Fluid Mechanics, Course in Theoretical Physics

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Fluid Mechanics, Course in Theoretical Physics

Fluid Mechanics, Course in Theoretical Physics
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Preface

The present book deals with fluid mechanics, i.e. the theory of the motion of liquids and gases.

The nature of the book is largely determined by the fact that it describes fluid mechanics as a branch of theoretical physics, and it is therefore markedly different from other textbooks on the same subject. We have tried to develop as fully as possible all matters of physical interest, and to do so in such a way as to give the clearest possible picture of the phenomena and their interrelation.
Accordingly, we discuss neither approximate methods of calculation in fluid mechanics, nor empirical theories devoid of physical significance. On the other hand, accounts are given of some topics not usually found in textbooks on the subject: the theory of heat transfer and diffusion in fluids; acoustics; the theory of combustion; the dynamics of superfluids; and relativistic fluid dynamics.

TOC

I. IDEAL FLUIDS
§1. The equation of continuity 1
§2. Euler's equation 2
§3. Hydrostatics 6
§4. The condition that convection is absent 8
§5. Bernoulli's equation 9
§6. The energy flux 10
§7. The momentum flux 12
§8 The conservation of circulation 14
§9. Potential flow 16
§10. Incompressible fluids 20
§11. The drag force in potential flow past a body 31
§12. Gravity waves 36
§13. Long gravity waves 42
§14. Waves in an incompressible fluid 44

II. VISCOUS FLUIDS
§15. The equations of motion of a viscous fluid 47
§16. Energy dissipation in an incompressible fluid 53
§17. Flow in a pipe 55
§18. Flow between rotating cylinders 60
§19. The law of similarity 61
§20. Stokes' formula 63
§21. The laminar wake 71
§22. The viscosity of suspensions 76
§23. Exact solutions of the equations of motion for a viscous fluid 79
§24. Oscillatory motion in a viscous fluid 88
§25. Damping of gravity waves 98

III. TURBULENCE
§26. Stability of steady flow 102
§27. The onset of turbulence 103
§28. Stability of flow between rotating cylinders 107
§29. Stability of flow in a pipe 1 1
§30. Instability of tangential discontinuities 114
§31. Fully developed turbulence 116
§32. Local turbulence 120
§33. The velocity correlation 123
§34. The turbulent region and the phenomenon of separation 128
§35. The turbulent jet 130
§36. The turbulent wake 136
§37. Zhukovskii's theorem 137
§38. Isotropic turbulence 140

IV. BOUNDARY LAYERS
§39. The laminar boundary layer 145
§40. Flow near the line of separation 151
§41. Stability of flow in the laminar boundary layer 156
§42. The logarithmic velocity profile 159
§43. Turbulent flow in pipes 163
§44. The turbulent boundary layer 166
§45. The drag crisis 168
§46. Flow past streamlined bodies 172
§47. Induced drag 175
§48. The lift of a thin wing 179

V. THERMAL CONDUCTION IN FLUIDS
§49. The general equation of heat transfer 183
§50. Thermal conduction in an incompressible fluid 188
§51. Thermal conduction in an infinite medium 192
§52. Thermal conduction in a finite medium 196
§53. The similarity law for heat transfer 202
§54. Heat transfer in a boundary layer 205
§55. Heating of a body in a moving fluid 209
§56. Free convection 212

VI. DIFFUSION
§57. The equations of fluid dynamics for a mixture of fluids 219
§58. Coefficients of mass transfer and thermal diffusion 222
§59. Diffusion of particles suspended in a fluid 227

VII. SURFACE PHENOMENA
§60. Laplace's formula 230
§61. Capillary waves 237
§62. The effect of adsorbed films on the motion of a liquid 241

VIII. SOUND
§63. Sound waves 245
§64. The energy and momentum of sound waves 249
§65. Reflection and refraction of sound waves 253
§66. Geometrical acoustics 256
§67. Propagation of sound in a moving medium 259
§68. Characteristic vibrations 262
§69. Spherical waves 265
§70. Cylindrical waves 268
§71. The general solution of the wave equation 270
§72. The lateral wave 273
§73. The emission of sound 279
§74. The reciprocity principle 288
§75. Propagation of sound in a tube 291
§76. Scattering of sound 294
§77. Absorption of sound 298
§78. Second viscosity 304

IX. SHOCK WAVES
§79. Propagation of disturbances in a moving gas 310
§80. Steady flow of a gas 313
§81. Surfaces of discontinuity 317
§82. The shock adiabatic 319
§83. Weak shock waves 322
§84. The direction of variation of quantities in a shock wave 325
§85. Shock waves in a perfect gas 329
§86. Oblique shock waves 333
§87. The thickness of shock waves 337
§88. The isothermal discontinuity 342
§89. Weak discontinuities 344

X. ONE-DIMENSIONAL GAS FLOW
§90. Flow of gas through a nozzle 347
§91. Flow of a viscous gas in a pipe 350
§92. One-dimensional similarity flow 353
§93. Discontinuities in the initial conditions 360
§94. One-dimensional travelling waves 366
§95. Formation of discontinuities in a sound wave 372
§96. Characteristics 373
§97. Riemann invariants 381
§98. Arbitrary one-dimensional gas flow 386
§99. The propagation of strong shock waves 392
§100. Shallow-water theory 396

XL THE INTERSECTION OF SURFACES OF DISCONTINUITY
§101. Rarefaction waves 399
§102. The intersection of shock waves 405
§103. The intersection of shock waves with a solid surface 410
§104. Supersonic flow round an angle 413
§105. Flow past a conical obstacle 418

XII. TWO-DIMENSIONAL GAS FLOW
§106. Potential flow of a gas 422
§107. Steady simple waves 425
§108. Chaplygin's equation: the general problem of steady two dimensional gas flow 430
§109. Characteristics in steady two-dimensional flow 433
§110. The Euler-Tricomi equation. Transonic flow 436
§111. Solutions of the Euler-Tricomi equation near non-singular points of the sonic surface 441
§1 12. Flow at the velocity of sound 446
§113. The intersection of discontinuities with the transition line 451

XIII. FLOW PAST FINITE BODIES
§114. The formation of shock waves in supersonic flow past bodies 457
§115. Supersonic flow past a pointed body 460
§116. Subsonic flow past a thin wing 464
§117. Supersonic flow past a wing 466
§118. The law of transonic similarity 469
§119. The law of hypersonic similarity 472

XIV. FLUID DYNAMICS OF COMBUSTION
§120. Slow combustion 474
§121. Detonation 480
§122. The propagation of a detonation wave 487
§123. The relation between the different modes of combustion 493
§124. Condensation discontinuities 496

XV. RELATIVISTIC FLUID DYNAMICS
§125. The energy-momentum tensor 499
§126. The equations of relativistic fluid dynamics 500
§127. Relativistic equations for dissipative processes 505

XVI. DYNAMICS OF SUPERFLUIDS
§128. Principal properties of superfluids 507
§129. The thermo-mechanical effect 509
§130. The equations of superfluid dynamics 510
§131. The propagation of sound in a superfluid 517

XVII. FLUCTUATIONS IN FLUID DYNAMICS
§132. The general theory of fluctuations in fluid dynamics 523
§133. Fluctuations in an infinite medium 526
Index 530