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: 461891
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
49. ' 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 coordinates 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 velocitypotential exists. Pressure equation 19
2123. Steady motion. Deduction of the pressureequation from the principle
of energy. Limiting velocity 20
24. Efflux of liquids ; vena contracta 23
24 a. 25. Efflux of gases 25
2629. 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
3639. 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
4246. Green's theorem ; dynamical interpretation ; formula for kinetic energy.
Kelvin's theorem of minimum energy 43
47, 48. Multiplyconnected regions ; ' circuits ' and ' barriers ' .... 49
4951. Irrotational motion in multiplyconnected spaces ; many valued velocity
potential ; cyclic constants ........ 50
52. Case of incompressible fluids. Conditions of determinateness of <£ . . 53
5355. Kelvin's extension of Green's theorem ; dynamical interpretation ; energy
of an irrotationally moving liquid in a cyclic space .... 54
5658. ' Sources ' and ' sinks ' ; double sources. Irrotational motion of a liquid
in terms of surfacedistributions 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 velocityfunctions. Twodimensional
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 ; streamlines . . 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 streamlines. Schwarz' method of conformal transformation . . 94 7478. Examples. Twodimensional 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 coordinates 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
9496. Stokes' streamfunction. Formulae in spherical harmonics. Streamlines
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
103106. Ellipsoidal harmonics for an ovary ellipsoid. Translation and rotation
of an ovary ellipsoid in a liquid 139
107109. Harmonics for a planetary ellipsoid. Flow through a circular aperture.
Streamlines 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 coordinates. Transformation of V2</> . . . 148
112. General ellipsoidal coordinates ; 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 coordinates 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 coordinates. 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 coordinates. Equations of a gyrostatic system . . 192 Kinetostatics. 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
' Vortexlines ' and ' vortexfilaments ' ; 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
Velocitypotential due to a vortex . .211
Vortexsheets 212
Impulse and energy of a vortexsystem ....... 214
Rectilinear vortices. Streamlines of a vortexpair. Other examples . 219 Investigation of the stability of a row of vortices, and of a double row.
Karman's ' vortexstreet ' 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 streamfunction of an isolated circular
vortex ; streamlines. Impulse and energy. Velocity of translation
of a vortexring 236
Mutual influence of vortexrings. Image of a vortexring 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 wavesystems ; 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; semidiurnal 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
198201. 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 semidiurnal 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 Tidegenerating 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 twodimensional 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 simpleharmonic wavetrain 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
238240. The CauchyPoisson waveproblem ; 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
242246. Surfacedisturbance 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. Wavepropagation in two horizontal dimensions. Effect of a local dis
turbance. Effect of a travelling pressuredisturbance; wavepatterns 429 256 a, 256 b. Travelling disturbances of other types. Shipwaves. Waveresistance.
Effect of finite depth on the wavepattern 437
257259. Standing waves in limited masses of water. Transverse oscillation in
canals of triangular, and semicircular section 440
260, 261. Longitudinal oscillations ; canal of triangular section ; edge waves . 445 262264. Oscillations of a liquid globe, lines of motion. Ocean of uniform depth
on a spherical nucleus 450
265. Capillarity. Surfacecondition 455
266. Capillary waves. Groupvelocity 456
267. 268. Waves under gravity and capillarity. Minimum wavevelocity. Waves
on the boundary of two currents 458
269. Waves due to a local disturbance. Effect of a travelling disturbance ;
waves and ripples 462
270272. Surfacedisturbance of a stream ; formal investigation. Fishline problem.
Wavepatterns 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
276280. Plane waves ; velocity of sound ; energy of a wavesystem . . . 476 281284. 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 soundwaves. Equation of energy. Determinateness
of solutions . 492
289. Simpleharmonic vibrations. Simple and double sources. Emission of
energy ............ 496
290. Helmholtz' adaptation of Green's theorem. Velocitypotential in terms
of surfacedistributions 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 ballpendulum ; correction for
inertia ; damping . . . . . . . . . .510
296298. Scattering of soundwaves by a spherical obstacle. Impact of waves on
a movable sphere; case of synchronism . . . . . .511
299, 300. Diffraction when the wavelength 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 wavefront ........... 521
302. Soundwaves in two dimensions. Effect of a transient source ; comparison
with the one and threedimensional cases ..... 524
303. 304. Simpleharmonic 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 soundwaves by a grating . . . 533
308. Diffraction by a semiinfinite 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 nonrotating 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
323325. Viscosity of fluids ; specification of stress ; formulae of transformation . 571 326, 327. The stresses as linear functions of rates of strain. Coefficient of viscosity.
Boundaryconditions ; 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
345347. Problems of periodic motion. Laminar motion, diffusion of vorticity.
Oscillating plane. Periodic tidal force ; feeble influence of viscosity
in rapid motions 619
348351. Effect of viscosity on waterwaves. 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 twodimensional 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 soundwaves 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 fullscale . . 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 streamline 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
378380. Application to a rotating system. Secular stability of Maclaurin's and
Jacobi's ellipsoids. The pearshaped 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 directioncosines 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 firstmentioned 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 coordinate 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
26] 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 spacederivatives 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 coordinate 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, 186792, 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 yzfave 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 coordinate 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 coordinate 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.
6t] 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 yzf&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
aM4:=° • «
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(ppo)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 surfacecondition. 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 directioncosines 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.
79] Boundary Condition 7
The general surfacecondition, of which these are particular cases, is that if F(x, y, z, t)