and the anemometers are situated comparatively close the one to' the other, yet the " gustiness " as indicated on the second is much greatei than that on the first. At the same time the average velocity of the wind in the second case is considerably reduced by the geneial effects of the nature of the exposure. Other factors which affect this variation of the wind are to be found in the undisturbed velocity of the wind in the upper air and in the temperature of the surface of the ground, i.e. in the time of day and in the season of the year. A great deal of light has been thrown on these variations of the wind near the surface by G. I. Taylor through his investigations of eddy motion in the atmosphere.f *The position of this hut is about a quarter of a mile directly eastwards fiom the position of the anemometer shown in fig 6 Fig 6 is compiled from records taken in the old position t ''Phenomena connected with Turbulence m the Lower Atmosphere ". Proa. Roy Soc. A, 94, p 137 (1918). * ' WIND STRUCTURE 265 From the aviator's point of view these variations are often oi prime importance. Beginning therefore at the surface, we shall endeavour to asceitain how the actual wind is related to the geostrophic or gradient wind for various exposures, and after- wards determine how these relations alter as we ascend higher into the atmosphere. When the hourly mean values of the surface wind velocities are examined for any land station, it is found that there is a diurnal 6 ^ f. k O 2 4 fe 8 1O 12, 14 16 ifi 2X) 22 24 hour Aberdeen- R&UJ Fig 5 Diurnal Variation in Wind Velocity for January and July, for the period 1881-1910 and also a seasonal variation in the velocities. Fig 5 represents this diurnal variation for the two stations, Aberdeen and Kew, for the months of January and July. On the other hand, no correspond- ing diurnal variation of the barometric gradient is to be found for these stations. The diurnal variation of the wind is evidently (D312) 10* 2 66 THE MECHANICAL PROPERTIES OF FLUIDS dependent upon the diurnal arid seasonal variations of temperature, and therefore the i elation of the suiface wind to the geostrophic wind is also dependent on these quantities. The curves in fig 5 show a maximum corresponding closely with the time of maximum Fig. 6 Relation between Geostropluc and Observed Surface Winds of Force 4 at Aberdeen Central circle shows position of the c,ty of Aberdeen with reference to the two river valleys and the sea Stippled area shows high ground Hatch nj ar^a shows town buildings The outer circle represents the grad ent wind The inner circle represents 43 per cent of the gradient wind The dotted line represents the observed wind temperature. Therefore if W represent the surface wind and G the geostrophic wind, the ratio W/G increases with increase of tem- perature, and vice versa. If then the surface layers be warmed or cooled from any cause whatsoever, we always find this effect on the latio W/G. The exposure of a station also has its effect on the ratio W/G. UB3TAT CO 00 OX oo CO CO IO C 50 VO 10 10 10 10 10 VO "^J" VO CO Tj- AVN'N o 10 oo IO CO vO CO CO CO ^ ^. S ^ A\M o\ O o\ IO !> >o VO M nt- VO O IO t^ ^ OX 3 sf- ^- (U W a I M.NM % O o\ VO vO J> VO vO v O O *> CO IO P-* O ^ ^ o ^ T)- ON OX H OX U O P IO IO vO IO VO CO IO o w - AA w VO ^ 00 0\ Ox H co v O <* rr-J PH S 10 10 "*" * 10 crj in w * a 2! MS 10 ^ H rt- H to ^t- vO CJ ON v IO CO Xh CO -^ CO ^ S 8 MSS 10 oo % H VO $ M <M Os. IO CO ^vh .0 CO I 1 ro tj- JH w 4} W o vO t^ H H CQ ^^ Q^ ro H r* gj en vO * 10 o > CO > ro Cn H en S ass $ ^ oo CO $ 5? co 10 rh i 1O CO IO ^ 3. gi rK (U ^^ w oo H ON w. OX CO CO Tt- 0< D i I >l Si en CO co 10 IO Tj- VO C ^ ^5 00 "rt asa o VO N p; o t> vO TO C^T C 10 rf- r~. c - ll g w 10 10 CO ^ OO M 10 t -, o d "^ j^w w IO vO IO IO vO vO IO vO i fj_ _ t_ H Q^ ^ > HH o 3NH oo ON CO 5? !o IO O !>- M f 00 10 1> If 5 10 T-J W S PH W CO o IO OX CO -( T)- C Tf ^ 3 H^ vO vO IO 10 10 t-- vO vO > O IO Tf- ^* 2 EH S JNN r*- CO t> IO 00 co vO OO 10 1C r vO ^ 00 < <u O Ox O jf^ o 00 M 00 IO ^ vr 10 O t > ON "1-! U \ l> IO 00 ts vO CO CO C o M- w o "d M* H 'o-a t3 0) Q 5 S << - > - - i ^"d 1 bfo ffiO to OO N o CO >o H 00 ON vO vO H Th N 1 Q $ l-i O aj d u " (U tU 02 S5 O S I C a JJ o J2 M 'S T) < J I I Slop K S j u & W o c3 -3 Holyh PQ Portl H Sp 267 2 68 THE MECHANICAL PROPERTIES OF FLUIDS Fig. 6 shows this effect on winds of force 4 at Aberdeen. The geostrophic wind is represented by the outer circle, while the dotted irregular curve gives the percentage which the suiface wind is of the geostrophic wind. An idea of the exposure of the station is afforded by the circular poition of the ordnance map of the dis- tiict placed at the centre of the figure. On the west side there is land, on the east, sea. To the south-west of the station lies the city, and we find that in this direction the surface wind has the lowest percentage, while in the north-easterly directions the pei- centages are largest. Towards the north-west lies the valley of the Don, and a fairly open exposure, the effects of which aie also well brought out in the figure. The effect of different exposures on winds of the same geo- strophic magnitude will be understood readily from an examination of Table I. It is evident, therefore, that no general lule can be given with regard to the value of the ratio W/G. It may have a wide lange from approximately unity downwards, depending on the time of day, the season of the year, and the exposure of the station. In the same way the deviation a of the surface wind fiom the dnection of the geostrophic wind is found to vary over a wide range. An example of this is afforded by Table II, wherein are set out the values for Pyrton Hill and Southport, as given by J S Dines in the Fourth Report on Wind Structure to the Advisoiy Committee on Aeronautics. It is necessary, therefore, in giving an estimate from the baiometnc gradient of the probable surface wind, as regaids either direction or velocity, that due attention be paid to the details mentioned above Occasionally the surface wind is found to be in excess of the gradient. This probably anses from a combination of a katabatic * effect with the effect of the pressure distribution, the katabatic effect more than compensating for the loss of momentum in the normal wind due to friction at the earth's surface. Data for the purpose of examining the ratio W/G over the sea are very limited. The following table, as given in the Meteoro- logical Office report for moderate or strong winds over the Noith Sea, will serve to show the deviation of the surface wind from the gradient wind, both in velocity and direction. * I e Katabatic or Gravity Wind when the suiface au ovci a slope cools at night or from any othei cause it tends to flow down the slope, this is especially pronounced on clear nights In ravmes, if snow-coveied and devoid of forests, this wind often reaches gale force. Such a wind is known as a katabatic wind. WIND STRUCTURE 269 w PQ MNN MNM MSM $* * as i 3N3 3NN 3 a Q.T3 O & a. p <* SI* El PnW I 270 THE MECHANICAL PROPERTIES OF FLUIDS $ I O O O CO m " 00 o H S a! ^ M H * 8 n ^ I O C* H co -"a 1 Iz 9 ON O u 1 Pi GJ T3 o *" rt _L Nw O S. vO ON 3 i s U! ?_ O N N vo rO" 5 co W ^4. M o ^ d 00 $ ^ ,CJ i w J> ON co 3 _ -4-J 1 TJ o S " fl g -9 R 00 8 r-r* Tj *4- c H ifcj ^ - ^f. ^J. (-y, ^ ,H Q -2 + W HH N | g ^ <! |> o 9 ON (St OO O ^ 01 M H N (U <U J3 g co ^" < ~~~- O ^ co o> vo r-> 'S. O "f g -a H- co H H M 5< o ^ rt a V o 1 CO N 'D fy"\ LJ W |*^^ 00 H J 3 ,, W NOON>H rh |8 -fcococscc ^ !_. M fh O 9 ON 00 ON ^ CO N M CN -Q M vO CO HH P D T3 O OS w^,^ N?? V 2 N j| 5 I 8 vO o o H ^ sS N O N O 3 06 M co cs ^ 4) ^j r i 4) C/) TT* rj s -5 r\ o o 9 N M i E 5 O *h M c4 S -t-j oo I) w pj rt Tj o S 9 10 vo ON O vo H M N CO n W O ~ ' ' ^ ^ ^ o o | H M N N -H O S >1 O CH 3 Q d >^~i W 2 JJ T; h d vO ^ fe- cr* 1 o o ct o S co g 1 & I H OO CO ON > ^1 M r^ f^l m cT w g fe tuO . : H 3 PH rt K< p -y w &o m 1 I 1 . . . a & JH rJ S O O Qj Q O Qj JH 17 " C >S D | H 1 S S S * i '.'. ', . ! 3 4 R ^ -P i 5 o ^ ^4-1 1 h$ 3 3 f -) W n; ^ ^ w w J 12; co co Jz; WIND STRUCTURE 271 Here again we see that no definite rule can be given for estimating ie surface wind from the geostrophic wind. It will be observed, )wever, that each quadrant exhibits certain dominant features and ust be considered therefore by itself. In this way a considerable nount of guidance is obtained by the forecaster in estimating the md over the sea from a given pressure distribution. We now pass to consider the actual wind in relation to the :ostrophic wind in the first half-kilometre above the surface. r H. Dines, in his investigation on the relations between pressure id temperature in the upper atmosphere, has found a high correla- m between the variations of these elements from their normal lues for heights from 2 Km upwards. Below the 2-Km. level e coi relation coefficients gradually diminish until at the surface actically no connection at all is found. Above the 2-Km. level 2 may regard the air as in an " undisturbed " condition, i e free Dm the effect of the friction at the earth's surface In this un- sturbed region the velocity and direction of the wind at any given ight are governed by the pressure and the temperature gradients ling at that height, while in the lower layers we find considerable viation from this law, evidently due to the effects of the surface the earth on the air m contact with it Several empirical foimulas have been given whereby the velocity the wind at any height in the lower layers of the atmosphere may calculated ftom that at a definite height, say 10 m , above the rth's surface Fiom observations, up to 32 m , over meadow- id at Nauen, Hellmann* confirmed an empirical formula v =/e/z^, nch agrees very nearly with a formula v = ktf suggested by -chibaldf from kite observations in 1888. The results of observa- ms up to 500 m., carried out m 1912 with two theodolites, aie yen by J. S. Dines in the Fourth Report on Wind Structure already fened to Here he has represented his conclusions by a series curves, and m doing so has grouped the winds into three sets: ) very light, wheie the velocity at 500 m. is less than 4 m. per cond, (2) light, with velocity between 4 m per second and 10 m. r second; and (3) strong, with velocity greater than 10 m per cond The curve for very light winds (see fig 7) shows that in is class the surface wind approaches the geostrophic value, which also marked for each group at the top of the diagram, much more osely than m any of the other groups. Cuives of this type enable * Meteor Zeitschnft, 1915. [ Nature, 27, p. 243. 272 THE MECHANICAL PROPERTIES OF FLUIDS one to judge of the aveiage behaviour of the wind in the lowest half-kilometre according to the pressure gradient at the suiface, When, however, curves are drawn for different hours of the day, 7 hr., 13 hr., 18 hr , these show differences among themselves even for the same surface gradient. A whole series of curves for various hours of the day and different seasons of the year would be necessary, therefore, before a complete solution of the problem could be obtained. As these formulas and curves just referred to are applicable under certain conditions only, and as the constants used differ for Lig rv ght Strom, 400 300 2.00 1OO 16 2.0 2 4 6 8 1O 12 14 V&L.QCITY IN METRES PER SECOND Fig 7 Change of Wind Velocity with Height within 500 metres above the suiface different times of the day and different seasons of the year, though they supply a rough working rule, yet a more exact solution of the problem is desirable. This has been supplied by the investigations of G. I. Taylor.* In his solution he regards the wind m the un- disturbed layer as equivalent to the geostrophic wind at the surface, while the region between the surface and the undisturbed layer is considered as a slab through which the momentum of the undis- turbed layer is propagated, as heat is conducted through a slab of material the two faces of which are kept at different temperatures. The momentum is propagated, according to the theory, by eddy motion, the surface of the earth acting as a boundary at which the momentum is absorbed The equation representing the propaga- tion is given as g . 9 , <"\ / i V I UH \ t \ p3tt /a ( = _(_), (4) * " Eddy Motion in the Atmosphere ", Phil, Trans. A, 215, p. i (1915) WIND STRUCTURE 273 iere p = the density and K the "eddy conductivity" of the . For small heights up to i Km or thereby, p and K are approxi- ttely constant. Therefore the equation, representing the distri- tion of velocities with height and time within this region, may written as pdufdt - Kpd 2 u/dz z ................. (5) The value of K is, according to Taylor,* roughly ^wd where w the mean vertical component of the velocity due to the turbulence, cl d represents approximately the diameter of a circular eddy. The value of AC differs, however, according to (i) the nature of 5 surface over which the air current is passing, (2) the season the year, and (3) the time of day. As both heat and momentum j conducted by the eddies, the value of K will be the same for th Values of K have accordingly been determined by Taylor as lows. 'i) Over the sea (determined fiom temperature ob-\ 3 X io 3 C.G S seivations over the Banks of Newfoundland)/ units. '2) Over grassy land (determined from velocity ob-^ 4 CCS servations by pilot balloons over Salisbury H , ' YM \ I U.I111S* Plain) . . . . . . . . . .) 3) Over land obstructed by buildings (detei mined } o 4 C C S from the daily range of temperature observa- j- tions at diffeienl levels on the Eiffel Tower)J The effect of the season of the year on the value of K is seen by sparing the values obtained from the Eiffel Tower observations January and June. Whole range, 18 to 302 m.: (1) January . . . 4 3 X io 4 C G S. units (2) June . . . . . 18 3 x 10* (3) Whole year . . . . io X io 4 ,, That K is also to a certain extent dependent upon the height ly be understood by comparing the values for the first stage, to 123 m., with those for the last, 197 to 302 m Mean value for the whole year. (1) Lowest stage . . . . . 15 x io 4 C G.S units (2) Highest stage . .. u X io 4 The reason for this variation is to be found in the method used calculating K. The nearer the ground the greater the daily *Proc Roy. Soc A, 94, p 137 (1917) 274 THE MECHANICAL PROPERTIES OF FLUIDS variation of temperature, and therefore the error arising from the method used in the calculation is proportionately greater for the lower stages than for those higher up. The value of K for the lowest stage is therefore not likely to be so accurate as that for the highest From, wind measurements Akeiblom* has deduced the value of K for the whole range of the Eiffel Tower The value found, 7-6 X io 4 C.G.S. units, is in fairly good agieement with Taylor's mean value, and the agreement is sufficient to show that K is the same both for heat and for momentum This theory of eddy conductivity has been applied by Taylor in order to furnish an explanation of the diurnal variation of the velocity of the wind at the surface and in the lowei layeis of the atmosphere. Two important conclusions have been reached He has shownf that when once the steady state has been reached, a state which previous theories claiming to explain this diurnal variation took no account of, a relation can be found between the undisturbed wind (i e. the geostrophic wind), the surface wind, and the angle between the direction of the isobars and that of the suiface wind. This relation takes the form W/G = cos a sin a, . (6) where W represents the suiface wind, G the geostrophic wind, and a the angle between their directions The accuracy of this i elation has been tested by comparing values of a observed by G. M. B. DobsonJ with the calculated values for certain winds over Salisbury Plain. Some of these results are given in Table IV TABLE IV. W/G = cos a sin a. Light Moderate Stiong Winds Winds Winds Observed value of W/G 0-72 o 65 o 61 a observed . 13 deg 21^ deg. 20 deg a calculated . 14 18 20 ,, * " Recherches sur les courants les plus bas de 1'atmospheie au-dessus de Pans ", Upsala Soc. Sclent Acta , 2 (Ser 4), 1908, No a. f" Eddy Motion in the Atmosphere ",Phil Trans A,215(i9i5). See note, p. 285. \Quar^our Roy Met. Soc,, 40, p 123 (1914), WIND STRUCTURE 275 The table shows that the agreement between observed and cal- lated values is very close; with a greater than 45, the equation, wever, no longer holds. The other conclusion, as shown by Taylor,* on the assumption at the lag in the variation in wind velocity behind the variation turbulence which gives rise to it is small, is that the daily nation in turbulence is sufficient to explain qualitatively, and to certain extent quantitatively, the characteristics of the daily varia- n in the wind velocity If the geostrophic wind G be reduced surface friction so that the direction of the surface wind is inclined an angle a to the "undisturbed" wind, then it is found that the -ce of the surface friction, or the rate of loss of momentum to the rface, is given by 2/cpBG sma, where B = *v/a>sin0//<:. As before, is the angular velocity of the earth and (f> the latitude The ation between this force of friction F and the velocity of the rface wind has also been examined by Taylor,f and found to be F = 0-0023 pW 2 . /. o 0023 W 2 = 2/cBG sma. If now numerical values be given to at and j> in K cosing /B 2 find that I 2O'4 / \o / \ ^- (cosa sin a) 2 , .(7) BG sma at = 0000073, and siriA = 077, since for Salisbury Plain = 50 N The values of i/BG can therefore be found for a series values of a. Also fiom the same equation we see that /c/G 2 is unction of a If we tabulate the values of /c/G 2 foi the same sei ics values of a, we can find then the relations between a and K These nous values are given in Table V (p. 276). Basing his discussion upon these values of the constants, lylor has constructed the curves given in fig 8. The abscissae Dresent the latio of the wind velocity at any height to the geo- ophic wind, while the ordmates give the ratio of the height to 5 geostrophic wind. If the geostrophic wind be 10 m per second, m the numbers for the ordinates will give the heights in dekameties, d those for the abscissae the velocities in dekametres per second, le shape of each curve is determined by the value of a chosen, :h curve having its a value attached to it. Consequently where * Proc Roy Soc. A, 94, p 137 (1917) \Proc. Roy. Soc. A, 92, p. 198 (1916). 276 THE MECHANICAL PROPERTIES OF FLUIDS TABLE V rs , , a, Degrees C G.S. uAits C.G.S. Units. 4 252 3*54 6 155 *'35 8 106 0-635 10 77-5 0-338 12 58-5 0-192 14 44 -8 0-116 16 34-9 o 069 18 27-3 0-042 20 21-9 0-027 22 167 0-0156 24 12 9 0-0094 26 99 o 0055 28 7-4 00031 30 55 o 0017 32 3-7 0-00085 34 2-6 o 00038 36 1-7 o 00016 the geostrophic wind and the deviation at the surface are known, the curves enable us to deteimine the wind velocity at any desired height. The curves may also be used to find the variation in velocity at a particular height under varying conditions of K The value of K for the open sea, for Salisbuiy Plain, and for Pans we saw to be 3 X io 3 , 5 X io 4 , and 10 X io 4 C G S. units respectively. If we take a = 10, then /c/G 2 = 0-338, which means that undei these three conditions G must have the values o 9, 3-8, and 5-4 m per second respectively Therefore the same geostrophic wind will suit dif- ferent curves if the value of K be altered, which it will be accoidmg to the exposure of the station, the season of the year, and the time of day. From the foregoing we see how the wind at the surface and in the lowest layers differs considerably from the geostrophic value. As we ascend above the surface, a nearer approach is made to the geostrophic values, for the effect of surface friction diminishes with height. Turbulence also diminishes as we ascend, its influence being on an average very little felt at 1000 m., though on occasions it may reach to 2000 m. WIND STRUCTURE 277 10. 70 12 60 Si ul EC UlUJ 3? u^ IQ H2 Oh ^ o too; mu. 3 -4 -5 6 -7 -8 -9 1-0 RATIO OF WIND VELOCITY TO GEOSTROPHIC VELOCITY(V/fci) Fig 8' -Curves showing Variation of Wind Velocity with Height according to the Theory of the Diffusion of Eddy-motion. (Taylor) 278 THE MECHANICAL PROPERTIES OF FLUIDS The spiral of turbulence affords another method of representing the variation with height of wind velocity in magnitude and diiec- tion In this method, first introduced by Hesselberg and Sverdrup * m 1915, when lines representing the wind velocity are drawn from the point at which the wind is measured, then their extremities He on an equiangular spiral having its pole at the extremity of the line which represents the geostrophic wind. Thus in fig 8a, if O be taken as the origin and OgX. the direction of the A;-axis, O^ represents the geostrophic wind G, OS the surface wind and L SOg the angle a between the two. The wind at any height Z is represented by OP, and is the resultant of the geostrophic wind G and of another component represented by gP of magnitude -\/2G smae~ B * and acting in a direction which makes an angle (a + Ite) with the X Fig 8a Equiangular spiral representing velocity and direction of wind at any height geostrophic wind. B has the same meaning as previously. An analysis of this method has been given by Brunt,f on the assumption that the coefficient K is constant and that the geostrophic wind is the same at all levels. In a note added to the paper, Brunt also deals with the case where K vanes inversely as the height or in- versely as the square of the height. The problem of K varying as a linear function of the height has been considered by S. Takaya J in a paper e< On the coefficient of eddy- viscosity in the lower atmo- sphere ". The solution enables the components of the wind to be calculated. The relations are equivalent to those found by Taylor (see Note i). It must not be concluded, though the mathematical analysis appears to indicate it, that whenever a test is made on the wind that the results will produce an equiangular spiral. The gustmess of the wind prevents this, so that only when the mean of a large number of ascents is dealt with may one expect the wind values to form the equiangular spiral * " Die Reibung in der Atmosphare " Veroff d Geophys Inst d Umv Leipzig, Heft 10, 1915 -\Q J Roy. Meteor Soc , 46, 1920, p 175 J Memoirs of the Imperial Marine Observatory, Kobe, Japan, Vol IV, No. i, 1930 WIND STRUCTURE 279 The next region to be considered stretches from the surface to leight of approximately 8000 m. Observations with pilot balloons indicate that the geostrophic ocity is -reached on an average below 500 m., while the direction lot attained until about 800 m. above the ground.* Each quadrant )ws its own peculiarities, however. Thus Dobsonf finds that for rth-east winds the gradient velocity is reached at 915 m., for ith-east below 300 m , for south-west about 500 m., and for 22 1912 29 1912 13 1913 5 1913 5 1913 21 1912 26 1913 7 5 O +5 +10 +15 +20 +25 VELOCITY IN METRES PER SECOND Fig 9 W to E Component of Wind Velocity on East Coast +30 +35 th-west below 300 m Also the winds in the north-east and ith-east quadrants show little or no increase after reaching the (Strophic value, those in the north-east often showing a decrease, lie those in south-west and north-west quadrants are marked by ontmual increase beyond the geostrophic value, the velocity in north-west being at 2500 m., 145 per cent of this value. The iation in direction also differs according to the quadrant. In north-east quadrant, even at 2500 m., Dobson finds that the ection is 6 short of the direction of the isobars On the other id, m the south-east quadrant the direction of the isobars is ched at 600 m , and above this level the wind veers still farther, e south-west winds behave somewhat similarly, only the gradient * For theoietical tieatment, see Note II, p 2850 fp.y R Met *$to,40,p 123(1914). 28o THE MECHANICAL PROPERTIES OF FLUIDS direction is not attained until 800 m. is reached, while in the north- west quadrant the direction follows the isobars at 600 m. In this last quadrant, however, no further veer occurs until 1200 m. is reached, when a further veer begins. On the average the deviation of the surface wind from the gradient decreases from north-east to north-west, passing clockwise, Dobson's mean values being 27, 24, 19, and 11 respectively. These results refer to an inland station. When we come to -20 5 2 1 34 7 -15 -1O -5 +O +5 +10 VELOCITY IN METRES PER SECOND Fig. 10 S to N Component of Wind Velocity on East Coast + 15 deal with a station on the coast we find even greater complications. Figs. 9 and 10, which represent the component velocities of a number of observations canied out with the aid of two theodolites at Aber- deen by the author,* serve to show the irregularities of these veloci- ties. A greater variation is shown in the west-east component than in the south-north, as is to be expected from the exposure of the station. On the whole, the west-east component shows a tendency to increase with height, while any east-west velocity gradually dies out. The south-north diagram gives mainly negative values, *Q.J. R. Met. Soe., 41, p. 133 WIND STRUCTURE 281 e. the winds observed had mainly a north-south component. This )mponent changes comparatively little in the first 4000 m., though igher up there is a tendency to increase indicated both for south- orth and north-south winds. This is in general agreement with ic results arrived at by Dobson. Cave, in his Structure of the Atmosphere in Clear Weather, has iven the results of observations carried out at Ditcham Park, /hen we consider his icsults for heights between 2500 m. and 500 m., we find that there is a decided increase with height in ic westerly components, the easterly components tending to die nt. On the other hand, both southerly and northerly components low an increase, the range at 7500 m being much greater than 2500 m., the actual values being from 20 m per second to -20 m. per second at the higher level, to 6 m. per second to -9 m. per second at the lower. As the result of investigation, Cave has divided his soundings i the troposphere * into five different groups, and has added a sixth >r winds in the stratosphere f These are: (a) i. " Solid " current; little change in velocity or direction. 2. No current up to great heights (&) Considerable increase in velocity. (c) Decrease of velocity m the upper layers. (d) Reveisals 01 gieat changes m direction (e} Upper wind blowing outward fiom centres of low pressure: frequently reversals at a lowei layer. (/) Winds m the stratosphere. In gioup (a) the giadient diiection and velocity are reached irly, and thereafter the wind remains nearly constant. There is Tactically no temperature gradient, and the pressure distribution t different heights is similar to that at the surface. Group (b) is mainly due to a westerly or south-westerly type, md represents the average conditions where depressions are passing istwards over the British Isles. There is here a marked tempera- ire gradient over the area. In group (c) are included mainly easterly winds, the pressure * Troposphere, i e the part of the atmosphere in which the temperature falls off ith inci easing altitude In latitude 55 it extends from the surface up to about Km , m the tropics it extends to about 17 Km. f Stratosphere, i.e the external layer of the atmosphere in which theie is no >nvection It lies on the top of the troposphere, and the height of its base above ie suiface varies from equator to poles (see troposphere). The temperature langes within it are m a horizontal direction. 282 THE MECHANICAL PROPERTIES OF FLUIDS distribution showing an anticyclone to the north. The gradient velocity is reached at about 500 m., the gradient direction at a point a little higher. Thereafter decrease in velocity takes place and occasionally a backing of the wind, though the latter does not invari- ably occur. With " reversals ", which are placed in group (d), the surface wind is almost always easterly; the upper, westerly or south-westerly Here we have a warm current passing over a colder, and the result is generally rain. Very often in summer theie is found a south-west current passing over a south-east, the two being associated with shallow thunderstorm depressions, the south-west current supplying the moisture to form the cumulo-nimbus clouds. From an examination of the winds in group (e), it is almost always found that the depressions from which the winds come advance in the direction of the upper air current. This is par- ticularly the case with north-westerly upper winds. With south- westerly upper winds we have very often conditions similar to those mentioned under (J), with corresponding results. Observations within the stratosphere are comparatively few, but, m general, they show that the wind within this region tends to fall off with increase in height, and that the direction is almost mvaiiably from some point on the west side of the north-south line. Several models, which show at a glance how the air currents change with height, have been constructed by Cave For a desciip- tion of these and a full account of his investigation the reader is referred to his book already mentioned. Let us now examine the wind structure in these upper legions of the atmosphere from the theoretical standpoint. We have already noted that the variations in the distribution of pressure in the upper atmosphere are closely correlated with the variations in the temperature distribution. Starting with the ordinary equation for the diminution of pressure with height and combining it with the characteristic equation for a permanent gas, we are able to find an equation giving the variation of pressure gradient with height. These equations are: and p/T = R/>, the latter giving - = -^ - - .................. (9) P P T WIND STRUCTURE 283 Iso if the horizontal pressure and temperature gradients be written o . <5np 3 JL = s, and -= = <?, respectively, then we have ox ox ds d z p 9# dxdz* e. from equation (8) ds dp , N = -^ r .................. ( I0 ) 9# ox 'herefore combining (9) and (10) we have for the change with eight in pressure gradient, = _ 9* ~" SP \p dx T dx To find numerical values we must substitute for p its value /RT. For dry air R = 2-869 X io 6 C.G S. units, while for air iturated with water vapour at 273 a its value is, 2 876 X io 6 C.G S. nits This differs only slightly from the value for dry air Also le uncertainties which arise in connection with the determination f the wind velocity in the upper air are greater than the variations i R, and therefore the value for dry air may be used on all occasions ithout any appreciable eiror With this value, and with g as 8 1 cm /sec. 2 , we have f now we express the variation in pressure in millibars per metre f height and take the giadients in pressure and temperature over oo Km., the rate of increase of pressure gradient per metre of eight in millibars per 100 Km. is 3M.Xxo-.Jg-J>, and S being expressed in millibars, T and Q in degrees absolute The variation in pressure gradient depends therefore on the ifference of the quantities Q/T and S/P. Now T falls from about 8o<7 at the surface to approximately 220<z at 9 Km., whereas P banges trom 1010 millibars to nearly 300 millibars within the 284 THE MECHANICAL PROPERTIES OF FLUIDS same lange We see, therefore, that S/P runs tlnough a consider- able range of values, while Q/T remains comparatively constant, The variation in pressure difference is therefore not constant within the region considered, but is likely to show positive values at first, then change through zero to negative values higher up. If the pressure difference remained constant up to 9 Km., then V/> would be constant, and the velocity of the wind would increase in inverse proportion to the density of the air as we ascend. Now Egnell believed that he found by observation of clouds that Vp actually was constant, and in consequence this law, Vp = a constant, has been termed Egnell's Law. We have seen, however, that the observations by pilot balloons do not confirm the law. The wind very often shows an increase in velocity with increase in height, especially winds with a westerly component, but this increase is generally less, even in the latter case, than in accordance with a uniform gradient. Equation (u) is theiefore much more m agiee- ment with the behaviour of the actual winds than a constant piessure gradient would be. The variation of wind with height can now be obtained by combining equation (u) with the relation s = zvpio sm<, or, vp = s/zco sin</> (12) Let -v be the component of the wind velocity paiallel to thejy-axis drawn towards the north, the x-axis being drawn towards the east. Then dv p s-* i.e. - 1 ^ 4 v dz Q]^ _ . _ , _ I __ O __!_ 2. ' s dx p dz 9T\ / 1 dp __ 8T dx/ \p dz T 8^ _- ^ SP i 1 8T s \p ~ s dv v dz 8p ds, . , - iy_L = ~ /2co siiiu) 8^ 3^ i 8p i ds p 8^ s dz' i 8w i ds i dp v dz s dz p dz _gP( ty__ i s \p dx T WIND STRUCTURE - 28 and as s/v = . dv i J _ $p _, p 9T " 9* " |_^ + /Z\ I 9^ J^J r9* 9T dp dT} , . \i ^r - / -*-} ' - (13) I nv* /i*y H*y /Sv I l^t/i^/ i/iia C/<^3 t/i-v J >ng the tf-axis the corresponding value will be du i (dp 9T dp 8T _ . j _ , _ _ *!, , _ re the negative sign must be used because if the piessure rease towards the north then the wind will be from the east. If then the wind be observed at vanous levels it is possible from se equations to calculate the separation of the isobars and iso- rms at the different levels. For this purpose it is necessary to >w the values of p and T for each level considered. In any parti- ar case the normal values of these quantities for the month in ich the observation takes place may be taken without any very LOUS error. We may then proceed to calculate the separation of isobars and isotherms at intervals of a kilometre m the following f The change of piessure difference we already expressed in the m d =- 343 Xio-{!-f}inCGS units 'hen the pressure be expressed in millibars and the temperature degrees absolute, the change of pressure-difference per kilometre icight may be written as A P/AT AP\ , . A* = 34'^r(-^r - -p-J, ........... (14) sre AP and AT are the horizontal changes per 100 Km m ssure and temperature respectively. The wind velocity W due this pressure difference AP can be found from the relation R T T W = K i AP = K^AP . ... (15) P P v ' U and V be the components of W from west to east and n south to north respectively, then the components of pressure 2 8b THE MECHANICAL PROPERTIES OF FLUIDS difference at any level as deduced from the wind observations ai i P " ANP== KT U and A W P = Similarly the components of temperature difference can fc expressed from equation (14) in the form r* i^ v t j and A w = T^-~ X T + A W P) Table VI (p. 287) is an example of the application of thes equations. Of the last four columns the first two give the separa tion in kilometres between the component isobars where th difference is I millibar, the second two between the componen isotherms where the difference is i C When the duection of th resultant isobars and isotherms for the various levels are calculate* we find the following directions Height in Km 01234 Isobar from 370 272 271 214 263 Isotherm from 15 271 210 334 This appears to indicate the approach of a waimer current fron the south-west at a height over 3 Km, The pressure distnbutioi at 7 hr. on the zSth gave a depression over Iceland with a smfaa temperature of 50 F. The increase of temperature indicated a the 3ooo-m. level is apparently due therefore to the warm air fion this depression pushing its way across the colder northerly current This seems to be in agreement with Bjerknes' theory* of the circu lation of air within a cyclone. The observations we have been considering hitherto refer to on< station only, so that we have obtained only a very small section of the isobars and isotherms for the different levels. If a number of obser- vations be made simultaneously at different stations over the Bntisl: *Q.y. Roy. Met, Soc , 46, p. 119. WIND STRUCTURE Tt- M ON M 10 ui CO + I CO o o 10 M ON O ON Pi. 10 ^O 1O CO ON VO CO ON N O O > * w 2 w I ex, O CO o o ON cO O H + 4 ON M O O fin O o + ON o O vp 00 O M N -0 00 o o + + xf H oj C H 2 H OO ON OO o r^. O H O ON r^. cb wg 288 THE MECHANICAL PROPERTIES OF FLUIDS Isles, say, then a series of maps may be drawn showing the air flo 1 at each level. These will afford an indication of the distiibution c pressure and temperature at the vai lous levels. In fig. 1 1 the piessm distribution at the surface at 18 hr. on yth September, 1922, is give in (a). The following four members of the series show approximate! the run of the isobars at the levels indicated as deduced fiom pile observations made at 17 hr., while the last of the series depicts th pressure distribution at the surface at 7 hr. on the following morning The separation of the isobars is 2 millibars m eveiy case. At th suiface the isobars run from north-east to south-west, but higher u the direction changes towards a noith to south direction. Thi appears to indicate a mass of rather waimer air towaids the wes 01 south-west, especially about the 6ooo-Jt. level. The velocities however, are compaiatively small and therefore a break-up of th system is not to be expected. Instead, as the 7 hr. chart of th following morning shows, there has taken place a fmthei development and the direction of the isobars at the suiface has now become mucl moie in accordance with the uppei air isobais of the pievious evening We must now consider the case of curved isobars. In th< expression foi the gradient wind determined neat the beginning o om suivey there were found to be two paits, one dependent upor the rotation of the earth, the other on the cuivaluie of the path Hitherto we have dealt only with the fiist part, but now we shal consider buefly the eflect of the cuivatuie ol the path upon the relation of the wind to the distiibution oi piessurc. In discussing the cii dilation of air m temperate latitudes, Shaw ai rives at the following conclusion * " Thus out of the kaleidoscopic featuies oi the circulation of an in temp ei ate latitudes two definite states soit themselves, each having its own stability. The fust rcpicsents air moving like a poition of a belt lound an axis thiough the earth's centre. It is dependent upon the earth's spin, and the gcostiophic component of the giadicnt is the important feature; the cuivatuie of the isobais is of small importance. The second repiesents air lotat- mg round a point not very far away: it is dependent upon the local spin, and the curvature of the isobars with the corresponding cyclo- strophic component of the giadient is the dominant consideration." Up to the present we have been considering only one point in the path of the air, and the lines of flow of the air at that point we have regarded as coincident with the isobars m the upper air and making a definite angle with them near the surface owing to the * Manual of Meteorology, Part IV, p. 236: WIND STRUCTURE 289 3,000 FEET / *\ 1O.OOO FEET Fig ii. Map showing the Pressure Distribution and Wind Direction at the Surface at 18 hr on 7th September (a), at 7 hr on 8th September (/), and (b to e) the app-oximate Direction of the Isobais at 1000 ft , 3000 ft , 6000 ft , and 10,000 ft , as deduced from Pilot Balloon Observations at 17 hr. on 7th September, 1933. (D312) 11 290 THE MECHANICAL PROPERTIES OF FLUIDS tuibulence in the atmosphere. When we come to consider a sue cession of states, however, we see that the paths of the air are no necessarily coincident with the lines of flow or the latter with th< isobars. In the case of the first state mentioned above by Shaw the isobars are straight and the paths of the air are coincident witl the lines of flow; but when the two states are superposed and a series of maps drawn giving the pressure distribution at definite intervals it is seen that the paths of the air are no longer coincident with the lines of flow or with the isobars. What then are the paths of air m a cyclone ? A partial solution of the problem may be reached after the follow- ing manner. It is a well- known fact of experience that one of the character- istics of a cyclone is that it travels across the map, and when the isobars are circular that the velocity of translation is rapid. We shall here confine ourselves theiefore to the examination of a circular, rapidly moving storm, termed the " normal " cyclone or cartwheel depression. If two horizontal plane sections of a normal cyclone be taken, including between them a thin lamina or disc of rotating air, and if this disc travel unchanged in a horizontal direction, then to obtain the actual velocities of any point on the disc the velocity of trans- lation must be combined with the velocity of rotation. If the velocity of translation be V and the angular velocity n, then the centre of instantaneous rotation will be distant from the centre of the disc a distance V/w. This centre of instantaneous rotation will travel in a line parallel to the line of motion of the centre of the disc The actual paths of the air particles are traced out by points attached to a circle which rolls along the line of instantaneous centres and whose radius = V/n Fig. 12 represents these trajec- tories. The figure shows, in one position, the circle of radius V/w, its centre 0, and the instantaneous centre O'. The circle rolls on I<ig 12 Trajectories of Air for a Normal Cyclone (from Shaw's Manual of Mai o o'oi>y) WIND STRUCTURE 291 line through O' perpendicular to OO'. The path of a particle a cusp, a loop, or neither, according as the tracing point is on, lout, or within the circle. From this we see that in the normal cyclone there are two centres, one, O (fig. 12), the actual centre of the rotating disc which is led the tornado centre, the other, O', the centre of instantaneous tion termed the kinematic centre. They lie on a line perpendi- r to the path of the cyclone, and are distant from each other by length V/n. [n such a system as this the isobars will not coincide exactly L the lines of flow. For the present neglecting the variation in ude and m density, and also the curvature of the earth's surface, shall regard the cyclone as moving along a horizontal plane. hat case the system of isobars will be obtained by compounding stem of circular isobars embedded in a field of straight isobars. centre of the circular system will not coincide with either of centres already referred to, but will be at a distance from the matic centre = V/(soj sm< + ri) and lie on the line joining the matic and tornado centres. This centre has been termed the 'mic centre, and is the centre of the isobanc system as drawn map It is therefore quite easily identified, but one must bear imd that it is not the only centre m a normal cyclone f we take the centie of the rotating disc as origin, with x and y towards the east and north respectively, then for an eastward :ity of translation V, the pressure will dimmish uniformly towards north at the rate 2pVo> sm</>, i.e. the field of pressure will be rented by r*' c y I dp I 2pVco sm<f)dy t J '* Jo e p' = pressure at any point, and p' Q = pressure at any point te #-axis; i.e. p' p f = 2/>Vte> sin^y ---- . .(16) 'or the circular field with its centre at the origin we have n^ -f- v z p cota/R)^r, f* f r I dp = I J P, Jo e p is the pressure at any point distant r from the origin, and e pressure at the origin. 292 THE MECHANICAL PROPERTIES OF FLUIDS If we neglect the curvature of the earth, then v z cotct/R = v z l Also v = rn. /P /-r dp = pn I (20) sin< -f- n) rdr, P Jo 2 2 By combining the two equations (16) and (17), we have for tl resultant field P P = ( 2 o) sm< + n) (x z + y z ) 2/>Vco siny.. .(18) 2* This represents a circular field of pressure round a point __ 2o) sin<^V n(2") sm<jk + w)' and P is the pressuie at the centie of this field and not at the origi Now the distance of the kinematic centre from the tornac centre which was chosen as oiigm is equal to V/n Theiefore tl distance of the kinematic centie from the dynamic centre is TT/ 20) sind> V TT / / ; i \ V/ _ . __ .[ x - == V/ (2cu sin^ + ). 20) sm.<f> -\- n n We see, therefore, that this combination of a field of straight isobai with a circular system embedded in it is sufficient to give the fid of picssure necessary to keep the disc lotating. In the normal cyclone it follows that the centie of low piessui being not the centre of the lines of flow, the wind possesses a defini counter-clockwise velocity at the centre of low pressure. When actual example of a lapidly moving circular storm such as that loth-nth September, 1903, is examined, we find that the syste actually does possess such a wind, and consequently this diveige wind, which has often been regarded as accidental, is in reality perfect agreement with the pressure system. Another feature whi< the normal cyclone possesses in common with an actual circul cyclone is the greater incurvature in the rear of the cyclone as cor pared with that in front. " From these considerations," says Shaw,* " we are led to acce the conclusion to be drawn from the conditions of the normal cyclon * Manual of Meteorology, Pait IV, p. 343 WIND STRUCTURE 293 mely, that the wind calculated from the gradient by the full formula mg the curvature of the isobars, gives the true wind in the free air t at the point at which the gradient is taken but at a point distant m it along a line at right angles to the path and on the left of it by 3 amount V/(2,zo sin^ + )." The calculated trajectories in the case of the normal cyclone K Fig 13. Trajectories of Air in Circular Storm, leth to i ith September, 1903 ve already been referred to and given in fig. 12. A comparison these with the actual trajectories for the storm of xoth-iith ptember, 1903 (fig. 13), shows at once the remarkable similarity tween the two sets, indicating still further that this actual cyclone i the normal cyclone are very close akin the one to the other. te trajectories of the September cyclone are repioduced from e Life History of Surface Air Currents* Here we have considered in a very fragmentary way only one m of stable rotation, namely the circular. Even then no account s taken of the discontinuity of velocity which must occur at the * Life History of Surface Air Currents, by W. N. Shaw and R. G. K. Lempfert, D., No. 174, London, 1906. 294 THE MECHANICAL PROPERTIES OF FLUIDS edge of the rotating disc in the normal cyclone, but it is possible show that this discontinuity can be accommodated by mchidn in the revolving column of air an outer region represented by tl law of the simple vortex with vr constant. The regions beyor what have hitherto been included in the revolving disc will al; form part of the cyclone, therefore, and not simply belong to tl environment. For further treatment of this subject the reader is referred i treatises on dynamical meteorology, such as Shaw's Manual < Meteorology, as in this brief study some of the intricacies only of tl problems of wind structure, rather than their solutions, have bee placed before him. NOTE I The equations of motion of air over the surface of the eart when the effect of eddy viscosity is taken into account aie, whe the steady state has been reached and the motion is honzonta dhi <j) + K ................... (0 ax* i j /-i j i v , \ o = + 2o>M snip ~ 2o>G sm<p + /c-~ ....... (2) CIS* On eliminating u from the above we find that the equation fo v becomes d*v . 4eo 2 sinV> 4- ^ v = o, <fe* ^ K Z d*v , -DA i, T> 2 <*> sin< or - + 4B% == o, where B a = r . Now v does not become infinite for infinite values of z, and th solution of the equation is therefore v A 2 e~^ smBx -f A 4 e~ Ba cosB^ .......... (3) On differentiating this value of v twice with respect to z, an< substituting in (2), we get u = G + A 2 e~ B;s cosEs 1 A 4 e~ B * smite ........ (4) Now G = value of the gradient wind velocity, and therefore for great heights u = G, i.e. the gradient wind velocity and v =s o. WIND STRUCTURE 295 The values of A 2 , A 4 are found by imposing suitable boundary iditions. Now at # = o these are r&Afe-i r*/ L u J*=o L *> *=o ere a = angle between the observed wind and the gradient id. From these conditions . _ tana(i -f- tana).-, A tana(i tana)^ A 2 , 9 . ^; A 4 . o , ^ tan 2 a + i tan^a -f- i The surface wind W = */[> 2 -f z; 2 ] 2==0 = s/V"+ (A 2 + G) 2 C 1 = . v tan 2 a(i tana) 2 -f- (i tana) s I + tan 2 a v / ' \ / .^(i tana) , , = G^ ---- ' = G(cos a srna) seca >r W/G = (cosa sma). NOTE II The theory of eddy motion also accounts for the observed fact t the magnitude of the gradient wind is reached at a level lower n that at which the gradient direction is attained The height at which the gradient direction is reached is found m equation (3) of Note I by putting v = o If Hj. be the ght, then o = A 2 smBHi f A 4 cosBH 1} A 1 therefore tanBH = -* A 2 Substitute the values of A 4 and A 2 already found, and we obtain tanBIL = tan (a I -f- tana \ 4 296 THE MECHANICAL PROPERTIES OF FLUIDS 7T Since a is positive and less than -, the smallest value of H x is got from 4 BH t = 5 4 The height H 3 , at which the value of gradient velocity is reached, is given by n z + fl 2 = G 2 , which, on substitution, becomes -BI-I,, _ (i + tana) cosBH 2 (i tana) sinBH 2 . . 6 . . 12) tana v From this equation BH 2 can be found in terms of tana The following table, given by Taylor, shows the values of BH T , BH 2> and Hi/Kg as a goes from o to 45. a. EHj BH 2 Ili/IIa o 235 078 30 20 2-70 1-04 26 45 3 IS 144 2'2 Now for Salisbury Plain Dobson found that the deviation a was, in a laige number of cases, 20; also that i for this devia- 800 metres ,, 2 tion was = 2 66. 300 metres This is in good agreement with the theoietical value 2-6, (D312) II* CHAPTER IX ubmarine Signalling and the Transmission of Sound through Water Although practically every other branch of science has had siderable technical application, that of acoustics has until the few years remained practically in the academic stage, and few n among scientific men gave seiious attention to it Bells, gs, whistles, sirens, and musical instruments have indeed been d from remote times both for enjoyment and foi signalling pm- es, but their development has mainly been on cnipnical lines, h but little assistance from the physicist. The Great War has, however, bi ought about a sinking change this as in many other directions, and acoustics is now becoming only an important branch of technology, but shows signs even developing into the engineering stage and giving us a new and verful method of powei transmission, to judge by the pioneci k of M. Constantinesco, who has already developed it for the ration of lock drills and riveting machines, and shown how it f be applied to motors and other machines. Few blanches of :nce now offer such possibilities to the inventor. Acoustic signalling is of especial impoitancc m connection with igation, as sound is the only foim of eneigy which can be tians- ted through water without great loss by absorption. The lela- sly high electrical conductivity of water renders it almost opaque light and to electromagnetic waves. The present article deals principally with acoustic signalling ier water, but certain allied problems, such as sound langing, ith sounding, and other applications to navigation, will also be ;fly referied to. As is well known, sound consists of a vibratory disturbance of latenal medium, such as a gas, solid, or liquid, and its phenomena 208 SUBMARINE SIGNALLING 299 e almost of a purely mechanical nature. When a bell or tuning >rk is struck it is thrown into vibration, and as any part moves rward it compresses the medium in front of it and also gives it forward velocity. As the vibration reverses so that the move- icnt is in the opposite direction, the mass or inertia of the medium jeps it moving forward, and a partial vacuum or rarefaction is oduced, until the vibration again reverses and forms a fresh unpression. These regions of compression and rarefaction there- re travel forward as a series of pulses or waves away from the urce in much the same way as ripples are formed on the surface a pond when a stone is dropped into it. In the case of the rface ripples, however, the leal motion of the water is partly up id down, or transverse to the direction of movement of the waves, iereas in the case of sound the motion of each particle of the edmm is mainly forwards and backwards along the line of propa- tion of the sound. Sound vibrations are therefore spoken of as igitudinal or in the direction of transmission, which differentiates em from all other kinds of vibrations, such as those of ordinary ives, light, or electromagnetic waves, which are said to be trans- rse. It at once follows from this that although many of the entific principles of optics can be and indeed have been success- ly applied to sound, there can be nothing in acoustics correspond- l to polarization in light. This point is made clcai at the outset, the phenomena of light are fairly generally known and are most tnulating to acoustic development. Fundamental Scientific Principles In older to understand the operation of modern acoustic trans- ttmg and receiving instruments properly, it will be well to start h a brief statement of certain scientific principles and definitions, ne of these are well known, but others requiie a few words of ilanation. Sounds are divided into musical notes and noises, and musical es are spoken of as differentiated by intensity, pitch, and timbre quality. A musical note is produced whenever the vibrations are i legular character, so that each wave is similar to the previous . The intensity or loudness of the note depends on the strengtL implitude of the vibration, its pitch on the number of vibrations second or frequency, and its timbre or quality on the form the vibration. The purest musical note is given by uniform 300 THE MECHANICAL PROPERTIES OF FLUIDS vibrations of a simple harmonic character, and if the wave-form is saw-toothed or shows any other variation from the sine form, the note is more or less piercing in quality, due to the presence of overtones or higher harmonics besides the fundamental pure tone. Noises are differentiated from musical tones by having no regular character, and are made up of a number of vibrations of different intensity and pitch. Speech may be described as a noise from the point of view of acoustic transmission and reception, on account of the variable nature of the vibrations; and also the sound from machinery, ships, &c. This is a serious difficulty as regards the detection and recognition of such sounds, as nearly all trans- mitting and receiving devices are more or less " selective " in char- acter, i.e. they respond better to certain definite frequencies and are relatively insensitive to others. Everyone knows that telephones or gramophones reproduce certain sounds better than others, and acoustic signalling, like wireless transmission, is far moie effective with " tuned " devices, which are, however, very insensitive to other frequencies. Velocity of Propagation An accurate knowledge of the velocity of propagation of sound is of great importance in connection with acoustic signalling, espe- cially as regards determination of range or position as in sound ranging. The velocity is very diffeient in different substances, as it depends on the elasticity and density of the matenal, and thcre- foie on its composition, pressure, and temperatuic We are here concerned chiefly with the velocities in air and in sea water, although the acoustic properties of other substances require consideiation when the transmitting and receiving devices aie being dealt with. For air at temperature f C. the velocity v = 1087 + i-Sit ft. per second. For sea water v = 4756 + i3'8* O'i2t z ft per second, accoid- mg to the latest determination of Dr. A. B. Wood, for sea water having a salinity of 35 parts per thousand; the velocity being increased by about 3-7 ft. per second for each additional pait per thousand in salinity. This gives a velocity of 1123 ft. per second in air, and 4984 ft. per second in normal sea water at a temperature of 20 C., so that the velocity in the sea is about four and a half times as great as in air. SUBMARINE SIGNALLING 301 Wave -length As above mentioned, acoustic waves from a vibrating source msist of a number of compressions and rarefactions following le another and all travelling with a velocity given above. The stance between one compression or one rarefaction and the next called the wave-length of the sound, and it is evident that if the equency of vibration is n cycles per second there will be a train n waves in a distance equal to the velocity, so that the wave- m ngth A = - . For example, if we take a frequency n of 500 ~- , the >rresponding wave-length in air and in sea water respectively 20 C. will be: In air = 2 246 ft., and in sea water _ = g.g68 ft. 500 T 500 yy The gi eater wave-length in water introduces somewhat serious fficulties as regards directional transmission and reception, as will >peai later Transmission of Sound through Various Substances As was mst shown by Newton, the velocity of sound in any sub- ance can be calculated fiom a knowledge of its elasticity of volume id its density. If K is the elasticity and p the density, it is easy to IK ove that the velocity of propagation of sound v = A/-. It must > ' lemembered, however, that with the rapid vibrations of audible unds the heating and cooling resulting from compression and refaction have no time to die away, and we must therefore take e adiabatic elasticity instead of the constant temperature or iso- eimal elasticity in the above formula. For gases the isothermal isticity is equal to the pressure, or about io 8 dynes per square ntimetre in the case of ordinary atmospheric pressure, and the labatic elasticity of air is 1-41 times this amount, while the density air p = 0-00129 8 m< P er cu bi c centimetre, so that the velocity = A/ 4 = 33,000 cm. per second, or about 1085 ft. per " 0-00129 cond, agreeing closely with the value obtained by direct experiment. The mathematical theory also enables us to calculate the amount 302 THE MECHANICAL PROPERTIES OF FLUIDS of acoustic powei transmitted by sinusoidal waves, and the pheno- mena which lesult when the sound passes from one medium mti another matters of considerable importance in connection wit] submarine signalling. It can be shown that the relation betweej the pressure P due to vibiation (i e. the alternating excess ovei th mean pressure), and the velocity V of a moving particle, at an point of the medium in the case of a plane wave of laige aiea com pared with the wave length, is given by the relation P = RV where R = v /c/o. This iclation being analogous to Ohm's Law i] Electiicity, the quantity R has been called by Bullie the " acousti resistance " * of the medium. The powei transmitted (w) per uni area of the wave front is^P raax V max , i e. P 2 max /zR, 01 RV 2 max /2 For a plane wave sinusoidal distuibance of frequency n peiiod per second, and writing co for 27772, we have V mvv = a>a, wher a is the amplitude of the displacement, so that w = axiP max / = Rw 2 # 2 /2 ergs pei square centimetre pei second For ordinary sea watei in which K 2-2 X io 10 dynes pei squar centimetre, and p 1-028, R = v itp = 14 X io 4 , so that foi frequency of 500 ~ and a displacement of o i mm., the powe transmitted would be 7 watts per squaie ccntimctie When sound passes from one medium into anothei, it can b shown that unless the two media have the same acoustic icsistanc there will be a certain amount of reflection at the interface II is the ratio of the acoustic lesistance of the second medium t that in the first, and the wave fronts aie paiallel to the inteifac which is large in compaiison with the wave length, Pa , ^ P/ = ^ Va = -L-V,, and V/ = ;;-!v, \vheie P t is the pressuie and V x the velocity in the oiigmal wave P 2 > V 2 liansmittcd PI V/ reflected If the second medium is highly resistant compared with th fiist, so that r is very large, P 2 = aPj, P/ = P : , V 2 = o, and V/ ~ V 3 so that the pressuie at the interface is double that in the origins wave and the velocity, being equal to (V x V/), is zero, since th movements in the direct and reflected waves aie equal and ii *H. BnlliS, Le G&ue Civil, z-^id and 3oth August, 2:919; "Modem Mann Problems in War and Peace", nth Kelvin lecluto to Institution of Electnci Engineers, by Dr. C V Drysdale. your. Inst Elect. Eng., 58, No. 293, Jub 1933, PP 591-3. SUBMARINE SIGNALLING 303 iposite directions. The wave is therefore totally reflected back in e first medium and there is no transmission. On the other hand, if the second medium has a very small oustic resistance compaied with the first, so that r is very small, P 2 = o, P/ = -P 15 V 2 = 2V 1} and V/ = - V x . this case the total piessure (P 1 -[- P/) at the interface is zero, d the velocity (Vj V/) is aVj, so that the velocity is doubled, it there is again no transmitted wave since P 2 = o, and the wave totally reflected with a leveisal of the velocity Vj. In the first se the surface is called a " fixed " end, and m the second a " free " id. If, finally, the two media have the same acoustic resistance, so at r i, P 2 = P 1? P/ = o, V 2 = V lf and V/ = o, d the wave passes on without any reflection This is the ideal ndition to be secured m transmitters and receivers When the two acoustic icsistances are not equal, it is easily shown at the ratio of the eneigy in the tiansmitted wave to that of the igmal wave, which we may call the efficiency of transmission (\ 9 T ' I \ ) . Now r+ij i watei we have seen that Rj = 14 X io 4 , and for air R 2 = 40 2e also table on p. 292), so that r = ~ 2 = 2 86 X io~ 4 , and l\i = ^ oooii, so that only a little ovei o-i per cent of (r -|- i) 2 e eneigy is tiansmitted This at once illustrates the difficulty in [ underwater listening, as the sound passing through the water must nerally pass into the air before falling on the drum of the ear. Again, the value ol R foi steel is about 395 X io 4 , so that on issing from water to steel r = 28 approximately, and the efficiency transmission is about 13 per cent, while from steel to air it is ily o 004 per cent. Hence for sound to pass from water thiough e side of a ship to the air inside, the efficiency would be only , per cent of 0-004 P er cent > or 0-00052 per cent, were it not for the ct that the plates of a ship are sufficiently thin to act as diaphragm, id thus allow a greater transmission than if they were very thick. L any case, however, the loss of energy is extremely great, and this is led to the practice of mounting inboard listening devices, either 304 THE MECHANICAL PROPERTIES OF FLUIDS directly on the sides of the ship or in tanks of water in contact with the hull, as will be described later. The following table of the acoustic properties of various media has been given by Bnllie\ Medium Steel . Cast iron Brass . . Bronze Lead ("Teak Wood I Fir i I Beech Water Rubber . . Air .. Vaiue of K (kg per sq mm ) 2 X io 4 o 95 X io 4 065 X io 4 o 32 X io 4 o 06 X io 4 0-16 X io 4 o 09 X io 4 06 X io 4 2 X I0 a Below i (vanable ac- cording to the nature of the rubber) 1 40 X io~ a Value of o (COS) Value of V - p (velocity metres per second) Values of R = V/cp in C G S units. 7-8 CJIOO 395 X"io* 70 3680 258 X io 4 84 2780 234 X io 4 88 1910 168 X 10* 114 735 82 5 X io< 086 4300 37 X io* o 45 4470 20 X 10* 08 2740 22 X 10* I 1410 14 X io 4 >i (appioxi- rnately) j Below 100 / Below I i X io 4 o 0013 328 o 004 x io 1 It should be noticed that the values of R for pine or beech wooc are not greatly different fiom that for water, so that sound shoulc pass from water to wood or vice veisa without great icflection Pressure and Displacement Receivers From what has been said concerning the theory of acoustic transmission, it is evident that sound may be detected eithci by the variations of pressure in the medium or by the displacements thej produce, in the same way as the existence of an electncal supply may be detected by the electiical pressure or by the cunem it produces. Acoustic receivers may thercfoie be classed as pies sure receivers, analogous to electrical voltmeteis, and displacemen leceivers corresponding to ammeters; but this classification is no a rigidly scientific one, as a receiver cannot be operated by picssuu or by displacement alone. We have seen that the power per uni area of wave front is |P max V max- , so that unless the receiver makes use of both the pressure and velocity of displacement it receives no energy and can give no indication. A perfect pressure receive: would, in fact, constitute a fixed-end reflector, and a perfect dis placement receiver a free-end reflector, in both of which cases w< have seen no -energy is transmitted. SUBMARINE SIGNALLING 305 The distinction between pressure and displacement receivers is, >wever, a useful one, just like that between a voltmeter and am- eter. A voltmeter is predominantly an electrical pressure-mea- ning device although it takes a small current, and an ammeter a irrent-measuring device although it requires a small P.D across 3 coils. Similarly a pressure receiver is one in which the dia- iragm is comparatively i igid and yields very little to the vibiations, hile a displacement receivei is one with a very yielding diaphragm, he distinction is of importance directly we consider directional ceivers, as the piessme in a uniform medium is the same in all rections while the displacements are in the line of propagation, i that a pressure receiver will give no difference of intensity on ling rotated into diffeient diiections if it is so small that it does 3t distoit the waves, wheieas a displacement receiver will give maximum when facing the source As regaids sensitiveness, however, it is evident that the best suits should be obtained when the receiver absorbs the whole the eneigy which falls upon it, which will only be the case when given alternating pressme on the diaphragm pioduces the same splacement as it does in the medium, so that the eneigy is impletely tiansmittcd into the leccivmg device without reflection, his will only be the case if the diaphicigm is in icsonance with e vibrations and the icceiving mechanism absoibs so much lergy as to give cutical damping In the case of a piopeily designed receivei which is small compaiison with the wave length, it will draw off energy om a giedtcr aiea of the wave fiont than its own area, just as wnelcss aenal may absoib eneigy fiom a fanly large region ound it. The piactical constiuction of undei water receivei s and hydio- loncs will be dealt with Litei , but it will be well at this point to ve some idea ol their essential features. The simplest form of ich a leceiver, which is analogous to the simple trumpet for air ception, is what is called the Broca tube, which consists of a ngth of metal tube with a diaphragm over its lower end. When is is clipped into the water, the sound from the water is com- unicatcd through the diaphragm to the air inside the tube, and e observer listens at the fiee end This is moderately effective, it not veiy sensitive or convenient, as it makes no provision for apHfymg the sound, and it is not easy to listen through long bent bes, so that the observer must generally listen only a few feet 3o6 THE MECHANICAL PROPERTIES OF FLUIDS above the water. Modern hydrophones are therefore neaily all of an electrical character, containing microphones or magnetophones from which electrical connections are taken to ordinary telephone receivers at the listening point. Microphones are generally used, as they are more sensitive, and there are two types of microphone which conespond approximately to pressure or displacement receivers lespectively. The former is termed the " solid back " type, in which a number of carbon granules are enclosed between a metal or carbon plate forming or attached to a dia- phragm and a solid fixed block of carbon at the back. If piessure is applied to the diaphiagm it compresses the granules and in ci eases their con- ductivity, so that a greater cunent passes fiom a battery through the microphone and the icceiveis and reproduces the sound through the piessme variations. In the " button " type of microphone, on the other hand, the carbon gianules aic en- closed in a light metallic box or capsule coveicd by a small diaphragm, and the whole aiiangemcrit is mounted on a largci diaphragm, so that ifs vibrations move the capsule as a whole and shake Fig i Non-dnec. U P tne granules, with only such changes of pies- tiond Hydrophone sure as resu it f 10 m the inertia of the capsule In this case it is the motion 01 displacement of the diaphragm which produces the vanations of icsistance in the microphone. The commonest type of simple hydrophone is diagrammati- cally shown in fig. i and illustrated m fig 18 It consists simply of a heavy circular metal case of disc form with a hollow space covered by a metal diaphragm to the centre of which a button microphone is attached. It is fairly sensitive but has no dnectional properties. Directional Transmission and Reception The problems of directional transmission and reception aie among the most important as regards acoustic transmission. As in the case of wireless telegraphy or telephony, acoustic transmis- sion suffers greatly from the difficulty that sound, like wireless waves, tends to ladiate more or less uniformly in all directions, with the SUBMARINE SIGNALLING 307 SOURCE isult that its intensity rapidly diminishes according to the inverse ]uaie law, and there is great difficulty as regards interference and r ant of secrecy. Again, as icgards reception, it is of comparatively ttle value to have a sensitive receiver which will detect the existence F a source of sound at a great distance if it gives no indication of le direction or position of the source. On this account the ques- on of directional transmission and reception is of at least equal npoitance to that of power- il transmitters and sensitive iceivers. This question of irectional transmission and sception has received a large nount of attention. The Binaural Method f Directional Listening. - Our own ears form a 51 y efficient directional ic- iiving system When a idden noise occms we istinctively tuin to wauls le source, and if we are iindfolded we can gene- lly tell with consideiable cuiacy the dnection liom Inch a sound comes. This due to the fact that, as ir two eais aic on op- )site sides of the head id about 6 in. apait, the und leaches one ear a tie soonei than the other, iless it anses fiom a point in a plane perpendicular to the ic joining the ears, i.e. clnectly in front of, behind, or above our k ad. Our eais are exceedingly sensitive to this minute differ- ce of time, and as this interval depends upon the direction, tting larger the more the source is on either side, we learn to timate the direction fairly closely, provided our two ears are nearly ually sensitive. This is known as the binaural (two-ear) method estimating direction, and it has been developed both for air and bmarinc listening For example, if we take two trumpets fixed a horizontal bar (fig. a), each of which is provided with a definite I- O ,'/*-* Fig a Dmauial Listening with Tuimpcts 3 o8 THE MECHANICAL PROPERTIES OF FLUIDS length of rubber tubing to an ear piece, we can detect and locate an aeroplane with considerable accuracy from the noise of its engines, as the trumpets magnify the sound, and the sensitiveness to direction may be increased by increasing the distance between the trumpets. When a sound is heard, the obseiver swings the bar carrying the trumpets round in the diiection indicated, and as he does so the sound appears to cross over from one ear to the other behind his head. The position at which this occurs is called that of binauial balance, and when this balance is obtained the bai is at right angles to the diiection of the source. The same principle can obviously be applied to undei water listening with two receivers, but in this case it should be noted that, as the velocity of sound in water is about foui and a half times that in air, the distance between the receivers must be increased in that propoition to obtain the same difference of time, and theiefoie equal binaural discrimination As this involves the use of a some- what long bar, which is troublesome to turn under water, iccouise is generally had to what is called a binaural compensatoi for detei- mining the direction. Retuining to our pair of trumpets in fig. 2, suppose that the source of sound is to the right of the median plane, and that the tubes from' the trumpets, instead of being of equal length, are of different lengths, so that the additional length of tube to the light- hand trumpet is equal to the extra distance fiom the source to the left-hand tiumpet. In this case it is evident that the delay of the sound in reaching the left-hand tiumpet is balanced by the cxtia delay between the right-hand tiumpet and the ear, and that binaural balance will be obtained although the source is on one side ot the median plane. It is therefore possible to obtain the dnection of a source with a fixed bar carrying the leceivers, piovided that airange- ments can be made for varying the length of the stethoscope or ear tubes, and such an arrangement is called a binauial compensatoi, the most simple form of which is shown in fig. 3. Here the two equal tubes fiom the trumpets are brought to the two ends of a long straight tube, which is, however, made of three sections, the middle one sliding in the two end portions. The middle tube is blocked at its centie, and is provided with two apertures from which two equal rubber tubes are taken to the ear pieces. When the centre section of tube is in its middle position the two lengths of air path from the trumpets to the ear pieces are the same, and binaural balance will therefore only be obtained when the source is \ Ml UK \\ ( '<)iMI'l Ns \ I ( )|v i ( Ik I )IM ( I l( >S \l I Is I I MM HLOl K L-AMHIf D BYUPI'FR F'LATh TRUMPET \^ IU U> Xv ^ TRUMPET EARPIECES FIG. 4($). PRINCIPLE ot AMIKICVN Gt)Mi'i NSVIOK SUBMARINE SIGNALLING 39 bsmd he median plane; but if the source is to the right of this plane, that the sound reaches the right-hand trumpet first, sliding the tre tube to the left increases the path from the right-hand upet and diminishes that from the left-hand one, so that balance be restored, and an index on the sliding tube will read off the action on a suitably engraved scale which can be divided from zd relation 2 d = b sin0, or sin0 = , where b is the distance between b trumpets, d the displacement of the central tube from its mid ition, and 6 the angle obliquity of the direc- i of the source from median plane. In order to carry out jctional listening on se lines with the atest convenience, a ular form of compen- )i has been designed he United States and de by the Automatic cphone Company 4 shows the ex- lal appeal ancc oi this apensator, and fig. 46 essential feature oi its constiuction Two concentnc ;ular giooves aie cut m a fixed plate, and are covered by a te which conveits them practically into circular tubes This te can be rotated above the fixed plate, and is piovidcd with ) projections which close the grooves but connect the inner and er ones togelhei by two cross channels. The sound from the } trumpets, entering the two ends of the outer groove, travels ind this groove to the stop and then passes thiough the mnels to the inner grooves, returning to its two ends, to which ; ear pieces are connected. It is evident that as the upper te is turned the difference of path between the two systems altered by" four times the distance through which the stop vels, and a pointer on the top plate indicates the direction on dial. This binaural principle is of such importance that it has been scribed at length, and many applications of it will be seen later, Fig 1 binaural Method with Rectilineal Compensator 3io THE MECHANICAL PROPERTIES OF FLUIDS but there are other methods of directional reception which may first be referred to. Sum -and -difference Method. In the case of electrical receivers the binaural method may be replaced by what is called the sum- and- difference method. Suppose, m fig. 5, that our two trumpets on the bar are replaced by two similar ordinary microphone receivers M. I and M a , and that these receiveis are connected to two telephone transformers Tj and T 2 , the secondaries of which can be M, SUM-AND-DIFFERE.NCE METHOD Fig. 5 connected in series as shown. It is evident that if the source is in the median plane, so that the sound reaches both receivers simul- taneously, they should be similaily affected and produce equal electromotive forces in the transformer secondaries. If these secondaries are connected so as to assist one another, a loud sound should be heard, but if one of them is reversed by the switch s the two electromotive forces should be equal and opposite, arid silence should result. But if this is the case, and the source moves to one or other side of the median plane, the sound will reach the two receivers at different times, and cancellation should no longer take place, so that the source should appear louder the greater the angle of reception from the median plane. By swinging the bar SUBMARINE SIGNALLING 311 til the sound vanishes, or at least becomes a minimum, the direc- n of the source is given just as in the binaural method, and in s position the sound will be a maximum when the transformers ust one another. This sum-and- difference method has the advantage over the laural method that it does not depend on the binaural sensitive- 3s of the observer, which may be very poor, especially in the case pel sons with partial deafness in one ear; and on this account ne observeis prefer it On the other hand, it reintroduces the jectionable featuie of swinging the bar unless a cornpensatoi is reduced between two sets of icceiveis which intioduces undesir- le complication. But in any case this sum-and-difleience method of great value in connection with electrical receivers, as it brings t a difficulty which has to be overcome before such receivers can used ior binaural listening It will be noticed that for silence be obtained with the difteience connection the sound must affect th receivers equally, but this is very larely the case with oidmary crophones, owing to difleicnccs in the propeities of their clia- ragms. In fact, if two such icceiveis aic placed close together as to receive the same sound, it is not uncommon to find very le difference between the sound heaid with the sum-and-difleience mections, and in this case such leceivers arc quite useless for lauial listening, which depends upon perfect similarity of icsponse. replacing the oidmaiy metal 01 caibon diaphragms by i ubber mbranes, howevei, much greater equality can be secured, and the n-and-difference method can be used in the test loom to test s equality and to select peifectly paued receivers either for bmauial for sum-and-diffeience diiection finding Directional Receivers. It has alicady been pointed out t although the picssure changes m an acoustic beam have no ection, the displacements take place in the direction of propaga- a, and that a displacement receiver should therefoie have direc- lal properties. This principle has not actually been employed directional listening to any extent, but Mr. B. S. Smith has ised a displacement receiver consisting of a small hollow sphere ttaining a magnetophone transmitter, the whole arrangement ng of neutral buoyancy. Such a sphere vibrates as if it were t of the water, and consequently gives maximum effect on the gnetophone when its axis is in the direction of propagation and o when it is perpendicular to it. The type of directional receiver which has been most employed Sis THE MECHANICAL PROPERTIES OF FLUIDS in practice, however, is of a balanced type, as shown in figs 6 and 7, It is similar to the non-duectional hydrophone (fig. i), except that instead of a thick hollow metal IJ 51 \ / \Q / x i. i / s If \ T -TI \ i, . 1 Fig 6 Bi-directional Hydrophone Fig 7 Polar Curves of Intensity for Bi-directional Hydrophone case it has simply a heavy brass ring with a cential diaphragm having a hollow boss at its centie in which the button microphone is fixed. Obviously if such an arrangement is placed so that its plane lies along the direction of piopagation, the pressuie falls upon both faces of the diaphragm equally and simul- taneously and no motion results, so that nothing can be heard in this position. When the hydro- phone is turned with one of its faces towaids the source, however, the back face is screened by the ring, and the sound reaches it later and with less intensity, so that there is a resultant effect. On turning such a hydrophone round, therefore, the sound is a minimum when the edge points towards the source, and rises to a maximum when turned through a right angle, the intensity for various angles of turning being shown in the polar diagram fig. 7. A similar effect is given by the Morris- Sykes directional hydrophone (fig. 8), which has two similar diaphragms on its two faces connected by a rod at their centres, on which the microphone is mounted. As the variations in pressure tend to move the two Fig 8 Morns-Sykes Hydrophone SUBMARINE SIGNALLING phragms in opposite directions, no movement of the bar is )duced and no sound heaid when the hydrophone is edge on. These forms of directional hydrophone are fairly effective, giving liily sharp minimum, but they do not entirely fill the requirements directionality, as it is evident that minimum is given when either *e of the disc points to the source, so that the source may be in icr of two diametrically opposite directions. For this reason they called bi-directional hydrophones; but it has been found possible get over this difficulty and to convert a bi- directional into a uni- / I Fig 9 Uni-directional Ilydioj lion and Polar Curv o of Intensity 'ctional hycliophone, by simply mounting what is called a iffle plate " a few inches away fiorn one face, as shown in fig. 9. s baffle plate may be made of layers of wood or metal 01 have ivity filled with shot m it, so that it tends to shield the sound n one face. Such a hydrophone gives the loudest sound when unbaffled face is turned towaids the source and the weakest ad when it is turned directly away from it, the intensity in ous directions being shown by the polar curve, so that there ow no ambiguity as to direction, and the device is then called m-directional hydrophone. It does not, however, give such nite indications of direction as the sharp minima of the bi-direc- al form, and it is therefore better to couple a uni-directional a bi-directional hydrophone at right angles to one another on 3 i4 THE MECHANICAL PROPERTIES OF FLUIDS the same veitical shaft. When maximum intensity is obseived on the former and a minimum on the latter, the diiection of the source is definitely given. Besides the foregoing methods of directional reception there are others, such as those of Professors Mason and Pierce, depending on the principle of acoustic integration first enunciated by Piofessor A. W. Porter, which leads to the use of large flat surfaces for iccep- tion, and the Walser gear in which the sound is brought to a focus by a lenticular device, as will be described below. As regards directional transmission, it may fiist be mentioned as a general principle of all radiation that transmission and icception are reciprocal problems, that good receivers make good transmitters, and that a directional receiver will make a directional tiansmitter with the same distribution of intensity in diflerent directions. For example, if, instead of listening by means of two ttumpels coupled by equal tubes to the ear, we bring the two tubes to a poweiful source of sound so that the sound escapes in an exactly similai manner from the two trumpets, an observer in median plane will hear this sound very loudly, but as he moves to one 01 othei side of this plane the sound will appear lainter. Similarly, by vibrating the diaphragm of a uni-dnectional hydrophone sound will be emitted chiefly in one direction, and by extension ol this principle a beam of sound may be sent in any direction we please PRACTICAL UNDERWATER TRANSMITTERS AND RECEIVERS We can now turn to the actual devices employed loi submarine signalling, and they may be described under the headings (a) trans- mitters, (b) receivers, and (c) directional devices. SUBMARINE TRANSMITTERS OR SOURCFS OF SOUND The simplest form of submarine tiansmitter is the submarine bell which has been used as an aid to navigation for many years. Originally suggested by Mr. Henry Edmunds in 1878, it was not until 1898 that it was taken up seriously as a practical navigational device by Mr. A. J. Munday and Professor Elisha Gray, who formed the Giay Telephone Company in 1899, an d employed a bell struck under water with a submerged telephone icceiver. After Professor Gray's death in 1901, the work was carried on by Mr SUBMARINE SIGNALLING iday, who stalled the Submarine Signal Company to take over >perations. Various forms of submarine bell were expej imented , but the form which was finally adopted is shown in fig. 10, consists of a bronze bell, weighing 220 Ib. and having a frequency 115 ~ in water, which is struck by a hammer generally operated ompressed air. A twin hose pipe is used to supply the corn- led air and to convey away the exhaust air from the appa- , and the strokes are legulated by ode valve ", which consists of a [ diaphragm actuating the main apply to the hammei mechanism type of bell is generally used on ships, in which case it is simply overboard to a depth of 18 to ., but m the case ol lighthouses e electric supply is available an ically operated bell ol the same is employed, which is hung on )od stand about 25 ft. high and . spread, standing on the bottom ly convenient position up to a or so from the lighthouse. In :ase the hammer is opeiated by Dulai iron armatuic atti acted to ectiomagnets on a common yoke, ole faces being coveied by coppci to prevent sticking by lesidual etization A foui-coie cable is led, two for supplying the 3! of operating cuirent, the othei )emg connected to a telephone nitter m the mechanism case, which enables the operatoi to f the bell is working propeily. The first of these electrically ted bells was laid down at Egg Rock, near Boston Haibour, d States, and a large number of pneumatically and electrically ed bells aie now in service round the British and American Fig 10 Submannc Signal Company's Licit lileuucally operated type 316 THE MECHANICAL PROPERTIES OF FLUIDS Electromagnetic Transmitters On account of the ease of the operation and control, electio- magnetic transmitters have been most popular, and they are now made up to large sizes transmitting hundreds of watts of acoustic power. They may be divided into two classes: (a) continuous, and (&) intermittent or impulse transmitters (a) Continuous Electromagnetic Transmitters. In all these transmitters alternating current is employed, of frequency corresponding to the natural vibration frequency of the vibiating system, and this current may be used either to energize a laminated electromagnet which acts on the diaphragm, or to traverse a coil in a powerful steady magnetic field, thus developing an alternating force which can be applied to the diaphragm. These two types of transmitter may be called the " soft-iron " and the " rnovmg- coil " types respectively. In the former type the frequency of the note is double that of the alternating current as the diaphragm is attracted equally when the current flows in either direction, but in the latter type the note frequency is the same as that of the current. The soft-iron type of continuous transmitter has been greatly developed by the Germans, and fig. n shows one of the most gener- ally used types constructed by the Signalgesellschaft of Kiel. The diaphragm D is provided with a boss at its centre, to which is fixed a casting carrying a laminated E-shaped iron core C nearly in con- tact with a similar block of stampings C' above The exciting coil encircles the inner pole of these stampings, as in the familiar core type of transformer, and produces a powerful attractive force at each passage of the current m either direction, so that the ficqucncy of variation of the force is double that of the current The upper block of stampings is not rigidly fixed, but is coupled to the lower block through the agency of four vertical steel tubes T with steel rods inside them, the lengths of these rods and tubes being such that the natural frequency of their longitudinal vibiations is equal to that of the diaphragm. The diaphragm is bolted to a conical housing with glands for the introduction of the supply cables. A transmitter of this type, having a total weight of about 5 cwt. and a diaphragm about 18 in. diameter, gives an acoustic radiation of 300 to 400 watts, the mechanical efficiency being about 50 per cent. A great objection to these moving iron transmitters is their inherently low power factor owing to their great inductance, which involves a large wattless exciting current. This can, of course, be I' K II ( ON I INI 01 S I I I < 1 KOM \(,M n< 1 K \NSMI I I I R \S ( I )\s I M ( III) h\ I 111 Sl( N \l ( I si I I S( II \l I Hi Kill IMC. 20 SIM.I i DiAi'iimc.M HI-DIKII- IIONVl I I\ DROl'llONI (ONMKIID INK) I'M DIKIt IIONVI iNSIUl'MINl BY AUDI I ION Ol HA! MI Pi \ II , Fat i D C, EXCITING COIL MAGNET SUBMARINE SIGNALLING 3x7 tilled by using a large condenser in parallel or series with the iting coils, but this is not a very satisfactory expedient, On this account the moving-coil type of transmitter has been cured, especially by the lericans, and its funda- ntal principle is diagram- tically shown in fig. 12. ie coil of wiie travel sed the alternating current is ached directly to the dia- ragm, and moves in the nular field of a powerful Dot magnet " excited by -ect current. This type s relatively little induc- ice, and therefore a high A MOVING COlt DIAPHRAGM -I'nnuple of Moving-coil Tinmmittcr wer-factor, but its con- ruction is mechanically difficult, as the coils of wire do not rm a rigid mass and are theieloie liable to cause great damping id loss of efficiency. This difficulty was very neatly got over by Fessenden in the nited States, and the Fessenden ansmitter is probably the most, ficient and poweiful of all electio- iagnetic tiansmitters The pnn- ple is exactly the same as above, tit, instead of mounting the coil irectly on the diaphragm so as to love with it, Fessenden employs a xcd coil which induces currents in copper cylinder by transformer stlon, and this copper cylinder is ttached to the diaphragm. Fig. 13 tiows a diagrammatic section of a essenden transmitter in which the irect- current electromagnet is bi iolar and encircles the copper cylin- der which is attached to the dia- hragm. The alternating current traverses a fixed coil wound on an aner iron core, the coil being wound in. two halves in opposite direc- ions to correspond with the two poles of the magnet, and this coil -Diagram of Feaaenclon Transmitter 3 i8 THE MECHANICAL PROPERTIES OF FLUIDS induces powerful currents in the copper cylinder which traverse the strong field of the magnet and impart longitudinal forces to it of the same frequency as that of the alternating current. The arrangemenl is therefore very rigid mechanically, and a high power-factor and efficiency are obtained at resonance, which is usually for a frequency of 500 ~. Transmitters of this type giving an acoustic radiation of 500 watts or more have been constructed, and are capable oj signalling under water to a distance of 300 miles or theieabout Moise signals can be senl by either of the above types of transmitter b) the aid of a suitable sig- nalling key. (b) Intermittent 01 Impulse Transmit- ters. Reference has al- ready been made to the " submarine bell, which \-: j was the fiist type oi intermittent submaimc transmitter and which '' can be opeiated electro- J '/ magnetically A more '/^//////////l simple type of impulse / / A// // // / //jr ,, , | transmitter is the cha- Fig 14 Diaphragm Sounder phl'agm SOimdcr of Ml B. S Smith, which has the advantage over the bell in that the staking mechanism is totally enclosed and therefore does not work m water. Fig. 14 shows a section of a sounder of this type, which is provided with an ordinary steel diaphragm with centre boss against which a cylin- drical hammer strikes. This hammer is withdiawn on passing direct current through the exciting coil, against the foice of a spnal spring, and upon the sudden interruption ol the current the spiin causes the hammer to strike the diaphragm with a single sharp blow thereupon rebounding and leaving the diaphragm free to vibrate A very powerful impulse, though of brief duration, owing to the heavy damping of the water, is produced in this way. Similar powerful impulse transmitters have been constructed ir which the hammer is operated pneumatically by compressed air ai a frequency of about a hundred blows per second, and this type oJ SUBMARINE SIGNALLING 319 ismittei can be used for signalling in the Morse code, by means a suitable pneumatic key. A simple tiansmitter has been specially designed by the hor for acoustic depth sounding, the object being to give a ies of single impulses to the water without vibration. Here the ?tic diaphragm is entirely done away with and its place taken a square laminated plate, which is attracted to an E-formed dnated magnet on passing direct current round an exciting coil B 2 ircling the centre pole. The atti active force is dynes per 877 tare centimetre, where B is the magnetic field in gausses, so that B = 15,000, the loice is about 14 Kgm. pei square centimetie, 1 a pole area of 140 sq. cm. gives a total foice of about 2 tons, order to impart this foice to the water, the pole faces and plate grooved, and india-rubber slnps mseited which are compressed the attraction of the magnet On switching on this tiansmitter a loo-volt cncuit the current uses compaiatively slowly, owing its great inductance, and the plate is giadually drawn up, but on Idenly bi caking the cunent the reaction ot the iubbci strips >ots the plate suddenly forward with an initial foice of about ons, and imparts a single sudden shock like an explosion to the ter The use of this transmittci will be explained in connection h acoustic depth sounding Submarine Sirens A number of forms oi submaime siren, in which plates 01 cylin- s provided with holes thiough which jets oi water pass when the tes 01 cylmdcis aie rotated, have been devised both in this country i in Germany, and are extremely powciful By suitably bevelling : holes, the watei piessure can, of course, be made to lotate the tes, but this is objectionable iiorn the signalling point of view, it involves a gradual miming up to speed and a consequent iation m the frequency of the note On this account the plate cylinder is usually rotated independently at a constant speed by electric motor, and signalling is effected by switching on and the higlb-pressure water supply. These sirens have not, how- :r, come greatly into use, as the electromagnetic transmitters are much more convenient, and they will therefore not be described detail. There are many other forms of acoustic transmitters, but the 320 THE MECHANICAL PROPERTIES OF FLUIDS above are most geneially useful for acoustic signalling or impuls transmission. For sound-ranging purposes small explosive charge are sometimes employed. RECEIVERS OR HYDROPHONES The C Tube The eaihest and most simple of all subaqueous acoustic icceivers as already mentioned was the Broca tube, consisting of a length of meta tube with a diaphragm sti etched ovej its lower end The Americans have improved this form of tube, by le placing the diaphragm by a thick- walled rubber bulb 01 teat, and have called it the C tube (fig 15) from Dr. Coohdge, its inventor It is fairly sensitive, but the amount oi energy communicated to the an within the bulb is very small by the principle of transmission given above, and it sutlers from the inconvenience of requiring the obsciver to listen at the end of a somewhat shoit tube. The advantages, as icgaids sen- sitiveness and convenience, of em- ploying miciophones were also appreciated by the Americans who enclosed microphones in hollow rubber bodies, and a combination of three such bodies was often floated on a triangular frame and employed for binauial listening Fig. 16 shows a double C tube anangement for binaural listening. As has al- ready been explained, binaural lis- tening on two receivers permits the direction of the source to be ascertained, but it is evident that the direction suffers from the same ambiguity as in the bi-directional hydrophone, as a source symmetrically situated on the other side of the line joining i! S Fig 15. C Tube SUBMARINE SIGNALLING 321 r two hydiophones would give the same difference of time of ral. By using tfcuee hydrophones arranged at the corners of equilateral triangle, and urallmg on each pair in , this ambiguity disap- s. The necessity for ectly pairing the micro- les by the sum-and-dif- tice method has been idy referred to. Magnetophones \lthough greatly inferior sensitiveness to micro- ties, magnetophones have e advantages for under- T listening, as they are from the vagaries of ^_ ^ ular microphones and ' be more easily paired binaurallmg As their itiveness can be enhanced most any extent by the Q, "J Srn Valve amplifiers, Fig 16 Double C Tube Bmaurnl Arrangement h cannot be employed microphones owing to the grating or " frying " noise pro- d by the granules, they can be made equally effective, lie Fessenden transmitter described on p. 305 can be used as a ;rful magnetophone receiver by exciting its magnet and listening he coils, which are supplied with alternating current when nutting, and it is commonly used as a receiver in signalling, as of course, in tune with the note of all such transmitteis This > tuning, however, renders it unsuitable for general listening oses. >ne of the most effective magnetophone devices for inboard ing is the " air-drive " magnetophone of Mr, B. S. Smith [7). It consists of a massive lead casing (4) fixed to the side of lip, carrying a thick mdia-rubber diaphragm (2) in contact with /ater. Close behind this diaphragm an ordinary Brown reed- telephone receiver (3) is mounted, so that the sound transmitted (DS12) , 322 THE MECHANICAL PROPERTIES OF FLUIDS from the water to the air behind it causes the diaphragm and iced of the receiver to vibiate and induces cunents m the receiver wind- ings ^This type of leceiver connected to a three- valve amplifiei and high-resistance telephones gives a fairly faithful reproduction of ordmaiy sounds; and if four of these receivers are mounted on the hull m positions fore and aft and port and starboard, the screening effect of the hull enables the direction of the source to be estimated from the relative intensities on the four receiveis a foui-way change-over switch being interposed between the receivers and the amplifier. Ship noises aie greatly diminished by fixing the lead ^^^*'^^/^ "'.;%,, 0| = fxx fvJ U-x^^H^ ~~ Fig 17 Air-drive Magnetophone ring to the plates with a rubber seating, as the gicat ineitia of the lead (4) prevents it from taking up the hull vibiations readily. Theie are many other forms of receivers, but the above are the principal ones which have been used for undei water acoustic reception. PRACTICAL CONSTRUCTION OF HYDROPHONES A few illustrations may now be given of the actual foims of some of the most generally used hydrophones. Fig. 18 shows the simplest form of non- directional hydrophone, of which a diagiam was given in fig. i , in which a heavy hollow bronze casting is pro- vided with a diaphragm on one side, to the centre of which a small " solid back " microphone is attached. Fig. 19 is an illustration of the double- diaphiagm bi-directional hydrophone, diagramrnatically shown in Rg. 8, and fig. 20 (see plate facing p. 316) shows a single- diaphragm bi-directional hydro- Ftitniff pagt )-- SUBMARINE SIGNALLING 323 ihone converted into a urn-directional instrument by the addition if a baffle plate, as m fig. 9. In order to be able to listen from a ship in motion and to reduce hip and water noises as much as possible, hydrophones, either of he rubber-block form or of one of the foregoing types, have been nclosed in fish-shaped bodies and towed through the water some iistance astern, and combinations of such bodies have been used or directional listening by bmaurallmg. The modern tendency, Microphone Fig 21 Reception by Hydrophone in Tanks owever, has been m the direction ol inboard listening, by securing Hcient acoustic insulation from the hull. The method of listening in tanks inside the hull, fust intio- iced by the Submarine Signal Company, has been greatly lopted by the Germans. Fig. 21 shows the disposition of a pair these tanks with the hydrophones inside. This device avoids ie great loss by reflection on passing from water to air, as has sen referred to above. A remarkably interesting and effective form of directional m- >ard listening device, however, is that known as the Walser gear, jvised by Lieutenant Walser of the French navy, in which the 3 2 4 THE MECHANICAL PROPERTIES OF FLUIDS sound is brought to a focus, as in a camera obscura, and the direc- tion determined by the position of this focus For this purpose a "blister", consisting of a steel dome A of spherical curvature and about 3 ft. 6 m. diameter, part of which is seen m fig 22, is fitted to the hull, and this steel dome is provided with a large number of apertures B into which thin steel diaphragms C are inserted. These Fig 22 Walser Apparatus diaphragms being on the spherical dome collect the sound and direct it to a focus at a distance of 5 or 6 ft A trumpet D, to which a stethoscope tube is attached, is mounted on an arm E turning on a vertical axis, so as to be able to follow the focus and point in the direction of the sound from whatever direction it comes. Two of these blisters are generally mounted somewhat forward on the two sides of the hull, and an observer seated between them applies the tubes from the two trumpets to his ears, so that he can follow the position of the source on either side, the direction being given on a scale when the maximum intensity is obtained. SUBMARINE SIGNALLING 325 DIRECTIONAL DEVICES Sound Ranging One of the most important acoustic applications in the War ras that of sound ranging for the detection of the position both of uns and of submarine explosions, the importance of which is bvious. There are two chief methods of location, which may be escribed as "multiple-station" and "wireless-acoustic" sound mgmg respectively, but the former, although less convenient, was le only one employed in the War, as it needs no co-opeiation on ie part of the sending station. Multiple -station Ranging. The multiple-station method Fig 23 Sound-ranging Diagram / / sound ranging depends on the principle that sound waves are tit out as spheres with centre at the source of sound. If three more receivers are therefore set up on a circle with centre at 5 source, the sound will arrive at all of them simultaneously, so it if the signals are all coincident the source must be at the centre the circle passing through the receivers. If, however, the source in any other position the signals will be received at different tunes, d if the differences of the times of reception are measured the sition of the source can be located by calculation, or graphically. A simple diagram (fig. 23) will make this method clear. Let 5 CD be four receivers in any accurately known positions and DC the position of an explosion to be located. If we draw a circle h P as centie through the receiver A, it is evident that when the md arrives at A it still has the distances bE to travel before arriving 326 THE MECHANICAL PROPERTIES OF FLUIDS at B, and cC and dD before arriving at C and D respectively, so that the times of arrival at B, C, and D are t, = t , and 2 3 = v v v behind that at A. Consequently if we can measure the time inter- vals /tj, a , and 3 , and multiply them by the velocity, we get the perpendicular distances of the station B, C, and D from the circle passing through the source, and if we draw circles round B, C, and D with radii to scale representing these distances, the centre of a circle tangential to these circles will be the position of the source P. The method of determining these time differences almost entirely employed during the War was by means of a multiple-stringed Einthoven galvanometer, four of these strings being connected to four microphones or hydrophones, while a fifth was connected to an electric clock or tuning fork, so as to give an accurate time scale. The image of the strings was focused on a continuous band of bromide paper, which was drawn through the camera and a deve- loping and fixing bath by means of a motor, so that it emeigecl fiom the apparatus ready for washing and drying, though the times could be read off instantly it appeared. To facilitate the reading off of the time intervals, a wheel with thick and thin spokes was kept revolving in front of the source of light by means of a " phonic motor " in sychromsm with a tuning fork, so that a number of lines were marked across the paper at intervals of hundiedths and tenths of a second. Fig. 24 is a reproduction of a sound-ranging recoid so obtained, on which the times of reception at four receivers are marked, and fig 25 a view of the Einthoven camera outfit employed The re- ceivers used in this case were simple microphones, mounted on diaphragms bolted on watertight cases mounted on tripods lowered on the sea bottom and accurately suiveyed, the microphones being connected by cables to the obseivmg station On account of the importance of sound ranging as a means of locating the position of a ship m a fog, efforts have been made to improve it still further, and to eliminate the photographic apparatus. The greatest achievements in this direction have been made by Dr. A. B. Wood and Mr. J. M. Ford at the Admiralty Experimental Station, who have devised what they call a < phonic chronometer " for indicating the time intervals directly on dials to an accuracy within one-thousandth of a second. The principle of the instru- ment is very simple, and can readily be understood by reference to fig. 26. A phonic motor with vertical spindle revolves with a 1'K, .25 - KlN I IIOV1 N C\MIK\ I OK SoiNI) R\N(.INK. FlG 26-TlIRM< DIAL PHONIC ClIRONOMGIliR Facing page 336 3 28 THE MECHANICAL PROPERTIES OF FLUIDS sist in this case of diaphragms with single-point contacts which aj thrown off on arrival of the shock, and remain broken until th< are restored by electromagnets. Each of these contacts is connecte to the electromagnet windings of the dials as shown, and it will I seen that as each contact is broken it breaks one of the circuits i either one or two of the dial mechanisms, and starts the pomte revolving until the breaking of another contact breaks the secorj winding and allows the small wheel to fly away from the revolvir wheel and against a brake which immediately stops it. After tl shock is received at all four hydrophones, therefore, the three dia indicate the time intervals between the ai rival at the first hydi( phone and that at the other three directly in thousandths of second, each thousandth representing a distance of about 5 ft , fro] which the graphical diagram shown in fig 23 can be constiucte and the position of the source indicated on a chart In order to obtain this position as readily as possible the wnl< has devised what he calls a sound-ranging locator (fig 28, sc plate facing p. 326). It consists of a long steel bar pivote at one end on a ball-bearing, the centie of which can be fixe on the chart exactly over the position of one of the hydiophonc Three thin steel bands are attached to the other end of this bar b means of keys, like the strings of a violin, and pass thiough slot in a sliding piece to graduated rods sliding through simiL ball-bearing swivels, which are fixed on the chait in positior corresponding to those of the other three hydrophones. Tl graduations on the sliding bars are maikcd in times to the sea, of the chart, so that by sliding them to the readings eoirespondin to the time differences indicated on the chronometer, each stn is lengthened by the amounts bE, cC, and dD in the diagiam fig. 2; and when the slotted slider on the main bar is pushed down an the bar turned until all the strips aie tight, the point from wluc they radiate indicates the position of the source on the chait withoi any calculation, and a marking point just under the edge of th slider can be depressed to prick the position. In order to secui accuracy, each of the strips is provided with a small tension indi cator which shows when the strip is strained to a definite tensioi Two strips only are shown in fig. 28, but any number ma be employed according to the number of receivers. A device on a similar principle has been put forward by Mi H. Dadounan in the United States.* * Physical Review, August, 1919. Pacing [n using the multiple-station method of sound ranging for iting navigation in foggy weather, a ship desirous of being rmed of its position calls up the nearest' sound-ranging station, :h instructs it to drop a depth chaige As soon as the record sceived on the Emthoven camera or phonic chronometer, the tion of the ship is worked out or marked by the locator and lessed to the ship. iYireless- Acoustic Sound Ranging. A method of sound ;ing which promises to be of much gieatei value foi navigation, which has not yet been fully developed as it was of little value in time, is the wireless-acoustic method proposed by Professor Joly. he original experiment of Collodon and Sturm in 1826, the city of sound in watei was determined by striking an under- >i bell and igniting a charge oi gunpowder simultaneously mowing the distance from the souice and observing the inteival ime between the flash and the sound of the bell the velocity determined, as light travels piactically instantaneously over any naiy distance Convcisely, if the time inteival and the velocity known, the distance oi the souice can be at once dctei mined, as be familiar method oi ascei taming the distance of a lightning i by noting the time between the flash and the tlumdei clap The intage oi employing an nuclei watei method is that sound is tians- cd moie clTectively thioiigh watei, and that theie aie no watei cnts compaiablc with winds to aflect the velocity appieciably Jnf 01 tunately a flash of light is of no value in a fog, but wireless 2s are little affected by it, and travel with the same speed as , so that if a wireless flash and an underwater explosion are lated simultaneously at a lighthouse or other known position, the ship is provided with a wireless equipment and a directional ophone, the distance of the station can be at once determined he ship by noting the interval between the two impulses. As velocity of sound in sea water is neaily a mile a second, the ince can be determined within a quarter of a mile by a simple -watch, and the direction of the source found by either the ;tional hydrophone or directional wireless, without any com- ication with the station. If the lighthouse or lightship simply s out wireless impulses simultaneously with the strokes of the narine bell at convenient inteivals, all ships in the vicinity can e their positions fiom time to time without delay or mutual ference, and if they aie within the range of two such stations can do so without any directional apparatus. (D312) 12, 330 THE MECHANICAL PROPERTIES OF FLUIDS The lecent developments m duectional wireless have lendered the application of sound ranging to navigation of less impoitance, but even now wireless direction finding is not always reliable, especially at sunrise and sunset; and there is also liability to error on steel ships owing to their distorting effect on the wireless waves. As hydrophones become increasingly employed on ships for listen- ing to submarine bells, &c , the ability to obtain accurate ranges by wireless acoustic signals will doubtless prove of great value Leader Gear Although not stiictly speaking an acoustic device, some mention should be made of the leader gear or pilot cables as an aid to navi- gation of harbours and channels in foggy weather For this pur- pose it is necessary to be able to follow some well-defined tiack with a latitude of only a few yards, so that sound langing is inadequate But if a submarine cable carrying alternating current of sonic frequency, say 500 ~, is laid along the desired track, and the ship is provided with search coils with amplifier and telephones, the alternating magnetic field pioduced by the cable induces alternating electromotive forces in the coils, and thus gives a sound in the tele- phones when the ship is sufficiently near the cable By using two inclined coils on the two sides of an non or steel ship it is lound that the sound is loudest when the telephones aie connected to the coil which is neaiei to the cable, so that the ship can be steeied along it, and keep a fairly definite distance to one side of it, so that vessels passing in opposite diiections will not collide This device, which was first put forward by Mr C A. Stephenson of Edinburgh in 1893, was icvived duiing the war by Captain J Manson, and is now coming into use both in this country and in the United States. An i8-mile cable has been laid by the Admnalty from Portsmouth Harbour down Spithead and out to sea. Acoustic Depth Sounding Another purely acoustic device which promises to be of con- siderable value to navigation is that of depth sounding by acoustic echoes fiom the bottom. If a ship produces an explosion neai the surface, the sound travels down to the bottom and is reflected back as an echo, and for each second of interval between the ex- plosion and the echo the depth will be half the velocity of sound or 2500 ft., say 400 fathoms. Various experimenters, notably SUBMARINE SIGNALLING t[. Marti in Fiance, Herr Behm in Geimany, and Officeis of the anerican Navy, have devised apparatus wheieby the time between ring a detonator or other small charge under the ship and the sception of its echo fiorn the bottom can be recorded on a high- peed chronograph, and veiy accurate results have been obtained. T Uh- DETONATOR' Fig 29 Behm's Acoustic Depth-sounding Method The method of Behm, called the " Echolot " or echo-sounding evice, now being developed by the Behm Echolot Co , Kiel, has ttained a high degiee of perfection, and is claimed to give inch- ations in a ship at full speed, and even in rough weather, to an ccuracy of within a foot. The transmitter consists of a tube irough which a cartridge is impelled by air picssurc into a holder xed on the hull a little above the water line. The caitndgc is red out of the holder on pressing the firing-key, and is shot towards 332 THE MECHANICAL PROPERTIES OF FLUIDS the Impulse Receiver, while a time fuse in the cartridge is airanged to explode a detonator just before the cartudge reaches the micio- phone. Both the Impulse and Echo Receivers are microphones, but the latter is scieened from the direct eflect of the detonator by being fitted on the opposite side of the ship. The explosion of the detonator causes a sudden drop in the current through the impulse receiver and weakens the cuiicnt passing round an electromagnet, and causes it to release an " im- pulse spring " which suddenly starts a pivoted disc in lotation with a uniform velocity until the weakening of the cunent thiough the brake magnet, due to the echo i caching the echo microphone, stops the disc. The angular motion of the disc is theieforc pio- portional to the interval between pressing the firing-key arid return of the echo, and a light minor on the disc spindle causes a spot of light to revolve round a translucent scale divided in depths, and to stop at the depth indicated It is claimed that this timing device is capable of indicating shoit mteival of time to an accuiacy of one-ten-thousandth of a second, corresponding to only 3 in in depth Three keys are provided on the indicator, one for restonng the indicator to zeio, one for filing the chaige and obtaining the depth, and the third for checking the inclicatoi against a standaid time interval A number of detonator charges can be stoied in the transmitter magazine, and fiied as requiiecl The whole appai- atus can be operated by a few dry cells, as the lamp is lit only at the moment of restoration, indication, or checking, and the colour of the light is varied at each opciation to eliminate nsk oi mistake It is stated that a lock with an upper suiface of only 2 sq. meties in area is sufficient to give a correct indication The Bntish Admnalty have icccntly developed a very simple and accurate echo sounding gear. Echo Detection of Ships and Obstacles By means of leader gear, sound langing, and echo sounding navigation in fogs may be made much safei and more regular, but there still lemams the gieat danger of collision in the open sea between ships, and especially with wrecks, rocks, and icebergs. As far as ships are concerned the difficulty is to some extent met already by signalling with sirens, but the curious blanketing and reflecting or lefi acting effect of fogs is a source of considerable confusion and danger. Undei water signalling does away with this difficulty almost entirely, and as hydrophone equipments become SUBMARINE SIGNALLING 333 ore common the nsk of collision between moving ships will rapidly imimsh. With a good directional hydrophone equipment an ordinary eamship can easily be detected and its direction determined up > a range of some miles merely by the noise of its engines. But i the case of wrecks, locks, and icebergs, which emit no sound, te danger is still very great, and nothing but an echo method will ^tect them. Unfortunately this is a difficult matter, as a ship or nail rock at a moderate distance is a very small target for an echo, that the echo is of very small intensity, and it may quite easily j masked by bottom echoes However, Fessenden, by the use of s powerful electromagnetic transmitter, succeeded as early as )i6 in obtaining echoes from distant obstacles, and by employing rectional transmitting and leceivmg devices, which concentrate the und in the desired direction, the strength of the echo can be creased, disturbances reduced, and the dnection and approximate nge of the obstacle determined. As eaily as 1912, just after the name disastei, a pioposal to employ echo detection for avoiding mlar dangers was put loiwaid by Mr, Lewis Richardson, and it ay be hoped that this method will ultimately eliminate the last the serious dangers of navigation Acousnc TRANSMISSION OF POWLR Beloie concluding this article, reference ought to be made to the mderful achievements of M Constantmesco, as showing the possi- lities of what may be called acoustic engineering. For the pur- ges of underwater signalling the power transmitted, although large comparison with what we have heretofore contemplated in con- ction with sound, raiely exceeds a hundred watts; and it has been t for M Constantmesco boldly to envisage the possibility of msmittmg large amounts of power by alternating pressures in iter of sufficiently high frequency to be described as sound waves. >r many yeais it has been customary to illustrate the phenomena alternating electric cm rents by hydraulic analogies, and the esent writer has even written a book in which such analogies ve been used as a means of giving a complete theory of the subject; t the obvious possibility of using such alternating piessures in ter for practical purposes was entirely missed until M. Con- mtmesco conceived it, and immediately the idea occurred it was ident that the whole of the theory was ready to hand from the 334 THE MECHANICAL PROPERTIES OF FLUIDS electrical analogies. In a surprisingly short time, therefore, M. Constantmesco has been able to devise generators, motois, and transformers capable of dealing with large amounts of power trans- mitted by hydraulic pipes in the form of acoustic waves of a frequency of about 50 '--'. The generator is, of course, simply a high- pressure reciprocating valveless pump, and the motor can be of similar construction, but by having three pistons with cranks at 120, three-phase acoustic power can be generated and employed in the motors. The first commercial application of M Constantmesco 's devices has been to reciprocating rock drills and riveters, for which this method is especially suitable, as the reciprocating motion is obtained simply from a cylinder and pistol without any valves whatever, and the power is transmitted by a special form oi flexible hydraulic hose pipe comparable with an electric cable It is not too much to say that M. Constantmesco 's ideas have opened up an entirely new field of engineering, and their development may have far-reaching effects. For a discussion of the theory of hydraulic wave transmission of power, see Chapter VI. Although this article is necessarily very incomplete, it will at least have served its purpose of showing the great importance of underwater acoustics, and there can be no doubt that a new depait- ment of scientific engineering has been opened up which has vast possibilities Developments in Echo Depth -sounding Gear. Since the first appearance of this volume, the chief advance in underwater acoustic devices has been in the improvement of echo depth-sounding devices which have proved their gieat value for navigation and appeal likely in time to become a standard feature of ship equipment Three diflcrent types of such gear are now manu- factured m this country: the Admiralty type by Messrs H. Hughes & Sons; the Langevin piczo-electric type by the Marconi Sounding Device Company, and the Fathometer gear, which has been de- veloped from the original Fessenden apparatus by the Submarine Signal Company. All these devices have now been made to give both a visual indication of the depth on a dial and a continuous record on a chart. The basis of all methods of acoustic depth sounding is the re- cording of the time taken for a signal to travel from the ship to the bottom of the sea and leturn, but they differ in the type of the signal SUBMARINE SIGNALLING 335 ad method of indication, and may be divided into impulse or some " methods and high frequency or " supersonic " methods a the formei class to which the Behm " Echolot " (see p. 331), le original Admiralty sonic gear, and the Fathometer belong, the gnal is in the form of a single powerful impulse provided by an splosive cartridge or an electromagnetic or pneumatic hammer nkmg a diaphragm; while in the latter a short train of high- equency vibrations is emitted from a quartz piezo-electiic oscillator, steel rod which vibrates at a high frequency when struck by a ammer, or by a magnetostriction oscillator which is the magnetic nalogue of the quartz oscillator The single impulse or sonic transmitter is practically non- irectional, i.e the disturbance tiavcls equally m all directions under ic ship. This has the advantage of making the indications practi- illy independent of any rolling of the ship, but it has many dis- ivantages Firstly, it is liable to give such a severe shock to the iceiver at the moment the impulse is sent out that it does not ;cover in time to icspond to an echo fiom a very shallow bottom, icondly, the greater part of the energy is wasted, thndly, the echo mst be very strong to be heard above the noises caused by the ship's lachmery and motion through the water, and iomthly, il may not ive true depths if the bottom is shelving steeply, as the fust echo received from the object which is nearest to the ship With the igh-frequency method the sound can be concentiated within a 3ne of any desired angle, so that the receiver can be fanly close to xe transmitter without sustaining any severe initial shock, and the >ceiver can be sharply tuned to the transmitted fiequency, so that is nearly deaf to any othei disturbances If the ship could be kept n. a perfectly even keel, the narrower the beam the better, as it ould be equivalent to a vertical sounding line, but on account of )llmg it is desirable that it should have an angle of something like ilf the maximum angle of roll. For a circular transmitter the 'imangle of the beam 9 is given by the relation sin 9 = i z~, wheic A the wave length of the sound and d the diameter of the tians- utter, so that we can obtain any beam angle we please by varying le diameter and frequency As regards receivers, a granular microphone is the most suitable ir the single impulse or some system, and it must be mounted at >rne distance from the transmitter and preferably on the other side the keel, so as to be shielded as much as possible from the initial 336 THE MECHANICAL PROPERTIES OF FLUIDS shock. This separation is however objectionable, as it seriously reduces the accuiacy of sounding in very shallow water, wheie it is frequently most important. The device, of course, indicates the dis- tance from the transmitter to the bottom and back to the receiver, and this varies very little when the depth imdei the keel is small compared with their separation. With the high-frequency system, however, the transmitter and receiver can be close together, so that this difficulty does not arise; and as both the quaitz and magneto- striction transmitters will also serve as receiveis, it is even possible to dispense with a separate receiver, as is done in the Marconi gear The essential function of the indicator is, of couise, the measure- ment of the time interval between the impulse and echo As the average velocity of propagation of sound in sea water is about 4900 ft. per second, and the sound has to travel the double distance to the bottom and back, each second of interval corresponds to a depth of 2450 ft. or about 400 fathoms; and if soundings are leqmred within an accuracy of one foot, the time must be measured within an ac- curacy of four ten thousandths of a second The most simple and reliable method of effecting this is by the contact method employed in the Admiralty some gear, m which the receiving earphones arc shunted by two brushes, which press on a revolving img which has a small gap in it, so that the phones are short-cucuited for all but an interval of one or two thousandths of a second in each i evolution The transmitter is actuated at a certain moment in each i evolution, and the two brushes are carried on an arm which can be turned by the observer until the short circuit is icmoved simultaneously with the arrival of the echo. The depth is then indicated by the position of the arm on a scale which can be divided in feet 01 fathoms Ducct visual indication is, of course, piefcrablc, and is seemed in the Fathometer gear by a revolving disc mounted close behind a giound- glass scale The disc has a narrow slot in it, behind which is a small neon lamp, and the echo when sufficiently amplified, causes this lamp to flash and show a momentary red streak on the scale at each revolution. In the Marconi gear the amplified echo is received by an oscilloscope or high-frequency galvanometer the beam fiom which falls on an oscillating mirror and shows a luminous streak on a giound- glass scale. When the echo is received the momentary kick of the galvanometer shows as a kink in this luminous streak at the corre- sponding depth on the scale. Messrs. Hughes have produced a direct-reading pointer indicator for the high ficquency Admiralty gear, which operates on the phase indicator principle. A revolving SUBMARINE SIGNALLING 337 lolenoid is fed with direct current and therefore produces a rotating nagnetic field, and a soft iron needle is momentarily magnetised by he current from the echo receiver, so that it sets itself along the ixis of the solenoid at that moment. Any of these devices enable the depth to be observed at intervals )f every few seconds even when the ship is running at full speed, Tvhich is an enormous advantage over the old lead line, which required he ship to be running dead slow Merely for ensuring safety in navi- gating shallow waters this is sufficient, but a great gain is secured by naking the apparatus record the depths continuously on a chart vhich gives a profile of the bottom along its course, as this enables i ship to locate its position with considerable precision if the con- FIR 30 I- Icctrormgnctic Hammer Transmitter for Sh illow Water Gear our of the bottom is accurately known. During the last few years ecorders have come into general use, and have been found very atisfactory The motor which actuates the tiansmitter contacts md the receiver mechanism is also employed to move a stylus uni- ormly across a band of paper which has been previously soaked in a ensitive solution (usually starch and potassium iodide, as in the arly Bain printing telegraph), and the amplified and rectified echo auses it to make a mark on the.paper at the moment it is received The paper band is moved slowly forward at a constant rate by the ame motor, or it can be driven from an electrical log so as to move >roportionately to the distance covered by the ship, and, as the tylus makes a mark for each echo, a practically continuous line is Irawn on the paper showing the variation of depth either with time >r distance. By simple contact devices the stylus can also be made o mark the paper at each five or ten feet or fathoms of depth, and t regular intervals of time or distance, so that the record is complete, 33 8 THE MECHANICAL PROPERTIES OF FLUIDS and can be reproduced directly m a hydro-graphic atlas. Fig. 31 shows such a record of a 15 minutes' run, with a shallow water magnetostriction set. After the above general description, the only features of the various gears which require special consideration are the tians- Ftg 32 Pneumatic Hamrnu Tiansmittci mitters. For the impulse or sonic transmitters the types employed in the shallow water Admiralty gear and the Fathometer gear are very similar, and the former is shown m fig. 30. A ring of iron stamp- ings, with internally projecting poles, is excited by coils on the poles, and the hammer consists of a tapered block of stampings, which is drawn into the gap between the poles and compresses a spiral spring which drives the hammer down against a diaphragm when the O> Facmg page 33& SUBMARINE SIGNALLING 339 rent is broken. For the deep water Admiralty gear, which has in used for depths of over 2000 fathoms, the hammer is operated 2umatically with an electromagnetic release (fig. 32). The Marconi high-frequency quartz transmitter, which also ves as the receiver, shown in figs. 33 (p. 338) and 34, has a thin er of quartz crystals H cemented between two steel discs F and G, 1 IK 34 Marconi Quaitz Ti. inarm tier e lower of which is usually m contact with the water while the upper highly insulated and connected to a high-voltage oscillator which ves a frequency of about 37,500 periods per second, producing ives about 4 cm long in the water, and a somewhat sharp beam i 01 dei to provide for the removal and replacement of the oscil- tor without diy-dockmg the ship, the housing is sometimes pro- dcd with a second resonant steel plate shown at the bottom of C Inch is clamped by a central flange and transmits the oscillations the water.* * Cdr. J. A. Slee, C.B.E., R.N., Journal Institution Electrical Engineers, Dec. 1931. 340 THE MECHANICAL PROPERTIES OF FLUIDS Quartz oscillators, although highly efficient, are somewhat costly and require special technique in construction, and hence efforts have been made to obtain a high-frequency impulse without employing crystals. One simple method, which is fairly effective, is to employ the ordinary hammer of the single-impulse transmitter, but to substitute a steel rod clamped at its centre like the lower plate in the Marconi transmitter for the diaphragm. This lod i i ..Toroidal winding _Thm sheet "paper, & cement End load c erncni eg to nickel Fig 35 Magnetostriction Scroll-type Oscillator vibrates with its resonance frequency and emits a short train of damped oscillations each time it is struck But within the last few years a great advance has been made by employing the principle of magnetostriction, i.e. the defoimation which takes place in magnetic materials when they are magnetized. This effect is most marked in pure nickel, and m certain nickel and cobalt steel alloys, and it enables oscillators of any frequency to be constructed very cheaply and by ordmaiy workshop methods It lends itself to very various forms of oscillator, but the two which have been found most convenient for echo sounding work are the " scroll " and " ring " types shown in figs. 35 and 36, In the former, a strip of nickel is simply wound up like a scroll of paper, and is SUBMARINE SIGNALLING 34* rovided with a simple toroidal winding like a gramme ring armature, /hen this winding is supplied with alternating current the scroll tpands and contracts axially, so that a disc cemented to one end ;rves as the emitting surface. The axial length of the scroll is made ich that its mechanical lesonance frequency is that required , Toroidal winding \ u -# -! d Annular nickel stampings 1'ig 36 Magnetobtnction Ring Oscillator isually about 15,000 periods per second), and the winding is fed with ternatmg current at a low voltage either from a valve oscillator or condenser discharging through an inductance, in either case at the 'sonant frequency In the disc type the oscillator is made up of a imber of ring stampings with holes round its inner and outer snphery These stampings are cemented together into a solid ock with an insulating cement, and a toroidal winding is wound irough the holes. When supplied with alternating current the 342 THE MECHANICAL PROPERTIES OF FLUIDS ring expands and contracts radially, and it is operated at its resonance frequency. As the sound is emitted m all directions m a plane parallel to the ends of the ring, it is surrounded by a sound reflector made of two thin metal cones with india-rubber " mousse " between v \ J (a) Watertight glarm Rubber mousbc Thm metal Fig 37 Magnetostriction Oscillatoia and RefkUors them, ay shown m fig 37, and the beam angle can be varied by choos mg the diameter of this reflector The transmitters will serve equalb well as receivers, but it is found preferable to mount two of then close together, one acting as transmitter and the other as receiver These magnetostriction transmitters and the associated recorder were designed by Drs. A. B Wood and F. D. Smith and Mr. J, A gg 5! w Q P Facing '$age 342 SUBMARINE SIGNALLING 343 Peachy,* of the Admiralty Research Laboratory, after a research magnetostriction by Dr. E P Harrison, and have been mcor- ited with great success in the latest forms of Admiralty depth- Iransmittmg key Recording stylus Depth marker Constant -speed motor Chemical recorder HT generator _ Rectifier Amplifier Condenser- Transmitting + keyK Fig 38 Magnetostriction Geneial Arrangement mdmg gear. As they only require very small power for shallow )ths without high- voltage oscillators, it has been possible to mstal m with recorders in small motor-boats for the hydrographic survey shallow rivers and estuaries, which has enormously increased the thty and rapidity of such surveys On the other hand, they have * Journal Institution of Electrical Engineer*, Vol 76, No 461, May, 1935, p 55 344 THE MECHANICAL PROPERTIES OF FLUIDS proved equally efficient for moderate depths and deep water geai, and soundings have been taken successfully in depths of 2000 fathoms with the transmitters and receivers in water-filled tanks in contact with the hull, and transmitting and receiving through the ship's plates. Fig 38 is a diagram of the motor-boat outfit with chemical recorder Figs. 39, 40 and 41 show external views of the Marconi, Fatho- meter, and Admiralty indicating and recording sets (Plates facing pp. 342, 346). Probably over two thousand naval and mercantile vessels have by now been equipped with echo depth-sounding gear of one 01 other of the above types, and the icports on them show their great value, accuracy and reliability For moderate depths down to 200 or 300 fathoms, an accuracy of a foot is obtainable, while the mag- netostriction motor-boat gear actually records depths to an accuiacy of three inches even when the boat is almost touching bottom The deep water set on the Discovery II was so satisfactoiy in Antarctic waters that line sounding was discontinued, and such sets have been of great service in cable-laying ships by enabling them to lay then cables on the least irregular bottoms The importance of continuous and recorded soundings for mercantile ships by facilitating their eitry to harbours and locating their positions at sea hah aheady been referred to, and its value will increase as more and moie iccords along the main trade routes become available Lastly, a icmaikable application of such gear has been found for fishing, and many ti awki s are now being equipped with it, as it has been found that the quantity and quality of catches depends greatly on the depth, and echo sound- ing gear enables the ship to follow a contour line of the depth de- sired. Echo depth-sounding has proved the most useful application of underwater acoustics and seems likely to be universally adopted The writer is indebted to the three firms above mentioned for particulars and illustrations in this section, and to the Journal of the Institution of Electrical Engineers for figs. 34, 35, 36, 37, and 38. II Hughes & Son, Ltd FIG 41 ADMIRALTY SUPERSONIC RECORDER Facing page CHAPTER X The Reaction of the Air to Artillery Projectiles Introduction All calculations of the motion of a projectile through the air are cted to one object to determine the position and velocity of the |ectile at any given time after projection in any prescribed manner, general the reaction of the air to a rotating projectile is very com- ated, the complication is considerably reduced, however, if the jectile can be made to travel with its axis of symmetry coincident i the direction of the motion of its centre of gravity. It is a ter of experience that by giving the projectile a suitable spin it be made to travel approximately in this manner for considerable ances; m such circumstances the reaction of the air is reduced to Qgle force, called the drag, which acts along the axis of the pro- lie and tends to retard its motion.* When this drag is known for roj'ectile of given size and shape the problem enunciated above )mes one of particle dynamics, and its solution for that pro- lie can be effected, at all events, numerically. The first and major of this chapter is devoted to the consideration of this drag. When the angle of elevation of the gun is considerable the cur- ire ol the trajectory mci eases too rapidly for these simple condi- s to hold. The motion then becomes complicated and the prob- becomes one of rigid dynamics in three dimensions; the trajectory twisted curve instead of a plane one, and the well-known pheno- lon of drift appears Similar complications arise when the pro- ile is not projected with its axis coincident with the direction of ion, or when the spin is insufficient to maintain this coincidence A couple of small magnitude due to skin friction also exists, it acts about the and tends to reduce the spin; its effect is generally negligible with modern ctiles. 345 346 THE MECHANICAL PROPERTIES OF FLUIDS In the second part of this chapter the component forces and couples of the reaction of the air in these circumstances are briefly con- sidered. THE DRAG Early Experiments: the Ballistic Pendulum Most early writers on ballistics* assumed that the resistance of the air (the drag) to the motion of projectiles was inconsiderable The first experimenter to attempt the determination of the air drag on projectiles moving at a considerable speed was Robins, who, in 1742, carried out experiments with his ballistic pendulum He found that the icsistance encountered was abnormally gi eater foi velocities greater than about noo ft. per second than for lesser velocities Following Robins, many experiments were performed with the ballistic pendulum, notably at Woolwich (by Hutton, T775-88) and Metz (by Didion, 1839-40), to determine the drag as a function of the velocity of the projectile f The method employed by Robins was, briefly, as follows: A gun was placed at a known distance from a heavy ballistic pendulum; the charge was caiefully weighed and the projectile was fired horizontally at the pendulum. The latter received the pro- jectile in a suitable block of wood, and the angle through which it swung was recorded Knowing the weights of the projectile and pendulum and the free period of oscillation of the latter, the velocity of the projectile at the moment of hitting could be calculated The expeiimcnt was repeated with the same charge, the distance between the pendulum and gun being varied from round to round. There resulted a series ol values of velocity at known distances from the gun, the retardation of the piojectile and hence the resistance of the air at these distances could be deduced By performing similai sets of expeiiments with various weights of charge, the drag could be determined as a function of the velocity of the projectile. The unceitamty of realizing the same muzzle velocity in each set of experiments with constant charge vitiated the reliability of the results. Hutton overcame this difficulty by hanging the gun hon- * The study of the flight of piojectiles f For a full account of these expei imenls see Robins, New Principles of Gunnery, 1761; Hutton, Tracts, 1812, especially Tract XXXIV; Dicuon, Lois de la resistance de I'air (Paris, 1857). REACTION OF AIR TO ARTILLERY PROJECTILES 347 itally from a suitable support, so that the gun itself became the D of another pendulum From the angle through which this tern swung on firing, the muzzle velocity of the projectile could calculated. For each round fired he thus obtained two values the velocity one at the muzzle, the other at a known distance m the muzzle Let v and v> 2 be these values, and let x be the distance between a and pendulum. Then, if m be the mass of the projectile, the Wl / \ s of energy m traversing the distance x is f % 2 v^ J. If R 2 > ' the mean value of the drag we therefore have r, mf o 2 N \ R = <y_ 2 <Q 2 ). 2X\ l * J avided that the distance x is sufficiently small, this value may be .en as the actual value of the drag for the velocity v = ^(V L -f- v z ) By varying the charge and the distance between the gun and ndulum Mutton determined the drag numerically as a function of 2 velocity From the time of Hutton to the present day, experiments con- cted on the Continent and in Ameiica to measure the lesistance the air have been based on this pimciple, namely, to determine ? velocity at two points on an approximately horizontal trajectory known distance apart A large number of instruments for mea- nng the velocity of a projectile at a given point have been invented irmg this time; the reader is refeired to Ealistica Experimental Aphcada, by Col Negrotto of the Spanish aimy (Madrid, 1920), r an up-to-date and exhaustive account of them. It should be ited here that few of these chronogiaphs were invented especially r the determination of the resistance of the air; there are, of course, any important uses for such instruments m gunnery. Since 1865 experiments on the resistance of the air conducted England have been based on a different principle. The method is first proposed by the Rev. F. Bashforth, B.D., sometime Pro- ssor of Mathematics at the Artillery College, Woolwich; it consists measuring the times at which a projectile passes a number of uidistant points along an approximately horizontal trajectory. hese times are then smoothed and differenced, and the velocity and tardation of the projectile at a number of corresponding points are Iculated by the method of finite differences. This method is 348 THE MECHANICAL PROPERTIES OF FLUIDS evidently more economical m expenditure of ammunition than that of foreign experimenters. The Bashforth Chronograph In 1865 Bashforth invented his now-famous electric chronograph,* by means of which he succeeded in measuring small intervals of time with an accuracy previously unattained in ballistic instiuments. The chronograph consists essentially of two electro-magnets, to the keepeis ol which two sciibeis aie attached by hnkwoik; these scribeis tiace continuous spual lines on paper fixed on a revolving cylindei The two spnals aie genet ated by a mechanical movement ol the iiameuoik supporting the electro-magnets in a direction piiiallel to the axis of the cylmclei The movement of each keeper is eonti oiled by a suitable spring, and any small movement of either is identified on the iccoid by a kink on the corresponding spiral tiace. One of the dcetio-magnets is connected with a clock and the cm lent is bioken momentarily every second; one of the spiral traces thus constitutes a time record. The othei electro-magnet is con- nected in seiies with screens placed at equal distances along the tiajectory of the projectile. At the moment the latter passes a screen the curient is bioken for a shoit inteival of time and a kink is made in the co i responding spual tiace. The mechanism by which the em tent is bioken is as follows. A boaul us suppoitecl in a honzontal position with its length at tight angles to the dhection of motion ol the piojcctilc Tiansverse grooves aie cut in the board at equal distances, somewhat less than the diameter ol the projectile. Haul spimg-wne staples are fixed in the board so that each ptong piojects upwards fiom a gioove On the near edge of the boaul a number of copper straps are fixed; each strap has two oval-shaped holes which are placed at the near ends of adjacent grooves, The prongs of the staples are bent down into the grooves and project through the oval-shaped holes; the arrangement k such that the butts of the staples and the copper straps alternate, so that a current may pass continuously through the staples and straps. The prongs terminate in hooks from which are suspended small * For ft full account of Bashforth's experiments consult his Rmsed Account oj thu Eptrimmt$ mad$ with the Sashforth Chronograph (Cambridge University Press, REACTION OF AIR TO ARTILLERY PROJECTILES 349 Fig i The Bashforth Chronograph and Screen Reproduced by courtesy from "Description of a Chronograph", by F Bashforth B D , Proceedings of the Royal Artillery Institution, 5, 1867 350 THE MECHANICAL PROPERTIES OF FLUIDS equal weights by means of fine cotton; the weights rest against a second horizontal board supported some distance below the first, and aie sufficiently heavy to maintain the prongs in contact with the bottom edges of the holes in the stiaps. When the projectile passes it will break at least one of the cottons; the corresponding prong will spring from the bottom to the top of its hole in the copper strap and so break the current momentarily. This mechanism constitutes a " screen ". The record, when removed from the cylinder and laid flat, con- sists of two paiallel straight lines with kinks in them; m the upper line the kinks correspond to the passage of the piojectile thiough successive screens; in the lower the kinks indicate seconds of time. With a suitable measuring appaiatus it is possible to read off the time intervals between the screens to four decimal places of a second. Bashforth continued his experiments until 1880, and pioduced a table giving values of the air drag for velocities up to 2780 ft per second. Contempoiary experiments were also conducted m Europe by Mayewski (Russia), Krupp (Germany), and Hojel (Holland), giving results in substantial agreement with those of Bashforth Later Experiments In the early years of the piesent century a Luge amount of work was done in England, Fiance, and Germany to obtain acrmate infor- mation concerning the air diag In 1906 the Oidnance Board used a method similai to that of Bashfoith, but having a moic accuuitc timing and recording device; the drag tor velocities up to 4000 it per second was determined. In 1912 O von Eberhaid, at Kiupp's,* made a large number of experiments with projectiles of various shapes and sizes. It is thought that these foim the most exhaustive set of expeiiments yet undertaken; the results, which aic ficqucntly used in this chapter, aie certainly the most complete yet published openly, In this method the velocity at two points on the trajectory was measured by means of a spark chionograph; the distance between the points varied from 50 m. for small projectiles to 3 Km. for those of large calibre The method of deducing the resistance was similar to that used by Hutton, and results for velocities up to 1300 m, per second were obtained. * Cf O von Eberhard, Artillenstische Manatshe/te, 69 (Beihn, 1912). REACTION OF AIR TO ARTILLERY PROJECTILES 351 Krupp's 1912 Experiments The velocity at a given point is measured, in this method, by irmg the projectile through two screens; one screen is placed a cieasured distance (a few metres) m front of the point, and the other he same distance behind it The spark chronograph measures the ime taken by the projectile to traverse the distance between the creens; the average velocity between them is deduced and is taken 3 be the actual velocity at the point. The distance being short o appreciable error arises from this assumption. Each screen consists of a square wooden frame across which fine oppci wire is stretched backwards and forwards continuously in r To 1st Screen- I To 2nd -j Screen* L, L, To 3rd } Screen, To 4tk Fig 3 The Spark Chronograph used in Krupp's 1913 Experiments ich a way that a projectile passing through is ceitam to break the ire. In the experiments four screens are used; one pair serves > measure the velocity at the beginning of the measuied range, the ther, the velocity at the end of it. The spark chronograph is shown diagrammatically m fig 2 A tetal drum A is rotated at high speed by means oi a suitable motor, ie speed being recorded by means of a Frahm tachymeter; readings within one revolution per second can be taken with this mstru- icnt. It is essential that the speed of the drum be constant during ie flight of the projectile over the range, and this instrument serves ie additional purpose of indicating the most suitable moment to e the gun. The surface of the drum is silvered and is coated with >ot except at one edge where a circumferential scale is fixed. There are four induction coils, I; their primary circuits, which >ntam batteiies, are connected respectively to the four screens; ie terminal of each secondary circuit is connected to the spindle of LC drum, the other terminals being connected to sharp platinum 352 THE MECHANICAL PROPERTIES OF FLUIDS points, P. A break in one of the primary circuits will cause a spa to pass across the small gap between the corresponding point and tl drum; the spark is enhanced by a condenser in parallel with tl secondary circuit. The mark on the drum made by the spark like a bright pin-point and is surrounded by a sort of halo; it is thi easily identified. The positions of the marks are read by means of a micioscoj mounted on the frame supporting the drum; this microscoj can be tiaversed parallel to the axis of the drum. To take a leadir the drum is rotated by hand until the mark made by a spark is i the field of the micioscope; it is then clamped. The mark is the brought to the zero line of the eyepiece by means of a fine adjustmen The microscope is then traversed to the edge of the dium and tl reading is taken from the circumferential scale The positions < the marks made by the other sparks are similarly measured Tl time intervals between the pairs of spaiks are then deduced wit the aid of the tachymeter leading The chronograph is cahbiated by bi caking all the pnmai circuits simultaneously and recoiding the iclative positions of tl marks made by the spaiks. With this instrument such a small time interval as 0-0017 se< can be measured with a probable erroi of 7 5 X io~ 6 sec., or 04 per cent. Expeiiments prior to the war thus fell into two types the Hutto type, in which the velocity of a projectile was mcasuicd at two point a known distance apait, and the Bashfoith type, in which the times c passing a number of equidistant points along the tiajectoiy wci recorded. With regard to the fiist type, unavoidable errois in the measuic ment of the velocity would vitiate the icsutts if the distance bet wee the points were too short; on the other hand the appioximale mctho of deducing the resistance as a function of the velocity cannot giv satisfactory results unless the distance is short. It would thus appes to be a difficult matter to choose suitable distances, and laws of re sistance based on methods of this type must be somewhat uncertair With regard to the second type, it is evident that, provided tib time readings are sufficiently smooth to ensure that differences c finite order vanish, the resistance and velocity at correspondin points can be deduced to a known degree of accuracy. Wher however, the observed times require appreciable alteration to mat EACTION OF AIR TO ARTILLERY PROJECTILES 353 smooth, considerable uncertainty attaches to the results deduced, 'o both types there is the objection that no account is taken of ble yaw * of the projectile. It is well known that all shells have yaw on leaving the muzzle of the gun, and it cannot be hoped it is always damped out sufficiently before reaching the points ilch observations are made. At high velocities very considerable nay develop, and, in particular, such obstacles as screens may to increase it. In any case the yaw does not remain constant g the flight of the projectile, hence, unless it is at all times jible, the resulting law of resistance cannot be consistent, i any method which depends on observations of a projectile in it is therefore necessary to make some provision for observing iw as well as the velocity (or time) at points on the trajectory, i yaw is small throughout the flight reliable results would be led; in cases of considerable yaw a method would have to be 2d of correcting for it in the analysis of the records before any ce could be placed on the deduced values of the resistance Cranz's Ballistic Kinematograph method in which provision is made for observing the yaw, at qualitatively, was devised by C Cianz, j- who carried out ex- ents which wcic contemporary with those of Ebertuird, it was, /er, applicable only to rifle bullets and to similar projectiles of mall calibre series of shadow photographs of the bullet was taken by means Ballistic Kmematograph at each end of a 20-m range. The ty at each end could be deduced from the positions of the ;sive images on the kinematograph film and the observed speed ich the film moved through the camera The occurrence of viable yaw could be detected at once from the photographs, le reliability of the records for the purpose of the experiments thus be estimated. The Solenoid Method ice the War technique has developed considerably in the rement of high velocity. A very successful method, developed Frank Smith, of measuring time intervals in experiments of he yaw is the angle between the axis of the shell and the duection of motion jntre of gravity 3i tt full account of Cianz's experiments see Artillenstische Monatshefte, 69 1912). 12) 13 354 THE MECHANICAL PROPERTIES OF FLUIDS the Bashforth type, consists in firing an axially-magnetized piojectile through the centres of a series of equidistant solenoids which are connected in a series with a sensitive galvanometer. The current induced in each solenoid reaches a maximum as the projectile approaches, falls rapidly to a minimum as the projectile passes through, and finally returns to its original value as the projectile emerges. The " signature " of the galvanometer is recorded photo- graphically on a rapidly moving film on which is also recorded the oscillations of a tuning-fork of known frequency. Experiments with High -velocity Air Stream During the war experiments were undertaken in a new direction. Instead of making observations on a projectile m flight, the thrust on a stationary projectile in a current of air moving at high velocity was directly measured.* The method has subsequently undergone considerable development m France and Amenca,f and at the National Physical Laboratory "[ The projectile (or a scale model) is supported by means of a thin steel spindle fixed to the centre of the base m prolongation of its axis; this spindle is attached at its olhei end to a mechanism designed to measure the thrust on the projectile. Compressed air issues from a reservoir through a suitable orifice, thus generating a high- velocity stream; the projectile is placed m the centie of the stream. The determination of the most suitable size and shape of the orifice was a matter of considerable difficulty, aftot a large number of trials an orifice was obtained which ensured a steady stream m the vicinity of the projectile. The temperature and velocity of the air in the stream are com- puted on the classical theory from the state of the air in the icseivoir before the orifice is opened. When the velocity is greater than that of sound m air a check on the computed value can be obtained by photographing the head wave caused by the projectile (see p. 357) and measuring its slope. The possibilities of such an experimental method are innumerable. Apart from the direct determination of resistance of air at all angles of yaw, its sphere of usefulness extends to the elucidation of many problems connected with the general reaction of the air on projectiles. * Experiments of various kinds on piojectiles had previously been can led out in wind channels, but the velocity of the air stream was at most 30 m, per second. t See " The Experimental Deteimmation of the Forces on a Piojectile ", by G. F Hull, Army Ordnance, Washington, May-June, 1921. f See the Annual Reports of the Dnectoi, N.P.L. for igzz and succeeding years. REACTION OF AIR TO ARTILLERY PROJECTILES 355 Considerable progress has been made in experimental ballistics nee the war. The time, the yaw, and the orientation of the axis of ie projectile at a series of points along a horizontal trajectory can e observed with considerable accuracy; results from such experi- tents co-ordinated with those of experiments with the high- velocity r stream, have placed our knowledge of the reaction of the air on projectile upon a very sound foundation. The Drag at Zero Yaw The resistance of the air to a projectile of given shape, moving ith its axis of figure coincident with its direction of motion, depends i the following arguments: the velocity, v, of the projectile; the calibre (i e. diameter), d, the characteristic properties of the air, chiefly the density, the the elasticity, and the viscosity Dimensional considerations lead us to the form R = P d z v 2 f(vja, vd/v) . . (i) r the resistance R of the air to projectiles of given shape, where p the density of the air, a is the velocity of sound in air, an index the elasticity, and v is the kinematic viscosity The function ;/, vd/v) in this expression is called the drag coefficient Since 2 a 2 has the dimensions of a toice it follows from equation (i) that 3 drag coefficient has no physical dimensions, its arguments must irefore be so chosen that they shall have no dimensions; vja and /v both satisfy this condition, and are the simplest arguments in ms of which the function can be expressed. The Drag at Low Velocities For velocities below the critical velocity it is well known that /(#/, vd/v) = A.v/vd ere A is a constant, the terms in v/a being negligible. This leads the expression R = Apdvv viscous drag. For velocities higher than the critical velocity we have a change physical conditions, the air behind the projectile breaks up into 35 6 THE MECHANICAL PROPERTIES OF FLUIDS eddies and the linear law of viscous drag no longer holds.* In su circumstances the resistance is found experimentally to be appro: mately proportional to the square of the velocity. For incompressible fluids an expression of the form Av/vd + B for the drag coefficient will usually fit experimental data for bod completely immersed, A and B being constants depending on t shape of the body When v is very small the fust tei m is large co] pared with the second, and the linear law for viscous drag reappea when, on the other hand, v is large the first term becomes srn compared with the second, and we have an approximate quadra law. An expression of the same form will also hold for the resistar of air to a projectile, provided the velocity is not sufficiently high cause compression of appreciable amplitude. No uppei limit velocity can be fixed for this law, since the amplitude of the compn sion will depend on the shape of the head of the projectile as well on the velocity; thus, for Krupp noimal shells, which have a mo or-less pointed head, the diag coefficient changes extremely slov with v even at a velocity of 215 m per second, showing that the i proximate quadratic law holds for these piojectiles at tins veloci whereas with cylindrical projectiles (fiat heads) the diag cocrfici< changes rapidly at this velocity (see fig. 5) There is little experimental evidence of the behaviour of 1 drag with variations of d for projectiles at these velocities r l results of wind-channel expenments confirm the loim given .ibc for the drag coefficient for velocities up to 30 m. pci second a it has generally been found that the drag coefficient is greater projectiles of small calibre than for those of laigc cahbic of the sa shape. The Drag at High Velocities For velocities greater than the velocity of sound the phys: conditions are again changed. At the nose of the projectile the air undergoes condcnsati The air being an elastic fluid, a condensation formed at any pr in it is transmitted in all directions with a velocity which is, in gene the velocity of sound. If, then, the projectile is travelling wit] velocity less than that of sound, the condensation of the air at * See Chapter III, p, 103. Facing page REACTION OF AIR TO ARTILLERY PROJECTILES 357 ose will be transmitted, as soon as it is formed, away from the nose i all directions If, on the other hand, the projectile is moving faster lan sound is propagated, the condensation of the air at the nose cannot e transmitted away from the nose in all directions; it can be trans- litted away laterally, but not forwards. The result is that the nose always in contact with a cushion of compressed air. Greatly (.creased pressure is thus experienced by the projectile when travelling ith a velocity greater than that of sound. Photographs of bullets moving with such velocities reveal the dstence of two wave fronts, somewhat conical in shape, one at the 2ad and the other at the base. In fig. 3 a photograph taken by Cranz his ballistic laboratory is reproduced. The wave front at the head can be accounted for by Huyghens' inciple; it is in fact the envelope of spherical waves which originate the head of the projectile at successive instants of time. If the nplitude of the condensation, when first formed, were small the ave fiont would be a cone of semi-angle Q, such that sioQ = ajv, . . ,(2) r hen v is less than a the spherical waves have no envelope, and, couise, no wave fiont is formed. In the actual state of affairs the amplitude of the condensation the nose is not small, but finite. The velocity at which it is opagated is theiefore greater than the normal velocity of sound, t points on the wave front near the nose we should therefore expect e angle Q to be greater than at more distant points where the aphtude has become considerably reduced The form of the tual wave fiont at the nose is therefore a blunted cone, and the liter the head of the projectile the more is the wave front unted. The foimation of the waves behind the projectile cannot be counted for in such a satisfactory manner. Loid Rayleigh * has own that the only kind of wave of finite amplitude which can be aintamed is one of condensation; his argument refers to motion in LC dimension only, but we see no reason for modifying the result len applied to motion in three dimensions. We therefore conclude at the wave at the base of the projectile is, like the head wave, one condensation. An examination of the photographs of bullets in ^ht veiifies this conclusion. * " Aerial Plane Waves of Finite Amplitude ", Scientific Papers, Vol. V. 35 8 THE MECHANICAL PROPERTIES OF FLUIDS The source of disturbance causing this wave might be identifi with the relatively high state of condensation of the air flowing u the rarefied region at the base The angle Q of the straight part of the wave appears genera to be less than that of the head wave; the difference angle is probably the geometrical consequence of placing t source of light close to the bullet; we have, in fact, a p< spective view of the waves. When the source of light very close to the bullet the consequent difference in angle m be consideiable. The tendency of the angle to diminish towards the apex of t wave is probably due to two effects. In the first place theie m be some variation in temperature of the air in the immediate neig bourhood. Close to the base the air may be cooler than at pou more distant; the wave may theiefore be propagated with less veloc in the vicinity of the base than at more distant points. In the seco place it seems certain that the air behind the projectile will hav< velocity gradient from the axis outwards Near the axis the air v be moving faster than at more distant points Of these two effects the fiist will tend to dimmish a, wh the second will cause an mcicasc m v m equation (2); 1 values of Q will therefore be less near the apex of the wave The change of sign of Q immediately behind the base is piobal due to change in direction of the air's motion in the immediate neig bourhood Lord Rayleigh has proved,* furthei , that such waves condensation cannot be maintained m the absence ot dussipati forces. It is therefoie evident that some tcim involving the viscosi such as vdjv, must be included m the diag coefficient. The Scale Effect There is some expciimental evidence of the dependence of the di coefficient on d, and hence on some such teim as vdjv. For cxamr- Cranzf quotes the following figures for the icsislancc pci squaie cer metre of cross section deduced from Krupp's 1912 experiments: (a) For cylindrical shell (flat heads). Calibie (cm.) foi v = 400 500 600 700 800 m./acc, 6-5 1-40 2-58 3-80 5-15 6-60 Kgm./cj lo-o 1-29 2-20 3-30 4-70 6-30 ,, * Loc. at. 'I Lehrbuch der Ballistik, Vol. I (Berlin, REACTION OF AIR TO ARTILLERY PROJECTILES 359 (6) For ogival * shell, 3 cahbres radius. Calibre (cm ) for v = 550 650 750 850 : 6 I -00 130 1-58 194 10 098 1-25 1-52 28 o 62 081 I 01 125 3 0-90 i 06 In the absence of a term involving d, such as vd/v, from the The coefficient the numbeis in each vertical column would be equal, drag discrepancies are, no doubt, partly due to differences in the yaw of the projectiles, but they cannot be wholly accounted for in this way. These results indicate that the drag per unit cross section (i.e 4R/7rd r2 ) for projectiles of small calibre is largei than that for those of large calibre. Didion noticed this so-called scale effect as early as 1856! and deduced a relation between R/d 2 and d, but later he abandoned it as it would not hold for all velocities encountered m gunnery This effect has not been confirmed in recent experiments and further evidence is needed before definite conclusions can be drawn. Dependence of the Drag on Density The assumption that R varies as />, other factois being constant, has considerable theoretical support, but up to the present the range of vanation of p m experiments has been extiemely small; we cannot therefore claim piactical veuft cation for this assumption When more work has been done with the high-velocity an stream more light may be thrown on this question, as considerable variations of air density are easily obtained m this method. The Function f(v/a, vdfv) At the present time no satisfactory mathematical expression for the drag coefficient has been derived from theoretical considerations We are therefore forced to accept values of the function deiivcd from experiment alone In ballistic calculations it has generally been assumed that the term vdfv could be neglected, that is to say, that the drag coefficient is independent of the calibre; the function f(v/a, vdfv) has, in consequence, been determined as a function of v/a only. * Sec p. 360, f Lots d"e la resistance de Vavr (Pans, 1857). 360 THE MECHANICAL PROPERTIES OF FLUIDS Shape of Projectile In our discussion we have so far considered the air resistance to projectiles of the same shape. Our next step is to consider the changes that occur in the resistance when the shape is altered All modern projectiles have a cylindrical body and a more-or-less pointed head. The head is usually ogival, that is to say, it is generated by the rotation of an arc of a circle about the axis of the projectile. The shape of the head is identified by the length of the radius (m calibres) of this arc. Thus in fig. 4 a head of 3 -calibres radius is depicted When the point is rounded the radius of -o 26 d the rounding is also stated m calibres Thu dotted head m fig 4 -- 3d would be described as a """ ~-- ----^ 3~cahbres radius head "T-L^-^ with a o 26-caIibre loimdcd point In fig 5 the diag co- _ _ efficients of a i^-cm. | piojeclile with flat-head Fig 4 Shape ofl lead and pointed heads of vaiious lengths are plotted against the latio, velocity of shell to velocity ol sound. The curves are deduced from British and ioieign experimental data The values of the drag co-efficient given m these curves may be used with any self-consistent system of units, foi example, if the fundamental units used aie the metic, kilogiamme and second, these values of the drag coefficient when used in equation (i) will give the drag in metre-kilogramme-second units of foice (i unit = 100,000 dynes) Again, if the ft.-lb. and second be used, these values of the drag coefficient used in equation (i) will give the drag in poundals. This property arises, of course, from the fact that f(vja) has no physical dimensions. The shape of the head, provided it is more-or-less pointed, does not appear greatly to influence the resistance at lower velocities, At velocities greater than about 350 m. per second, however, the effect of the length of the head is appreciable. At velocities greater than about 750 m. per second it appears that the shape of the point REACTION OF AIR TO ARTILLERY PROJECTILES 361 is more important than that of the rest of the head Thus the resis- tance is less, at these velocities, for a sharp-pointed 3 -calibres radius head than for a 5-5 calibres radius head with a blunted point. For k he same shape of point the resistance is less, at velocities greater han about 350 m per second, for a long head (e.g. 5-5 calibres adius) than for a short one (e g. 2-cahbres radius). 07 06 05 04 Fai.itedShett 02 ( ^ g- , >. -^ 1 calibre * ^v ^*^*^ -2 ca fibres 01 / 3 calibres v/a. 2.0 so to Fig 5 The Drag Coefficients for is-cm Projectile with Various Shapes of Head, plotted as a Function oi Velocity Experiments and trials lead to the conclusion that at high velocities long head with a sharp point encounters considerably less resistance lan a short head. For example an 8-calibres radius head experiences ily about half the resistance of one of a-cahbres i adius. Little Ivantage appears to be gained, however, by lengthening the head iyond 8-cahbres ogival radius; thus 10- and iz-calibres radius 2 ads are only slightly more effective than those of 8-calibres radius reducing the air resistance. (D312) 13 t 362 THE MECHANICAL PROPERTIES OF FLUIDS The Base The importance of the shape of the rear pait of a body moving in a resisting medium has been realized for many years; the torpedo, the racing automobile, and the fusilage of an aeroplane are examples of " stream-lining " familiar to all. The suggestion has frequently been made that artillery projectiles should have a tapeied (so-called " stream-line ") base with a view to reducing the air resistance. Experiments with rifle bullets have shown, however, that the stability is so seriously affected that any possible advantage gained by a pointed base is entirely eclipsed by the effects of a lapidly developing yaw. In recent times a compromise has been effected in a shape known as the " boat-tail ", which is illustiated m fig. 6. The base is tapered for a short distance and is then tut off squaie. The stability of the piojectile is not appreciably affected by this modi- Fig 6 Boat-tail Projectile fication of the base. Bullets ot this shape were tried in Fiance as fai back as 1898, but they were found to have no gieat advantage over the flat base. It has, however, been shown recently that, although such a base has no particular advantage at high velocities, it has appreci- able superiority over the flat base at velocities below about 450 rn per second. In the trials mentioned above, the French cxpeu- mented at high velocity over short ranges and so failed to discovci the merits of this shape. Some extremely interesting and suggestive results aic iccorclcd by G. P. Wilhelm * of comparative experiments with bullets having the boat-tail and the flat base. The following table gives A summary of these icsults: MUZZLE VELOCITY, 1500 FELT PER SFCOND Angle of Range (Yards), Range (Yards), Departure. Flat Base Boat-tail 20' 200 220 o 40' 360 410 1 o' 500 580 i 20' 630 730 1 40' 740 840 * In " Long Range Small Arms Firing ", Part VII, Army Ordnance, Washington, March-April, 1923. REACTION OF AIR TO ARTILLERY PROJECTILES 363 MUZZLE VELOCITY, 2600 FEET PER SECOND Angle of Range (Yards), Range (Yards), ' Departure Flat Base Boat-tail. 20' 570 600 o 40' 930 990 1 o' 1050 1200 5 o' 2250 2700 10 o' 2900 3600 15 o' 3250 4200 20 o' 3500 4650 It is clearly seen that for velo- ^ / cities lower than 1500 ft. per IT second the boat-tail bullet con- ^, siderably out-ranges the flat-base ** Diillet. On the high-angle trajec- ories with a muzzle velocity of z6oo ft per second the velocity )f the bullet is, during the greater )art of its flight, less than 1500 ft. )er second, in fact it is generally mly a few hundied feet per , . 06 lecond; on the low-angle trajec- ones, on the other hand, the r elocity is less than 1500 ft. per 11 1 r- i r 04 econd only in the rmal stages or he flight. The appreciable gam n range by the boat-tail bullet Q ired with high muzzle velocity 05 o 6 os 10 v/a t high angles, and the Small Or Fig 7 The Dms Coefficient deduced from 111 .1 i .1 High-speed Air-struim I 1 xpt-nments egligible gain at low angles, thus n^Grmc- fVi 1r<rr*ntKpeic tKdt thf Curve A Foi s-cal -rad head and flat base OnlirmS tile nypOtneSlS mat me CutveB ^. Same hend as A, boat-tail base, taper, hor>A f\f tV>P KIOP 1 of OTPfltfM* '5 Curve C Same head as A, boat-tail biae, nape Or me DaSC IS Oi grcdici tape , )7 o CwveD Same head as A, boat-tail -nnm-fonr-p. if Inw tVinn at fiitrh base, taper, 9 Dotted Curve The diag co- Bportance at low man at nign c[Ilclcnt glven m flg s y\ at base). elocities The results of experiments with ic high-speed air stream are interesting in this connection. In g, 7 some of the results of experiments conducted by the Ord- ance Department of America are reproduced.* * From " Experimental Determination of Forces on a Projectile ", by G. F ull, Army Ordnance, Washington, May-June, 1921. 364 THE MECHANICAL PROPERTIES OF FLUIDS These curves tend to show that foi velocities below 350 m. pet second the drag is greatly influenced by the shape of the base; on the other hand, as we have already seen, provided it is more or less pointed, the actual shape of the head has very little influence on the drag at these lower velocities From these results we may fairly conclude that the greater part of the air resistance to pointed projectiles at these velocities is due to the drag (suction) at the base, and that this drag is appreciably reduced by boat-tailing The divergence of curve A in fig. 7 from the dotted curve is not clearly understood It is possible that the assumptions made in deducing the velocity of the air stream aie not altogethei sound and lead to values which are too high, it is also possible that the tod sup- porting the model (p. 354) may materially affect the air flow at the baseband so modify the drag We have seen that at high velocities the drag is gieatly affected by the shape of the head, whereas no appieciablc effect is produced by modifying the base. The probable explanation of this is not fai to seek. At velocities greater than 750 m per second (the so-called " cavitation " velocity of air) the vacuum at the base must be of high order, and, as the velocity of the projectile increases, it must tend asymptotically to a perfect vacuum. We should therefore expect that the component of the air resistance due to the base is tolerably constant at these high velocities, wheieas the total icsistance is rapidly increasing with velocity The component due to the base, with increasing velocity, soon becomes a small part ot the total resistance, and theiefore any possible modification of it, clue to shape of base, can have little influence on the whole Our observations on the effect of shape of pointed piojectiles may now be conveniently summarized At velocities less than about 350 m. per second the drag at the base contributes the gi eater part of the air resistance, so that the shape of the base is of greater im- portance. At velocities between about 350 m. and 750 in. per second we have an intermediate stage in which the shape of the head gradu- ally gams ascendancy. At velocities greater than about 750 m. per second the greater part of the resistance is due to the head, and the shape of the latter is of greater importance than the shape of the base. Before leaving the subject of shape we must refer to some extremely interesting experiments designed to determine the pressure distribu- tion on the head of a piojectile moving with high velocity. REACTION OF AlR TO ARTILLERY PROJECTILES 365 The Pressure Distribution on the Head of a Projectile The pressure at any point of a body moving through a fluid :onsists of two components the static pressure, which is the pressure >f the fluid when the body is at rest, and the dynamic pressure, which s due to the motion. The sum of these two at any point is the total >ressure at that point and is essentially positive; the dynamic pressure nay be either positive or negative. A series of experiments was carried out by Bairstow, Fowler and laitree to determine the dynamic pressure at various points on the lead of a shell moving with high velocity.* The fundamental idea >f the experiments is the use of a set vice time-fuze j- as a manometer o determine the pressure under which the powder is burning Projectiles were fitted with hollow caps which entnely enclosed he fuze, each cap had a number of holes drilled m it at the same hstance from the nose, the pressure on the fuze was thus practically qual to thai at the holes. The projectiles were fired along the same trajectory at various uze settings and times to buist weie observed, a relation between aze setting and time was thus obtained, whence was deduced a elation between late of burning and time Since rate of burning is a function of the pressure on the fuze, by ompanson with laboiatory experiments it is possible to convert iis relation to one of pressuie and time Knowing the velocity of le piojectile at various times of flight on the trajectory, it is thus ossible to deduce the pressure in terms of the velocity By repeating the expenment with other caps of the same size ad shape with holes at other distances from the nose, the pressure istnbution over the head is obtained at a number of velocities. The results of the experiments are leproduced m fig 8 The rdmates are values of p/pv 2 , as this quantity has no physical dimen- ons the values given may be used with any self-consistent system of nits. The abscissae are distances from the nose of the projectile bservations were made at four positions on the head, indicated by * For a full account of these experiments see " The pressure distribution on the 'ad of a shell moving at high velocities ", Proc Roy Sac. A, 97, 1920 f " A service tirne-fuze contains a train of gunpowder, which is ignited, by a 'tonatoi pellet on the shock of discharge flora the gun The ' setting ' of the ze can be varied so that a length of powdei train depending on the setting is burnt 'fore the magazine of the fuze is ignited and the shell exploded The setting is >ecified by a number which defines the length of composition burnt on an arbitral y ale The time of burning is taken as equal to the time interval between the firing the gun and the bursting of the shell," Loc. cit. 366 THE MECHANICAL PROPERTIES OF FLUIDS Contin lous curves ( - ) are drawn through points marked according to the values of via to which they reftr, obtained from firing trials, and, for the nosu of the shell, (torn cal- culation Dotte 1 curves (----) are extrapolations of the continuous cuives, which are needed for mtegiatmg up the nessurc on the head Dash and dot curve ( ) is drawn throunh points marked X, determined by wind channel experiments, at a velocity of 40 ft -sec , 01 value of (via) of o o f v/a=/2 v/a=JO^A #0=0*4 6' (Reproduced from Proc Roy. Soc A, 97, 1930 ) . V/cli/Q of shell Beginning of Cylindrical fartian INCHES Fi. 8. Distribution of Pressure on tha G-cal.-rad Head of a i 3-In Shell the points A, B, C, and D. The value op/pv* at the nose was calcu- lated from Rayleigh's formula * in eaclf case. pv* 7 i_7 $ (7 i)a*lv*-> 7Z) ^( deduced from the foimulte given m hia Scientific Papers* Vol. V p. 610, REACTION OF AIR TO ARTILLERY PROJECTILES 367 These curves reveal most emphatically the necessity of a sharp Doint at the nose of the projectile. Compared with the pressure encountered at the nose the pressures at other points of the head are juite small. The authors integiated numerically the observed pressures over he head in each case, and derived values of the drag coefficient for 125 SO 05 10 15 2.0 i S 30 35 V'u. Fig 9 Drag in Kilogrammes Weight on a 3 3-in 6 C R H Projectile the dynamic icsistance on the head. From these results we have computed the actual dynamic lesistance on the head; it is plotted igamst velocity in fig 9. The total drag on projectiles of this shape is also shown (ap- proximately) in the figure. By subtracting the head resistance from the total resistance we have derived an approximate curve of the drag at the base The horizontal dotted line indicates the drag due to a complete vacuum at the base in this case, and the dotted exten- sions of the curves represent a tentative extrapolation of the results of the experiments. 368 THE MECHANICAL PROPERTIES OF FLUIDS The Effect of Yaw on the Drag Before approaching the complicated reaction of the air to a yawing projectile it will be convenient to consider, briefly, the effect of yaw on the drag; we now define the latter as the force exerted on the projectile in the opposite direction to the relative motion of the an and the centre of gravity of the projectile. If 8 be the angle of yaw the drag coefficient will now take the foim f(v/a, vd/v, B), and, in this notation, the drag previously considered takes the form * f(vja, vd/v, o) The manner m which the drag coefficient vanes with yaw at veiy low velocities has been determined experimentally m wind channels The results of such experiments on a 3-m piojcctilc with a 2-calibres- radius head and o i5-calibie rounded point arc given in fig io;f the ordmates aie values of the ratio f(vfa, vd/v, $)/f(vfa, vd/v, o) The velocity at which the experiments were conducted was 40 it pel second (vja o 04). We have seen that the drag on a body moving in air is appioxi- mately piopoitional to the square of the velocity,'] piovided that the shape is such that the condensation m Iront is ol small amplitude, for example, this quadiatic law holds loi pointed projectiles lor values of v\a not greater than 0*65 when the yaw is zeio We might icasonably expect the quadratic law to hold ioi yawing pointed projectiles within the same limit ot velocity, provided that the yaw is such that the air is encounteicd point fust, for the con- densation would be of the same order as when the yaw is zci o Within this limit for yaw, with values of v/a less than 0-65, we should theicfoic expect that the ratio f(v fa, vd/v, ^)f(vja, vd/v, o) is independent of v * Approximately. Except in the case of lesulls quoted from tur-stienm experi- ments we cannot be certain that the values of the drag coefficient hhheito used are for zero yaw All we can affum is that they aie the values foi very small 01 /eio yaw. f This cuive is derived fiom one given in " The Aeiodynamics of a Spinning Shell ", by Fowlei, Gallop, Lock, and Richmond, Phil. Trans, A, 591, 1920. J Except, of course, for such low velocities that the diag is due to viscosity alone. REACTION OF AIR TO ARTILLERY PROJECTILES 369 and therefore a function of 8 only. The limit of 8 for the projectile under consideration appears to be about 45. Experiments with the high-velocity air stream tend to verify the independence of velocity of this ratio within the limits mentioned, 12,5 75" 50 30 45" G0 75 90 Yaw Fig 10 The Ratio f(vla, vdjv, fi)//(w/a, vdjv, o) plotted against the yaw & for via => o 04 but at present the number of lesults available is insufficient to justify our drawing definite conclusions. The dotted curve in fig. 10 gives, approximately, the ratio between the total plane areas encountered by the projectile with yaws 8 and zero. The former is, of course, equal to the area of the shadow cast 370 THE MECHANICAL PROPERTIES OF FLUIDS by the projectile, in a paiallel beam of light inclined at angle 8 with the axis, upon a plane normal to the beam; the latter is simply the cross-sectional area of the projectile. The two curves in the figure are coincident for small angles of yaw, but rapidly diverge as the yaw increases; both curves appear to have a maximum at about the same value of the yaw. We have at present no knowledge of the effect of yaw on the drag at higher velocities; when more work has been done with the high- velocity air stream it is hoped that our knowledge in this direction will have been considerably extended The Drag Coefficient; Concluding Remarks In our consideration of the drag coefficient we have limited the number of arguments of the function to three only, namely v/a, vd/v, and 8. There appear to be two other possible arguments, namely y, the ratio of the specific heats, and ajd, where a is the effective diameter of the molecules of the air. Variations of y are so small in practice that no evidence of its effect is available. Expressions deduced from thermo dynamic theory, by various authors, for the drag in one-dimensional motion may give an indication of the manner in which it occurs * There is at present no evidence of the necessity of the aigument aid. If further experimental results show that the argument vd/v does not adequately account for the " scale " effect (e g. with vaiymg v), it will of course be necessary to include some other argument involving d, such as cr/d Finally, in the case of a projectile moving In air, as distinct from one which is stationary in an air stream of constant velocity, there is the question of retardation. It is just possible that some such argu- ment as rd{v*, r being the retardation of the projectile, may be re- quired to co-ordinate the results of air-stream experiments with those of experiments on a projectile moving in air, but it is difficult to see how the retardation can ever have an appreciable effect on the drag, except, perhaps, when the velocity is in the neighbourhood of the velocity of sound in air. * See for example the footnote on p. 346; also Vieille, Comptes Rendus, 130 (1900), and Okmghaus, Monatsh.fiir Mathem. . Phys. t 15 (1904). REACTION OF AIR TO ARTILLERY PROJECTILES 371 REACTION TO A YAWING, SPINNING PROJECTILE The most complete specification in existence of the system of >rces acting on a projectile is that given by Fowler, Gallop, Lock, id Richmond in " The Aerodynamics of a Spinning Shell ", Phil, "rans A, 591 (1920). In this paper the authors describe expeiiments inducted to deteimme numerically the principal reactions, other ian the drag, to which a spinning shell is subjected The experiments are confined to the study of the angular oscil- itions of the axis of the shell relative to the direction of motion of le centre of gravity. The projectile is fired horizontally through a sries of cardboard targets fixed, veitically, along the range at 30 ft. ad 60 ft. intervals Initial disturbances at the muzzle give rise to scillations of the projectile of sufficient amplitude for measurement; le details of the oscillations aie obtained by measuring the shape of le holes made in the caids If, on passing thiough a caid, the shell ; yawing, the resulting hole will be elongated, the length of the >nger axis of the hole determines the yaw; the orientation of this xis determines the azimuth of the plane containing the axis of the hell and the direction of motion; these two angles determine the irection of the shell's axis completely The range containing the cards is so short, and the velocity so igh, that the effect of giavity is negligible If, then, we ignore amping forces, the angular motions of the axis of a top and the xis of a shell' are identical, provided that (i) the top and shell have the ame axial spin and axial moment of inertia; (2) the tians verse moment f inertia of the top about its point of support is equal to the trans- erse moment of inertia of the shell about its centre of gravity; (3) be moment of gravity about the point of the top is equal to the cioment of the force system on the shell about its centre of gravity, lie foimal solutions of the two problems are then identical. From the periods of the oscillations of the axis of the top we can [educe the moment of the disturbing couple, and -vice versa', similarly he moment of the force system on the shell can be deduced. The lamping forces can then be determined from the nature of the decay >f the oscillations. The reactions are described by the authors as follows: 372 THE MECHANICAL PROPERTIES OF FLUIDS The Principal Reactions When the shell, regarded for the moment as without axial spin, has a yaw 8, and the axis of the shell OA and the direction of motion OP remain in the same relative positions, the force system can by symmetry be represented, as shown in fig. ir, by the following com- ponents, specified according to aerodynamical usage: (1) The drag R acting through the centre of gravity O, in the direction PO. (2) A component L, at right angles to R, called the cross-wind force, which acts through O m the plane of yaw POA, and is positive when it tends to move in the direction from P to A (3) A moment M about O, which acts A m the plane of yaw, and is positive when it tends to increase the yaw. The following foims aie assumed for L and M: L = pv z d z sinS/ L (?;/<2, 8). M = pv*d 3 sin8/ M (;/fl, 8). >L These equations are of the most natural forms to make / L and / M of no physical Fig u. dimensions. The form chosen is suggested by the aerodynamical treatment of the force system on an aeroplane Since L and M, by symmetry, vanish with 8, the factor smS is explicitly included in these expiessions in order that / L and / M may have non-zero limits as 8 o. The Damping Reactions The yawing moment due to yawing In practice the dncction of the axis of the shell, relative to the direction of motion, changes fahly rapidly. By analogy with the treatment of the motion of an aero- plane, we assume, tentatively, that the components of the force system R, L, and M are unaltered by the angulai velocity of the axis, but that the effect of the angular motion of the latter can be repre- sented by the insertion of an additional component, namely, a couple H, called the yawing moment due to yawing, which satisfies the equation H REACTION OF AIR TO ARTILLERY PROJECTILES 373 here w is the resultant angular velocity of the axis of the shell, 'he form is chosen to make / H of no physical dimensions and is the nly one suitable for the purpose. The couple is assumed to act in such a way as directly to diminish i (see fig. 12). It is suggested by, and is analogous to, the more nportant of the " rotary derivatives " in the theory of the motion f an aeroplane. The coefficient / H may be expected to vary considerably \vith '/a, and it may depend appreciably on other arguments such as id\v and 8 The effect of the axial spin, The spin N gives rise to certain dditional components of the complete force system. Fig 12 There will be a couple I which tends to destioy N, and, when the shell is yawing, a sideways force, which need not act thiough the centre of gravity, analogous to that producing swerve on a golf or tennis ball. This force must, by symmetry, vanish with the yaw; it is assumed to act normal to the plane of yaw (any component it may have in the plane of yaw is inevitably included m either R or L). The complete effects of the spin N can therefore be represented by the addition to the force system of the couples I and J and the force K, acting as shown in fig. 12. To procure the correct dimensions we may assume that these reactions have the forms I = pvNd% J = pvNd* sm8/ r K = pvNd* sin8/ 8 . 10 M o fc REACTION OP AIR TO ARTILLERY PROJECTILES 375 The coefficients /j,/ p / K may depend effectively on a number of lables which we can make no attempt to specify in the present te of our knowledge. It will be seen that this specification is equivalent to a complete stem of three forces and three couples referred to three axes at ,ht angles. Owing to the complex nature of the reactions the thors considered it essential to construct the specified system Vertical Vertical Radial Seals afdeqrees Fig 14 Examples of Path of Nose of Shell relative to the Centre of Gravity (FromP/H/ Trans A, 591, 1930) stead of attempting to analyse a complete system of three forces id three couples and to assign each component to its proper causes. The experiments were designed to determine L * and M and to ve an indication of the magnitude of the chief damping couple . It was not possible to determine I, J, and K; all three are resumably small compared with H and very small compared with > and M; no certain evidence that they exist was given by the qpenments. The results of the experiments are exhibited in fig 13, which is ^produced from the paper. The units in which the coefficients are ^pressed are suitable for use with any self-consistent system. The * Values of L were deduced from values of M for shell of the same external lape, with centres of gravity at different positions along the axis of figure. 376 THE MECHANICAL PROPERTIES OF FLUIDS curve of the drag coefficient is also given; in comparing this with / L and jf M it must be remembered that the latter should be multiplied by sinS. The shape of the curve for/ L is rather unexpected; the curves for / M appear to exhibit the same tendencies as that for the drag coefficient. The values of/ H deduced fiom the experiments were rough; they varied from about 1-4 to 5-0. It was impossible to deduce any details concerning the variations of / H with velocity, but the right order of magnitude is represented by these limits In comparing / H with / M it must be remembered that the former is multiplied by zvd, whereas the latter is multiplied by the much larger quantity v. The general features of the motion of the axis of the shell and of the damping are shown in the examples, reproduced from the paper cited, in fig. 14. SUBJECT INDEX (See also Name Index) coustic properties of matenals, 304 coustics under water, 298-344 ir, clouds in, 40 Displacement, 9 Resistance coefficients, 207. Viscosity, 118 vogadro's hypothesis, 43. alhstic pendulum, 348-51. arometei, the first, 2-3 aum hydiometer, n eanngs, cylindrical, 142-6. Flexible, 155-6 Lubrication, 133-58 Pivoted, 148-55 Self-adjusting, 139-42 Width, 146-9 ernoulh's equation, 57-9, 96 maural method of sound icception, 306-10 oiler tubes, gases in, 183 oyle, barometer experiment, 2-3 Law, derivation of, 42 rass, acoustic propeities, 304 ndgman, high-pressure expenments, 1 6-8 ronze, acoustic properties, 304. ubble, collapse of, 75 allendar's equation, 49 apillary tubes, action in, 25. Rayleigh equation, 27 Viscometer tubes, 112-6 'arbon bisulphide, viscosity, 118. 'ast iron, acoustic propeities, 304. 'astor oil, compressibility, 219 Viscosity, 1 1 8 hannels, flow in, 179-80 harles's law, derivation of, 42-3. lausius equation, 47-8 louds, suspension of, in air, 40. eofficient of viscosity, 32, 103-4, 117-9 ollision, prevention of, by echoes, 332-3 Compiessibility of liquids, 15-9, 23, 212, 218-9 Conduction of heat and viscosity, 99- 100 Contact angle, meaning of, 25. Continuity, equation of, 60- 1. Coohdge tube, 320 Critical point, 44 Critical temperatuie, 44 Critical velocity, 165-72 Crystals, elasticity of, 4-5 Cube, piessure on, 4-5 Cup-and-ball viscometer, 125-8 Cylinders and wires, icsistance of, 214-7 Deformation and rigidity, 3-4. Density, 7-13 Diaphragm soundei ,318 Dietenci equation, 47-8 Dimensional homogeneity, 187-203 Diiectional acoustics, 306-12 Directional sound receivers, 311-3 Displaced air, correction factoi, 9 Drops, deteimination of weight of, 28-9 Dynamical similarity, 193-203 Echo depth-sounding geai, 335-44 Echo, detection of ships by, 332-3 Eddies and turbulence, 166 Elasticity of crystals, 4-5 Elasticity, phenomena due to, 218-36 Electrical measurement of pressure, 17 Energy at surface of liquids, 19-29 EotvSs equation, 23 Equations of motion, 36 Equations of state, 4149 Errors in soap film experiments, 244-7. Explosions under water, 76 Fessenden sound transmitter, 317, 333. Flow, measurement, 172-8 Stream-line, 160-90 Sudden stoppages, 219-21, 377 378 THE MECHANICAL PROPERTIES OF FLUIDS Flow-meters, 173-8 Diaphragm, 175-6 Pitot tube, 176-9 Venturi, 172-5 Fluids, definition, 1-2, 6. Motion, 56-101. Pressure, 17 Solids irm^ersed in, 33. Friction in pipes, 221-3. Gas, perfect, definition of, 6. Gas equation, 3. Gases, liquids, and solids, classification, 1-2 Glycerine, viscosity, 118 Heat transmission, flow effects, 180-90. Hydrodynamics. See Stream-line Mo- tion Hydrodynamical resistance, 191-217 Hydrometers, calibration defects, 10-2 Hydrophones, 305-6, 320-4 Bi-directional, 312. C-tube, 320-1. Constructional details, 322-3. Magnetophones, 321-2. Morns-Sykes, 312 Uni-directional, 313 Incompressible fluid, equations of, 36 Irrotational stream-line motion, 62, 65-71- Kinetic theory of gases, 41-4. Laminar motion, 103 Lead, acoustic properties, 304. Liquids, compressibility, 13-9 Definition, 6. Molecular viscosity, 39 Surface energy, 19-29 Surface tension, 19-29 Liquids, gases, and solids, classification, 1-2 Lubrication, 128-59 Bearings, 133-58 Bibliography, 158-9 Inclined planes, 131-3. Viscous, 129-31 Lubricating oils, compressibility, 219 Viscosity, 118-9. Matter, states of, 1-2 Mercury, viscosity, 118 Mineral oils, viscosity, 118 Mohr's balance, 12 Molecular viscosity of liquids, 39. Motion, equations for incompressible fluids, 36 Motion and heat transmission, 180-90 Motion of fluids, mathematics of, 56- 101. See also Stream-line Motion, Laminar Motion, Vortex Motion, Wave Motion Naval architecture, model experiments, 209-1 i Nitrogen, Amagat's experiment with, 44 Oils, compressibility, 219. Viscosity, 118-9 Olive oil, compressibility, 219 Osmosis and osmotic pressure, 50-4, Ostwald's viscometei, 39 Parallel planes, viscous flow between, 119-25 Permeability, 50 Petroleum, compressibility, 219. Pipes, critical velocities, 168-72. Elasticity, 223-6 Flow resistance, 198-203. Flow stoppage, 226-7 Friction, 221-3 Opening valves, 228-9 Pitot tube, 59 Poiseuille's equation, 37-8 Power transmission, acoustic, 333-4 Prandtl's analogy, 239-42 Pressure, 56 Electrical measuiement, 17. Viscosity, 117-9 Projectiles, air density, 359 Ballistic pendulum, 346-8 Base shape, 362-4 Bashforth chronograph, 348 Boat-tail bullet, 362-4 Cranz's ballistic kmematogi aph, 353 Drag coefficient, 359, 370 Kiupp's 1912 experiments, 351-3 Piessure distribution on, 365-7. Reaction of an on, 345-76 Scale effect, 358-9 Shape, 360-4 Spark chronograph, 350-3 Spinning, 371-6 Yawing, 371-6 Pyknometei, descuption of, 9-10 Rape oil, viscosity, 119 Rayleigh's equation for capillary tubes, 27 Redwood's viscometer, 116 Resistance and compressibility, 212 Resistance of square plates, 211-2. Resistance of submeiged bodies, 206- 9 Resistance of wires and cylinders, 214-7 Rigidity and deformation, 3-4 Rubber, acoustic properties, 304. Scale effects, 211-3 Shearing of a cube, 4-5. SUBJECT INDEX 379 hips, resistance of, 209-11. irens, undei water, 319. km fuction, 203-6 oap film stress deteimmations, 237-52 Contour mapping, 244-7. PrandtPs analogy, 240 Sheai stress in twisted bar, 240. Toique on twisted bar, 241. Twisting of bais, 237-9. Warping of sections, 238 olid, scientific definition of, 5 olids, liquids, and gases, classification, 1-2 olids and fluids, mtei action of, 195 olutions and solvents, 50-4 ound, nature of, 299. ound ranging, 325-33 Depth sounding, 330-2 Leader geai, 330 Multiple station system, 325-9 Wixeless acoustic method, 328-30 ound icceivers, 304-6 ound transmission, 301-4, 316-8 ound velocity, 300 Wave-length, 301 ources and sinks, 75-81 pecific gravities, air displacement, 9. perm oil, compressibility, 219 Viscosity, 1 1 8 pheie in viscous fluid, 33 quaie plates, resistance of, 211-2 tate, equations of, 41-9 teel, acoustic properties, 304 tethoscope, action, 306-9 Cokes' foimula, 41 tone's viscometer, 114-6 tream function, definition of, So tream-line forms, typical, 216-7 tream-lme motion, 57-85, 160-90 Application to naval aichitecture, 91 Axial symmetry, 75-81 Bernoulli's equation, 57-9. Circulation, 61 Continuity equation, 60-1 . Critical velocity, 165-72 Equations, 83-5 Hele Shaw's expenments, 162-5 Iirotational, 62, 65-71. Stability, 1 6 1-2 Steadiness, 63-5 Tracing stream -lines, 81-3. Tubes, 162 Turbulent flow, 161-90 Two-dimensional, 60-2 Velocity potential, 71-4 Vortex rings, 75 Vorticity, 61-2 ream-tube, definition of, 57 ress determinations from soap films See Soap Film ruts, strearn-lme, 216-7 ibmanne signalling, 298-334 Submeiged bodies, resistance of, 206- 9- Sum-and-difference method of sound reception, 310-1. Surface tension, 19-29. Tempeiature effects, 3, 20-2, 39, 44 Thermodynamics of compression, 15-8. Timber, acoustic propeities, 304 Torsion, examination by soap films, 237-54 Trotter oil, viscosity, 119 Tubes, converging, 171-2 Stream-line motion in, 162 Viscosity in, 109-12 Turbulent and stream-line flow, 160-90 Two-dimensional motion, 60-2 Valves, sudden closing of, 219-21, 226 Sudden opening, 228-9 Van der Waals' equation, 46, 48 Velocity, critical, 16572 Measurement, 172-80 Ventun meter, 172-5 Viscometeis, 35-9 Capillary tubes, 112-6. Cup-and-ball, 125-8 Redwood, 116 Secondary, 116-7 Stone's, 114-6 Viscosity, 31-41, 102-28 Bibhogiaphy, 158-9 Bounding surfaces, 107-8 Coefficients of, 32, 103-4, 117-9 Equations, 97-9 Laminar motion, 103 Lubrication, 102-59 Measuiement, 34-6 Paiallel planes, 119-25 Piessure vanation, 117-9 Relative motion, 123-5 Relative velocities, 105-7 Solids, effect of, 33 Temperature effects, 39 Tubes, flow-in, 109-12 Two-dimensional cases, 99-101 Velocity giadient, 32 Voitex motion, 85-9 Isolated vortices, 87-9 Persistence, 85-7 Rings, 75, 85-9 Water, acoustic properties, 304 Compiessibihty, 218-9. Density, 9 Viscosity, 1 1 8 Water-hammer, 219-23. Wave motion, 89-97. Canal waves, 89-91 . Deep-water waves, 91-4. Group velocities, 94-7. 380 THE MECHANICAL PROPERTIES OF FLUIDS Superposed liquids, 93-4 Transmission of energy, 229-36. Wind structure Altitude and velocity, 276-81, 28, Anemometer records, 263-4. Anti-cyclone, 260-1. Clouds, cumulo nimbus, 282 Cyclone, 260-1, 286-94 Eddy theory, 272-7. Egnell's law, 284 Geostrophic component, 263. Giadient wind, 260-1. Rain, cause of, 282 Stiatospheie, 281 [-9. Strophic balance, 262 Surface winds, 266-71. Troposphere, 281. Wind variation, 263-6. Wires, heat dissipation fiom, 183 Wires and cylinders, resistance of, 214-7. ii f NAME INDEX Akerblom, 274. Allen, 191. Archbutt, 158 Archimedes, 8-n. Avogadro, 43 Bairstow, 203, 213, 365 Barnes, 168-9 Bashforth, 347-52 Bassett, 36. Baume', n Beauchamp Tower, 158 Behm, 331-2 Bennett, 23 Bernoulli, 57-8, 63, 65, 90-1, 96, 165 Bessel, 148 Booth, 203-13. Borda, 59 Boinstem, 63 Boswall, 159 Boyle, 2-3, 7, 42, 44 Bndgman, 16-9 Bailie", 302-5 Bnlloum, 158 Broca, 305, 320 Bryant, 186 Buckingham, 213. Bunsen, 115 Callendai, 49 Canovetti, 212 Carey Foster, 17. Carothers, 158 Cave, 281-2. Charles, 43 Clausius, 48-9 Clement, 169 Cleik Maxwell, 19. Coker, 168-9 Collodon, 339 Constantmesco, 13, 230, 23 6 . 2 9S> 333~4 Cook, 218. Coohdge, 320 Cranz, 353, 357 Crombie, 264. Dadounan, 328. Hershall, 172. Daicy, 1 68 Hun, 158 Datta, 230 Hodgson, 175 Davis, 1 8 Hojel, 350 Deeley, 158. Hoskmg, 158 "De F", 159 Hull, 354, 363. Didion, 346, 359. Hunsaker, 213. Dietenci, 48 Hutton, 346-7, 352 Dines, 212, 268, 271 Huyghens, 357 Dobson, 274, 279-81 Hyde, 119, 159, 218-9 Eberhard, 350, 353 Jacobs, 35 Eden, 170, 216 Joly, 329 Edmunds, 302 Jordan, 183. Egnell, 284 Eiffel, 212 Emthoven, 326, 329 Ekman, 169 Eotv&s, 23 Kmgsbuiy, 159 Kobayashi Toras, 159 Krupp, 350-60 Ewmg, 49 Ladenburg, 35, Lamb, 36, 164 Faust, 159 Lanch ester, 159 Fenanti, 155 Landholt, 218 Fessenden, 317, 321, 333 Lasche, 158 Ford, 326 Lees, 203 Fowlei, 365, 368, 371 Fiahrn, 351 Froude, 203, 205, 209-10 Lempfert, 293 Lock, 368, 371. Lorentz, 186. Love, 238. Gallop, 368, 371 Gibson, 35, 171, 177, 180 Macleod, 22 200-1, 223, 225, 250 Goodman, 158 Manson, 330, Marks, 18 Giassi, 218 Marshall, 186. Gray, 314 Marti, 330 Griffith, 242 Martin, 159. Gnffiths, 252-3. Grmdley, 171 Giimbel, 159, Martini, 219. Mason, 314 Matthews, 23. Maxwell, 108, 119, 133 HSlstiom, 7 Mayewski, 350 Hartree, 365. Michell, 146-50 Hele Shaw, 162-4 Mitchell, 23. Hellmann, 271 Morns, 312 Helmholtz, 100 Morse, 318-9, 330. Hersey, 159 Munday, 314-5. 381 THE MECHANICAL PROPERTIES OF FLUIDS potto, 347- nst, 23. vbigm, 159. vton, 33, 301. holson, 183. iselt, 183. n, 302. nghaus, 370. nell, 170, 184, 200-3 ions, 218. dn, 9. fer, 51 ee, 314 t, S9> *72 emlle, 37-9, 104-5, C2, 1x6, 158, 200, 234 er, 52, 3H- itmg, 43-4, 230 vdtl, 239-40 icke, 219, kine, 81. Rayleigh, 27, 102, 108, 146, 159, 187, 357-8. 366. Redwood, 116 Reynolds, 128-9, 139, 142, 146, 158, 161, 165-9, 181-3, 186, 200 Richards, 23. Richardson, 333. Richmond, 368, 371, Ritchie, 250. Robms, 346 St. Venant, 238 Shaw, 262, 288-96. Shore, 159 Smith, 311, 318, 321. Soenneker, 186 Sommerfeld, 142-5, 158 Sprengel, 9 Stanford, 10 Stanton, 170, 181-7, I9 1 . 200-3, 212-3 Stephenson, 330, Stokes, 33, 158, 164 Stone, 114, 159. Stoney, 159 Stuim, 329. Sykes, 312. Tait, 5, 218 Tayloi, 185-6, 242, 252-3, 264, 272-3, 275 Thomson, 218, 230. Torncelh, 58 Twaddell, 10 Van der Waals, 46, 48. Van't Hoff, 51 Ventun, 172. Vieille, 370. Walser, 314-23. Watson, 230 Webster, 36, Weston, 227 Wilhelm, 362 Wood, 300, 326. Young, 5, 48- Zahm, 205,