hydrodynamics

Sir Horace Lamb

ias long been the chief storehouse of information of all workers in hydrodynamics . . ." NATURL

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HYDRODYNAMICS

BY

SIR HORACE LAMB, M.A, LL.D., Sc.D., F.R.S.

HONORARY FELLOW OF TRINITY COLLEGE, CAMBRIDGE ; LATELY PROFESSOR OF MATHEMATICS IN THE VICTORIA UNIVERSITY OF MANCHESTER

SIXTH EDITION

NEW YORK DOVER PUBLICATIONS

First Edition 1879 Second Edition 1895 Third Edition 1906 Fourth Edition 1916 Fifth Edition 1924 Sixth Edition 1932

FIRST AMERICAN EDITION 1945

BY SPECIAL ARRANGEMENT WITH

CAMBRIDGE UNIVERSITY PRESS

AND THE MACMILLAN CO.

Library of Congress Catalog Card Number: 46-1891

Manufactured in the United States of America

Dover Publications, Inc.

180 Varick Street

New York 14, N. Y.

PREFACE

THIS may be regarded as the sixth edition of a Treatise on the Mathematical Theory of the Motion of Fluids, published in 1879. Subsequent editions, largely remodelled and extended, have appeared under the present title.

In this issue no change has been made in the general plan and arrangement, but the work has again been revised throughout, some important omissions have been made good, and much new matter has been introduced.

The subject has in recent years received considerable developments, in the theory of the tides for instance, and in various directions bearing on the problems of aeronautics, and it is interesting to note that the "classical" Hydrodynamics, often referred to with a shade of depreciation, is here found to have a widening field of practical applications. Owing to the elaborate nature of some of these researches it has not always been possible to fit an adequate account of them into the frame of this book, but attempts have occasionally been made to give some indication of the more important results, and of the methods employed.

As in previous editions, pains have been taken to make due acknowledg- ment of authorities in the footnotes, but it appears necessary to add that the original proofs have often been considerably modified in the text.

I have again to thank the staff of the University Press for much valued assistance during the printing.

HORACE LAMB

April 1932

CONTENTS

CHAPTER I

THE EQUATIONS OF MOTION

ART. PAGE

I, 2. Fundamental property of a fluid 1

3. The two plans of investigation ........ 1

4-9. ' Eulerian ' form of the equations *of motion. Dynamical equations.

Equation of continuity.* Physical equations. Surface conditions . 2

10. Equation of energy 8

10 a. Transfer of momentum 10

II. Impulsive generation of motion 10

12. Equations referred to moving axes 12

13, 14. 'Lagrangian' form of the equations of motion and of the equation of

continuity ........... 12

15, 16. Weber's transformation 14

16 a. Equations in polar co-ordinates 15

CHAPTER II

INTEGRATION OF THE EQUATIONS IN SPECIAL CASES

17. Velocity -potential. Lagrange's theorem 17

18, 19. Physical and kinematical relations of (f> 18

20. Integration of the equations when a velocity-potential exists. Pressure- equation 19

21-23. Steady motion. Deduction of the pressure-equation from the principle

of energy. Limiting velocity 20

24. Efflux of liquids ; vena contracta 23

24 a. 25. Efflux of gases 25

26-29. Examples of rotating fluid ; uniform rotation ; Rankine's ' combined

vortex ' ; electromagnetic rotation 28

CHAPTER III

IRROTATIONAL MOTION

30. Analysis of the differential motion of a fluid element into strain and

rotation 31

31,32. . ' Flow ' and ' circulation.' Stokes' theorem 33

33. Constancy of circulation in a moving circuit . . . . . 35

34, 35. Irrotational motion in simply- connected spaces ; single- valued velocity-

potential 37

viii Contents

ART. PAGE

36-39. Incompressible fluids ; tubes of flow. <\> cannot be a maximum or mini-

mum. The velocity cannot be a maximum. Mean value of $ over

a spherical surface 38

40, 41. Conditions of determinateness of 0 41

42-46. Green's theorem ; dynamical interpretation ; formula for kinetic energy.

Kelvin's theorem of minimum energy 43

47, 48. Multiply-connected regions ; ' circuits ' and ' barriers ' .... 49

49-51. Irrotational motion in multiply-connected spaces ; many- valued velocity-

potential ; cyclic constants ........ 50

52. Case of incompressible fluids. Conditions of determinateness of . . 53

53-55. Kelvin's extension of Green's theorem ; dynamical interpretation ; energy

of an irrotationally moving liquid in a cyclic space .... 54

56-58. ' Sources ' and ' sinks ' ; double sources. Irrotational motion of a liquid

in terms of surface-distributions of sources 57

CHAPTER IV

MOTION OF A LIQUID IN TWO DIMENSIONS

59. Lagrange's stream -function .62

60. 60 a. Relations between stream- and velocity-functions. Two-dimensional

sources. Electrical analogies ........ 63

61. Kinetic energy 66

62. Connection with the theory of the complex variable .... 66

63. 64. Simple types of motion, cyclic and acyclic. Image of a source in a circular

barrier. Potential of a row of sources 68

65, 66. Inverse relations. Confocal curves. Flow from an open channel . . 72

67. General formulae ; Fourier method 75

68. Motion of a circular cylinder, without circulation ; stream-lines . . 76

69. Motion of a cylinder with circulation; 'lift.' Trochoidal path under

a constant force 78

70. Note on more general problems. Transformation methods ; Kutta's

problem 80

71. Inverse methods. Motion due to the translation of a cylinder; case of

an elliptic section. Flow past an oblique lamina ; couple due to

fluid pressure 83

72. Motion due to a rotating boundary. Rotating prismatic vessels of

various sections. Rotating elliptic cylinder in infinite fluid ; general

case with circulation 86

72 a. Representation of the effect at a distance of a moving cylinder by a

double source . 90

72 b. Blasius' expressions for the forces on a fixed cylinder surrounded by an

irrotationally moving liquid. Applications ; Joukowski's theorem ;

forces due to a simple source 91

73. Free stream-lines. Schwarz' method of conformal transformation . . 94 74-78. Examples. Two-dimensional form of Borda's mouthpiece ; fluid issuing

from a rectilinear aperture ; coefficient of contraction. Impact of a stream on a lamina, direct and oblique; resistance. Bobyleffs problem 96

79. Discontinuous motions . 105

80. Flow on a curved stratum 108

Contents ix

CHAPTEE V

IRROTATIONAL MOTION OF A LIQUID : PROBLEMS IN THREE DIMENSIONS

ART. PAGE

81,82. Spherical harmonics. Maxwell's theory of poles 110

83. Laplace's equation in polar co-ordinates 112

84,85. Zonal harmonics. Hypergeometric series . 113

86. Tesseral and sectorial harmonics 116

87,88. Conjugate property of surface harmonics. Expansions . . . 118

89. Symbolical solutions of Laplace's equation. Definite integral forms . 119

90, 91. Hydrodynamical applications. Impulsive pressures over a spherical

surface. Prescribed normal velocity. Energy of motion generated . 120 91 a. Examples. Collapse of a bubble. Expansion of a cavity due to internal

pressure 122

92, 93. Motion of a sphere in an infinite liquid; inertia ' coefficient. Effect of

a concentric rigid boundary . . . . . . . .123

94-96. Stokes' stream-function. Formulae in spherical harmonics. Stream-lines

of a sphere. Images of a simple and a double source in a fiscal

sphere. Forces on the sphere .125

97. Rankine's inverse method 130

98, 99. Motion of two spheres in a liquid. Kinematical formulae. Inertia

coefficients 130

100, 101. Cylindrical harmonics. Solutions of Laplace's equation in terms of

Bessel's functions. Expansion of an arbitrary function . . .134 102. Hydrodynamical examples. Flow through a circular aperture. Inertia

coefficient of a circular disk . . . . . . . 137

103-106. Ellipsoidal harmonics for an ovary ellipsoid. Translation and rotation

of an ovary ellipsoid in a liquid 139

107-109. Harmonics for a planetary ellipsoid. Flow through a circular aperture.

Stream-lines of a circular disk. Translation and rotation of a

planetary ellipsoid 142

110. Motion of a fluid in an ellipsoidal vessel 146

111. General orthogonal co-ordinates. Transformation of V2</> . . . 148

112. General ellipsoidal co-ordinates ; confocal quadrics 149

113. Flow through an elliptic aperture . . 150

114,115. Translation and rotation of an ellipsoid in liquid; inertia coefficients . 152

116. References to other problems 156

Appendix: The hydrodynamical equations referred to general ortho- gonal co-ordinates 156

CHAPTER VI

ON THE MOTION OF SOLIDS THROUGH A LIQUID : DYNAMICAL THEORY

117,118. Kinematical formulae for the case of a single body 160

119. Theory of the 'impulse ' . 161

120. Dynamical equations relative to axes fixed in the body . . . .162

121, 121 a. Kinetic energy ; coefficients of inertia. Representation of the fluid

motion at a distance by a double source 163

122, 123. Components of impulse. Reciprocal formulae 166

Contents

ART.

124.

125.

126.

127-

-129.

130.

131.

132-

134.

134

a.

135,

136.

137,

138.

139-

-141.

142,

143.

144.

PAGE

Expressions for the hydrodynamic forces. The three permanent transla- tions ; stability 168

The possible modes of steady motion. Motion due to an impulsive couple 170

Types of hydrokinetic symmetry 172

Motion of a solid of revolution. Stability of motion parallel to the axis.

Influence of rotation. Other types of steady motion . . .174

Motion of a ' helicoid ' 179

Inertia coefficients of a fluid contained in a rigid envelope . . . 180 Case of a perforated solid with cyclic motion through the apertures.

Steady motion of a ring ; condition for stability . . . .180 The hydrodynamic forces on a cylinder moving in two dimensions . . 184 Lagrange's equations of motion in generalized co-ordinates. Hamiltonian

principle. Adaptation to hydrodynamics .187

Examples. Motion of a sphere near a rigid boundary. Motion of two

spheres in the line of centres 190 /

Modification of Lagrange's equations in the case of cyclic motion ;

ignoration of co-ordinates. Equations of a gyrostatic system . . 192 Kineto-statics. Hydrodynamic forces on a solid immersed in a non- uniform stream 197

Note on the intuitive extension of dynamic principles .... 201

CHAPTER VII

145.

146.

147.

148,

149.

150.

151.

152,

153.

154,

155.

156.

157.

158,

159.

159

i.

160.

161-

163.

164.

165.

166.

166

a.

167

VORTEX MOTION

' Vortex-lines ' and ' vortex-filaments ' ; kinematical properties . . 202 Persistence of vortices ; Kelvin's proof. Equations of Cauchy, Stokes,

and Helmholtz. Motion in a fixed ellipsoidal envelope, with uniform

vorticity ............ 203

Conditions of determinateness ......... 207

Velocity in terms of expansion and vorticity ; electromagnetic analogy.

Velocities due to an isolated vortex . 208

Velocity-potential due to a vortex . .211

Vortex-sheets 212

Impulse and energy of a vortex-system ....... 214

Rectilinear vortices. Stream-lines of a vortex-pair. Other examples . 219 Investigation of the stability of a row of vortices, and of a double row.

Karman's ' vortex-street ' 224

Kirchhoff's theorems on systems of parallel vortices .... 229 Stability of a columnar vortex of finite section ; Kirchhoff's elliptic

vortex 230

Motion of a solid in a liquid of uniform vorticity 233

Vortices in a curved stratum of fluid 236

Circular vortices ; potential- and stream-function of an isolated circular

vortex ; stream-lines. Impulse and energy. Velocity of translation

of a vortex-ring 236

Mutual influence of vortex-rings. Image of a vortex-ring in a sphere . 242 General conditions for steady motion of a fluid. Cylindrical and spherical

vortices 243

References 246

Bjerknes' theorems . 247

Clebsch's transformation of the hydrodynamical equations . . . 248

ART.

168.

169-

-174.

175.

176.

177-

-179.

180-

-184.

Contents xi

CHAPTER VIII

TIDAL WAVES

PAGE

General theory of small oscillations ; normal modes ; forced oscillations . 250 Free waves in uniform canal; effect of initial conditions; measuring of

the approximations ; energy 254

Artifice of steady motion . . . . 261

Superposition of wave-systems ; reflection 262

Effect of disturbing forces ; free and forced oscillations in a finite canal . 263

Canal theory of the tides. Disturbing potentials. Tides in an equatorial

canal, and in a canal parallel to the equator; semi-diurnal and

diurnal tides. Canal coincident with a meridian ; change of mean

level ; fortnightly tide. Equatorial canal of finite length ; lag of the

tide 267

Waves in a canal of variable section. Examples of free and forced

oscillations ; exaggeration of tides in shallow seas and estuaries . 273 Waves of finite amplitude ; change of type in a progressive wave. Tides

of the second order 278

Wave motion in two horizontal dimensions ; general equations. Oscilla- tions of a rectangular basin ........ 282

Oscillations of a circular basin ; Bessel's functions ; contour lines. Elliptic

basin ; approximation to slowest mode ...... 284

Case of variable depth. Circular basin 291

Propagation of disturbances from a centre ; Bessel's function of the second

kind. Waves due to a local periodic pressure. General formula for

diverging waves. Examples of a transient local disturbance . . 293

198-201. Oscillations of a spherical sheet of water ; free and forced waves. Effect

of the mutual gravitation of the water. Reference to the case of a sea

bounded by meridians and parallels 301

Equations of motion of a dynamical system referred to rotating axes . 307 Small oscillations of a rotating system ; stability 'ordinary' and 'secular.' Effect of a small degree of rotation on types and frequencies of

normal modes 309

Approximate calculation of frequencies . . . . , . .313

Forced oscillations 316

Hydrodynamical examples ; tidal oscillations of a rotating plane sheet of

water ; waves in a straight canal . . . . . . .317

Rotating circular basin of uniform depth ; free and forced oscillations . 320

Circular basin of variable depth 326

Examples of approximate procedure ....... 328

Tidal oscillations on a rotating globe. Laplace's kinetic theory . . 330

Symmetrical oscillations. Tides of long period 333

Diurnal and semi-diurnal tides. Discussion of Laplace's solution . . 340

Hough's investigations ; extracts and results 347

References to further researches 352

Modifications of the kinetic theory due to the actual configuration of the

ocean ; question of phase . . 353

225, 226. Stability of the ocean. Remarks on the general theory of kinetic stability . 35£ Appendix : On Tide-generating Forces 358

185,

186.

187,

188.

189,

190.

191,

192.

193. 194-

-197.

202,

203.

204-

-205 a.

205 b.

206.

207,

208.

209-

-211.

212.

212

a.

213,

214.

215-

-217.

218-

-221.

222,

223.

223,

a.

224.

xii Contents

CHAPTER IX

SURFACE WAVES

ART. PAGE

227. The two-dimensional problem ; surface conditions 363

228. Standing waves ; lines of motion 364

229. 230. Progressive waves ; orbits of particles. Wave- velocity ; numerical tables.

Energy of a simple-harmonic wave-train 366

231. Oscillations of superposed fluids 370

232. Instability of the boundary of two currents 373

233. 234. Artifice of steady motion 375

235. Waves in a heterogeneous liquid 378

236, 237. Group- velocity. Transmission of energy . 380

238-240. The Cauchy-Poisson wave-problem ; waves due to an initial local eleva- tion, or to a local impulse 384

241. Kelvin's approximate formula for the effect of a local disturbance in

a linear medium. Graphical constructions 395

242-246. Surface-disturbance of a stream. Case of finite depth. Effect of inequali- ties in its bed 398

247. Waves due to a submerged cylinder 410

248, 249. General theory of waves due to a travelling disturbance. Wave- resistance 413

250. Waves of finite height ; waves of permanent type. Limiting form . . 417

251. Gerstner's rotational waves 421

252. 253. Solitary waves. Oscillatory waves of Korteweg and De Vries . . 423

254. Helmholtz' dynamical condition for waves of permanent type . . 427

255, 256. Wave-propagation in two horizontal dimensions. Effect of a local dis-

turbance. Effect of a travelling pressure-disturbance; wave-patterns 429 256 a, 256 b. Travelling disturbances of other types. Ship-waves. Wave-resistance.

Effect of finite depth on the wave-pattern 437

257-259. Standing waves in limited masses of water. Transverse oscillation in

canals of triangular, and semi-circular section 440

260, 261. Longitudinal oscillations ; canal of triangular section ; edge- waves . 445 262-264. Oscillations of a liquid globe, lines of motion. Ocean of uniform depth

on a spherical nucleus 450

265. Capillarity. Surface-condition 455

266. Capillary waves. Group-velocity 456

267. 268. Waves under gravity and capillarity. Minimum wave-velocity. Waves

on the boundary of two currents 458

269. Waves due to a local disturbance. Effect of a travelling disturbance ;

waves and ripples 462

270-272. Surface-disturbance of a stream ; formal investigation. Fish-line problem.

Wave-patterns 464

273, 274 Vibrations of a cylindrical column of liquid. Instability of a jet . . 471

275 Oscillations of a liquid globe, and of a bubble 473

Contents xiii

CHAPTER X

WAVES OF EXPANSION

ART. PAGE

276-280. Plane waves ; velocity of sound ; energy of a wave-system . . . 476 281-284. Plane waves of finite amplitude; methods of Riemann and Earnshaw. Condition for permanence of type ; Rankine's investigations. Waves

of approximate discontinuity 481

285, 286. Spherical waves. Solution in terms of initial conditions . . . 489

287, 288. General equation of sound-waves. Equation of energy. Determinateness

of solutions . 492

289. Simple-harmonic vibrations. Simple and double sources. Emission of

energy ............ 496

290. Helmholtz' adaptation of Green's theorem. Velocity-potential in terms

of surface-distributions of sources. Kirchhoff's formula . . . 498

291. Periodic disturbing forces 501

292. Applications of spherical harmonics. General formulae .... 503

293. Vibrations of air in a spherical vessel. Vibrations of a spherical stratum 506

294. Propagation of waves outwards from a spherical surface ; attenuation

due to lateral motion 508

295. Influence of the air on the oscillations of a ball-pendulum ; correction for

inertia ; damping . . . . . . . . . .510

296-298. Scattering of sound-waves by a spherical obstacle. Impact of waves on

a movable sphere; case of synchronism . . . . . .511

299, 300. Diffraction when the wave-length is relatively large : by a flat disk,

by an aperture in a plane screen, and by an obstacle of any form . 517

301. Solution of the equation of sound in spherical harmonics. Conditions at

a wave-front ........... 521

302. Sound-waves in two dimensions. Effect of a transient source ; comparison

with the one- and three-dimensional cases ..... 524

303. 304. Simple-harmonic vibrations ; solutions in Bessel functions. Oscillating

cylinder. Scattering of waves by a cylindrical obstacle . . . 527 305. Approximate theory of diffraction of long waves in two dimensions.

Diffraction by a flat blade, and by an aperture in a thin screen . 531

306,307. Reflection and transmission of sound-waves by a grating . . . 533

308. Diffraction by a semi-infinite screen ... .... 538

309, 310. Waves propagated vertically in the atmosphere; 'isothermal' and 'con-

vective' hypotheses . . . .541

. 547

. 554

556

558

311, 311a, 312. Theory of long atmospheric waves

313. General equations of vibration of a gas under constant forces.

314, 315. Oscillations of an atmosphere on a non-rotating globe . 316. Atmosphere tides on a rotating globe. Possibility of resonance

xiv Contents

CHAPTER XI

VISCOSITY

ART. PAGE

317, 318. Theory of dissipative forces. One degree of freedom; free anu forced

oscillations. Effect of friction on phase 562

319. Application to tides in equatorial canal ; tidal lag and tidal friction . 565

320. Equations of dissipative systems in general ; frictional and gyrostatic

terms. Dissipation function 567

321. Oscillations of a dissipative system about a configuration of absolute

equilibrium ' 568

322. Effect of gyrostatic terms. Example of two degrees of freedom ; dis-

turbing forces of long period 570

323-325. Viscosity of fluids ; specification of stress ; formulae of transformation . 571 326, 327. The stresses as linear functions of rates of strain. Coefficient of viscosity.

Boundary-conditions ; question of slipping 574

328. Dynamical equations. The modified Helmholtz equations; diffusion of

vorticity - . 576

329. Dissipation of energy by viscosity 579

330, 330 a. Flow of a liquid between parallel planes. Hele Shaw's experiments.

Theory of lubrication ; example 581

331, 332. Flow through a pipe of circular section; Poiseuille's laws; question of

slipping. Other forms of section 585

333, 334. Cases of steady rotation. Practical limitations 587

334 a. Examples of variable motion. Diffusion of a vortex. Effect of surface- forces on deep water 590

335, 336. Slow steady motion ; general solution in spherical harmonics ; formulae

for the stresses 594

337. Rectilinear motion of a sphere ; resistance ; terminal velocity ; stream-

lines. Case of a liquid sphere ; and of a solid sphere, with slipping 597

338. Method of Stokes ; solutions in terms of the stream -function . . . 602

339. Steady motion of an ellipsoid 604

340. 341. Steady motion in a constant field of force 605

342. Steady motion of a sphere ; Oseen's criticism, and solution . . . 608

343. 343 a. Steady motion of a cylinder, treated by Oseen's method. References to

other investigations 614

344. Dissipation of energy in steady motion; theorems of Helmholtz and

Korteweg. Rayleigh's extension 617

345-347. Problems of periodic motion. Laminar motion, diffusion of vorticity.

Oscillating plane. Periodic tidal force ; feeble influence of viscosity

in rapid motions 619

348-351. Effect of viscosity on water-waves. Generation of waves by wind. Calming

effect of oil on waves .......... 623

352, 353. Periodic motion with a spherical boundary ; general solution in spherical

harmonics 632

354. Applications ; decay of motion in a spherical vessel ; torsional oscillations

of a hollow sphere containing liquid 637

355. Effect of viscosity on the oscillations of a liquid globe .... 639

356. Effect on the rotational oscillations of a sphere, and on the vibrations of

a pendulum 641

357. Notes on two-dimensional problems . 644

Contents xv

ART. PAGE

358. Viscosity in gases ; dissipation function 645

359, 360. Damping of plane waves of sound by viscosity ; combined effect of

viscosity and thermal conduction 646

360 a. Waves of permanent type, as affected by viscosity alone . . . . 650

360 b. Absorption of sound by porous bodies 652

361. Effect of viscosity on diverging waves 654

362, 363. Effect on the scattering of waves by a spherical obstacle, fixed or free . 657

364. Damping of sound-waves in a spherical vessel 661

365, 366. Turbulent motion. Reynolds' experiments ; critical velocities of water

in a pipe ; law of resistance. Inferences from theory of dimensions 663

366 a. Motion between rotating cylinders 667

366 b. Coefficient of turbulence ; 'eddy' or 'molar' viscosity .... 668

366 c. Turbulence in the atmosphere ; variation of wind with height . . 669

367, 368. Theoretical investigations of Rayleigh and Kelvin 670

369. Statistical method of Reynolds 674

370. Resistance of fluids. Criticism of the discontinuous solutions of Kirchhoff

and Rayleigh 678

370 a. Karman's formula for resistance 680

370 b. Lift due to circulation 681

371. Dimensional formulae. Relations between model and full-scale . . 682 371a, b, c. The boundary layer. Note on the theory of the aerofoil .... 684 37 Id, e, f, g. Influence of compressibility. Failure of stream-line flow at high speeds 691

CHAPTER XII

ROTATING MASSES OF LIQUID

372. Forms of relative equilibrium. General theorems ..... 697

373. Formulae relating to attraction of ellipsoids. Potential energy of an

ellipsoidal mass 700

374. Maclaurin's ellipsoids. Relations between eccentricity, angular velocity

and angular momentum ; numerical tables 701

375. Jacobi's ellipsoids. Linear series of ellipsoidal forms of equilibrium.

Numerical results 704

376. Other special forms of relative equilibrium. Rotating annulus . . 707

377. General problem of relative equilibrium ; Poineard's investigation. Linear

series of equilibrium forms ; limiting forms and forms of bifurcation.

Exchange of stabilities 710

378-380. Application to a rotating system. Secular stability of Maclaurin's and

Jacobi's ellipsoids. The pear-shaped figure of equilibrium . . 713

381. Small oscillations of a rotating ellipsoidal mass; Poincar^'s method.

References 717

382. Dirichlet's investigations; references. Finite gravitational oscillations

of a liquid ellipsoid without rotation. Oscillations of a rotating

ellipsoid of revolution 719

383. Dedekind's ellipsoid. The irrotational ellipsoid. Rotating elliptic cylinder 721

384. Free and forced oscillations of a rotating ellipsoidal shell containing

liquid. Precession 724

385. Precession of a liquid ellipsoid 728

List of Authors cited 731

Index 734

HYDRODYNAMICS

CHAPTER I

THE EQUATIONS OF MOTION

1. The following investigations proceed on the assumption that the matter with which we deal may be treated as practically continuous and homogeneous in structure ; i.e. we assume that the properties of the smallest portions into which we can conceive it to be divided are the same as those of the substance in bulk.

The fundamental property of a fluid is that it cannot be in equilibrium in a state of stress such that the mutual action between two adjacent parts is oblique to the common surface. This property is the basis of Hydrostatics, and is verified by the complete agreement of the deductions of that science with experiment. Very slight observation is enough, however, to convince us that oblique stresses may exist in fluids in motion. Let us suppose for instance that a vessel in the form of a circular cylinder, containing water (or other liquid), is made to rotate about its axis, which is vertical. If the angular velocity of the vessel be constant, the fluid is soon found to be rotat- ing with the vessel as one solid body. If the vessel be now brought to rest, the motion of the fluid continues for some time, but gradually subsides, and at length ceases altogether; and it is found that during this process the portions of fluid which are further from the axis lag behind those which are nearer, and have their motion more rapidly checked. These phenomena point to the existence of mutual actions between contiguous elements which are partly tangential to the common surface. For if the mutual action were everywhere wholly normal, it is obvious that the moment of momentum, about the axis of the vessel, of any portion of fluid bounded by a surface of revolution about this axis, would be constant. We infer, moreover, that these tangential stresses are not called into play so long as the fluid moves as a solid body, but only whilst a change of shape of some portion of the mass is going on, and that their tendency is to oppose this change of shape.

2. It is usual, however, in the first instance to neglect the tangential stresses altogether. Their effect is in many practical cases small, and, inde- pendently of this, it is convenient to divide the not inconsiderable difficulties of our subject by investigating first the effects of purely normal stress. The further consideration of the laws of tangential stress is accordingly deferred till Chapter XI.

The Equations of Motion

[chap. I

If the stress exerted across any small plane area situate at a point P of the fluid be wholly normal, its intensity (per unit area) is the same for all aspects of the plane. The following proof of this theorem is given here for purposes of reference. Through P draw three straight lines PA, PB, PC mutually at right angles, and let a plane whose direction-cosines relatively to these lines are I, m, n, passing infinitely close to P, meet them in A, B, C. Let p, Pi, P2, Pz denote the intensities of the stresses* across the faces ABC, PBG, PC A, PAB, respectively, of the tetrahedron PABC. If A be the area of the first-mentioned face, the areas of the others are, in order, IA, mA, raA. Hence if we form the equation of motion of the tetrahedron parallel to PA we have px . lA = pl . A, where we have omitted the terms which express the rate of change of momentum, and the component of the extraneous forces, because they are ultimately propor- tional to the mass of the tetrahedron, and therefore of the third order of small linear quantities, whilst the terms retained are of the second. We have then, ultimately, p—p\, and similarly p = p2 = p3, which proves the theorem.

3. The equations of motion of a fluid have been obtained in two different forms, corresponding to the two ways in which the problem of determining the motion of a fluid mass, acted on by given forces and subject to given conditions, may be viewed. We may either regard as the object of our investigations a knowledge of the velocity, the pressure, and the density, at all points of space occupied by the fluid, for all instants; or we may seek to determine the history of every particle. The equations obtained on these two plans are conveniently designated, as by German mathematicians, the 'Eulerian' and the 'Lagrangian' forms of the hydrokinetic equations, although both forms are in reality due to Eulerf.

The Eulerian Equations.

4. Let u, v, w be the components, parallel to the co-ordinate axes, of the velocity at the point (x, y, z) at the time t. These quantities are then functions of the independent variables x, y, z, t. For any particular value of t they define the motion at that instant at all points of space occupied by

* Reckoned positive when pressures, negative when tensions. Most fluids are, however, incapable under ordinary conditions of supporting more than an exceedingly slight degree of tension, so that^ is nearly always positive.

f " Principes generaux du mouvement des fluides," Hist, dc VAcad. dc Berlin, 1755.

" De principiis motus fluidorum," Novi Comm. Acad. Petrop. xiv. 1 (1759).

Lagrange gave three investigations of the equations of motion; first, incidentally, in

2-6] Eulerian Equations 3

the fluid; whilst for particular values of x, y, z they give the history of what goes on at a particular place.

We shall suppose, for the most part, not only that u, v, w are finite and continuous functions of x, y, z, but that their space-derivatives of the first order (du/dx, dv/dx, dw/dx, &c.) are everywhere finite*; we shall understand by the term 'continuous motion,' a motion subject to these restrictions. Cases of exception, if they present themselves, will require separate examina- tion. In continuous motion, as thus defined, the relative velocity of any two neighbouring particles P, P' will always be infinitely small, so that the line PP' will always remain of the same order of magnitude. It follows that if we imagine a small closed surface to be drawn, surrounding P, and suppose it to move with the fluid, it will always enclose the same matter. And any surface whatever, which moves with the fluid, completely and permanently separates the matter on the two sides of it.

5. The values of u, v, w for successive values of t give as it were a series of pictures of consecutive stages of the motion, in which however there is no immediate means of tracing the identity of any one particle.

To calculate the rate at which any function F (x, y, z, t) varies for a moving particle, we may remark that at the time t + 8t the particle which was originally in the position (x, ?/. z) is in the position (x + u8t, y + v8t, z + w8t), so that the corresponding value of F is

F(x + u8t, y + v8t,z + iv8t, t + 8t) = F+u8td-^ + v8t~- + w8t~ + 8t%- .

17 ox oy oz dt

If, after Stokes, we introduce the symbol D/Dt to denote a differentiation following the motion of the fluid, the new value of F is also expressed by F+DF/Dt.8t, whence

DF dF dF dF dF

Bt=Tt+UTx + Vdy + Wdz ' (1)

6. To form the dynamical equations, let p be the pressure, p the density, X, T, Z the components of the extraneous forces per unit mass, at the point {x, y, z) at the time t. Let us take an element having its centre at (x, y, z), and its edges 8x, 8y, 8z parallel to the rectangular co-ordinate axes. The rate at which the ^-component of the momentum of this element is increasing is p8x8y8z DujDt; and this must be equal to the ^-component of the forces

connection with the principle of Least Action, in the Miscellanea Taurinensia, ii. (1760) [Oeuvres, Paris, 1867-92, i.]; secondly in his "Memoire sur la Theorie du Mouvement des Fluides," Nouv. mem. de V Acad, de Berlin, 1781 [Oeuvres, iv.]; and thirdly in the Mecaniquc Analytique. In this last exposition he starts with the second form of the equations (Art. 14, below), but translates them at once into the ' Eulerian' notation.

* It is important to bear in mind, with a view to some later developments under the head of Vortex Motion, that these derivatives need not be assumed to be continuous.

4 The Equations of Motion [chap, i

acting on the element. Of these the extraneous forces give pBxByBzX. The pressure on the yz-fave which is nearest the origin will be ultimately

that on the opposite face

(p + \dp\dx . 8%) By Bz. The difference of these gives a resultant dp/dx. BxByBz in the direction of ^-positive. The pressures on the remaining faces are perpendicular to x. We have then

p Bx By Bz yc = pBxByBz X ^-BxBy Bz.

Substituting the value of DujDt from (1), and writing down the sym- metrical equations, we have

du du du du _ Y 1 dp dt dx dy dz pdx'

•(2)

dv dv dv dv _ v 1 dp

dt dx dy dz pdy'

dw dw dw dw _ 7 1 dp dt dx dy dz p dz

7. To these dynamical equations we must join, in the first place, a certain kinematical relation between u, v, w, p, obtained as follows.

If Q be the volume of a moving element, we have, on account of the constancy of mass,

Dt \Dp 1 DQ .

-Pm+QwrQ w

To calculate the value of 1/Q .DQ/Dt, let the element in question be that which at time t fills the rectangular space BxByBz having one corner P at {%, y, z), and the edges PL, PM, PN (say) parallel to the co-ordinate axes. At time t + Bt the same element will form an oblique parallelepiped, and since the velocities of the particle L relative to the particle P are du/dx . Bx, dv/dx.Bx, dw/dx.Bx, the projections of the edge PL on the co-ordinate axes become, after the time Bt,

(l+pSt)8*, d^ti.Zx, d^ St. Sec, \ dx ) dx dx

respectively. To the first order in Bt, the length of this edge is now

and similarly for the remaining edges. Since the angles of the parallelepiped

* It is easily seen, by Taylor's theorem, that the mean pressure over any face of the element 5x by 5z may be taken to be equal to the pressure at the centre of that face.

6-t] Equation of Continuity 5

differ infinitely little from right angles, the volume is still given, to the first order in Bt, by the product of the three edges, i.e. we have

1 DQ dii dv dw (G>.

or QDi = dx+ dy+ dz~ ( }

Hence (1) becomes

_s^®4;+S)=° ^

This is called the 'equation of continuity.'

rvu - du dv dw //1X

1 he expression a" "*" a ^~2~' ' '

which, as we have seen, measures the rate of dilatation of the fluid at the point (x,y,2), is conveniently called the 'expansion' at that point. From a more general point of view the expression (4) is called the 'divergence' of the vector (u,v,w); it is often denoted briefly by

div (u, v, w). The preceding investigation is substantially that given by Euler*. Another, and now more usual, method of obtaining the equation of con- tinuity is, instead of following the motion of a fluid element, to fix the attention on an element BxByBz of space, and to calculate the change pro- duced in the included mass by the flux across the boundary. If the centre of the element be at (x, y, z), the amount of matter which per unit time enters it across the yz-f&ce nearest the origin is

and the amount which leaves it by the opposite face is

f pu + \ '- Bx j ByBz. BxByBz,

The two faces together give a gain

d .pu dx

per unit time. Calculating in the same way the effect of the flux across the remaining faces, we have for the total gain of mass, per unit time, in the space BxByBz, the formula

(d .pu 3 . pv d . pw\ j j j.

Since the quantity of matter in any region can vary only in consequence of the flux across the boundary, this must be equal to

^(p BxByBz),

* I.e. ante p. 2.

6 The Equations of Motion [chap, i

whence we get the equation of continuity in the form

^+9_£V_^ + ^ = 0 (5)

dt ox Oy oz v

8. It remains to put in evidence the physical properties of the fluid, so far as these affect the quantities which occur in our equations.

In an 'incompressible' fluid, or liquid, we have Dp/Dt= 0, in which case the equation of continuity takes the simple form

a-M4:=° «

It is not assumed here that the fluid is of uniform density, though this is of course by far the most important case.

If we wish to take account of the slight compressibility of actual liquids, we shall have a relation of the form

p = /e(p-po)lpo, (2)

or plp0 = l+p//e, ..(3)

where k denotes what is called the 'elasticity of volume.'

In the case of a gas whose temperature is uniform and constant we have the ' isothermal ' relation

PlPo = p/po> (4)

where p0, p0 are any pair of corresponding values for the temperature in question.

In most cases of motion of gases, however, the temperature is not constant, but rises and falls, for each element, as the gas is compressed or rarefied. When the changes are so rapid that we can ignore the gain or loss of heat by an element due to conduction and radiation, we have the 'adiabatic' relation

PlPo = (plpo)y, (5)

where po and p0 are any pair of corresponding values for the element con- sidered. The constant 7 is the ratio of the two specific heats of the gas ; for atmospheric air, and some other gases, its value is about 1*408.

9. At the boundaries (if any) of the fluid, the equation of continuity is replaced by a special surface-condition. Thus at a fixed boundary, the velocity of the fluid perpendicular to the surface must be zero, i.e. if l> m, n be the direction-cosines of the normal,

lu + mv + nw = 0 (1)

Again at a surface of discontinuity, i.e. a surface at which the values of u, v, w change abruptly as we pass from one side to the other, we must have

l(ux u2)-\-m (v1—v2)+ n(w1 w2) = 0, (2)

where the suffixes are used to distinguish the values on the two sides. The same relation must hold at the common surface of a fluid and a moving solid.

7-9] Boundary Condition 7

The general surface-condition, of which these are particular cases, is that if F(x, y, z, t)