Fluid Mechanics (eBook)

Landau and Lifshitz: Course of Theoretical Physics, Volume 6

L D Landau, E. M. Lifshitz (Autoren)

eBook Download: PDF | EPUB

2013 | 2. Auflage
554 Seiten
Elsevier Science (Verlag)
978-1-4831-6104-4 (ISBN)

Infos

Fluid Mechanics, Second Edition deals with fluid mechanics, that is, the theory of the motion of liquids and gases. Topics covered range from ideal fluids and viscous fluids to turbulence, boundary layers, thermal conduction, and diffusion. Surface phenomena, sound, and shock waves are also discussed, along with gas flow, combustion, superfluids, and relativistic fluid dynamics. This book is comprised of 16 chapters and begins with an overview of the fundamental equations of fluid dynamics, including Euler's equation and Bernoulli's equation. The reader is then introduced to the equations of motion of a viscous fluid; energy dissipation in an incompressible fluid; damping of gravity waves; and the mechanism whereby turbulence occurs. The following chapters explore the laminar boundary layer; thermal conduction in fluids; dynamics of diffusion of a mixture of fluids; and the phenomena that occur near the surface separating two continuous media. The energy and momentum of sound waves; the direction of variation of quantities in a shock wave; one- and two-dimensional gas flow; and the intersection of surfaces of discontinuity are also also considered. This monograph will be of interest to theoretical physicists.

Inhalt

Front Cover 1
Fluid Mechanics 6
Copyright Page 7
Table of Contents 8
Prefaces to the English editions 12
E. M. Lifshitz 14
Notation 16
CHAPTER I. IDEAL FLUIDS 18
§1. The equation of continuity 18
§2. Euler's equation 19
§3. Hydrostatics 22
§4. The condition that convection be absent 24
§5. Bernoulli's equation 25
§6. The energy flux 26
§7. The momentum flux 28
§8. The conservation of circulation 29
§9. Potential flow 31
§10. Incompressible fluids 34
§11. The drag force in potential flow past a body 43
§12. Gravity waves 48
§13. Internal waves in an incompressible fluid 54
§14. Waves in a rotating fluid 57
CHAPTER II. VISCOUS FLUIDS 61
§15. The equations of motion of a viscous fluid 61
§16. Energy dissipation in an incompressible fluid 67
§17. Flow in a pipe 68
§18. Flow between rotating cylinders 72
§19. The law of similarity 73
§20. Flow with small Reynolds numbers 75
§21. The laminar wake 84
§22. The viscosity of suspensions 90
§23. Exact solutions of the equations of motion for a viscous fluid 92
§24. Oscillatory motion in a viscous fluid 100
§25. Damping of gravity waves 109
CHAPTER III. TURBULENCE 112
§26. Stability of steady flow 112
§27. Stability of rotary flow 116
§28. Stability of flow in a pipe 120
§29. Instability of tangential discontinuities 123
§30. Quasi-periodic flow and frequency locking 125
§31. Strange attractors 130
§32. Transition to turbulence by period doubling 135
§33. Fully developed turbulence 146
§34. The velocity correlation functions 152
§35. The turbulent region and the phenomenon of separation 163
§36. The turbulent jet 164
§37. The turbulent wake 169
§38. Zhukovskii's theorem 170
CHAPTER IV. BOUNDARY LAYERS 174
§39. The laminar boundary layer 174
§40. Flow near the line of separation 180
§41. Stability of flow in the laminar boundary layer 184
§42. The logarithmic velocity profile 189
§43. Turbulent flow in pipes 193
§44. The turbulent boundary layer 195
§45. The drag crisis 197
§46. Flow past streamlined bodies 200
§47. Induced drag 202
§48. The lift of a thin wing 206
CHAPTER V. THERMAL CONDUCTION IN FLUIDS 209
§49. The general equation of heat transfer 209
§50. Thermal conduction in an incompressible fluid 213
§51. Thermal conduction in an infinite medium 217
§52. Thermal conduction in a finite medium 220
§53. The similarity law for heat transfer 225
§54. Heat transfer in a boundary layer 227
§55. Heating of a body in a moving fluid 231
§56. Free convection 234
§57. Convective instability of a fluid at rest 238
CHAPTER VI. DIFFUSION 244
§58. The equations of fluid dynamics for a mixture of fluids 244
§59. Coefficients of mass transfer and thermal diffusion 247
§60. Diffusion of particles suspended in a fluid 252
CHAPTER VII. SURFACE PHENOMENA 255
§61. Laplace's formula 255
§62. Capillary waves 261
§63. The effect of adsorbed films on the motion of a liquid 265
CHAPTER VIII. SOUND 268
§64. Sound waves 268
§65. The energy and momentum of sound waves 272
§66. Reflection and refraction of sound waves 276
§67. Geometrical acoustics 277
§68. Propagation of sound in a moving medium 280
§69. Characteristic vibrations 283
§70. Spherical waves 286
§71. Cylindrical waves 288
§72. The general solution of the wave equation 290
§73. The lateral wave 293
§74. The emission of sound 298
§75. Sound excitation by turbulence 306
§76. The reciprocity principle 309
§77. Propagation of sound in a tube 311
§78. Scattering of sound 314
§79. Absorption of sound 317
§80. Acoustic streaming 322
§81. Second viscosity 325
CHAPTER IX. SHOCK WAVES 330
§82. Propagation of disturbances in a moving gas 330
§83. Steady flow of a gas 333
§84. Surfaces of discontinuity 337
§85. The shock adiabatic 341
§86. Weak shock waves 344
§87. The direction of variation of quantities in a shock wave 346
§88. Evolutionary shock waves 348
§89. Shock waves in a polytropic gas 350
§90. Corrugation instability of shock waves 353
§91. Shock wave propagation in a pipe 360
§92. Oblique shock waves 362
§93. The thickness of shock waves 367
§94. Shock waves in a relaxing medium 372
§95. The isothermal discontinuity 373
§96. Weak discontinuities 375
CHAPTER X. ONE-DIMENSIONAL GAS FLOW 378
§97. Flow of gas through a nozzle 378
§98. Flow of a viscous gas in a pipe 381
§99. One-dimensional similarity flow 383
§100. Discontinuities in the initial conditions 390
§101. One-dimensional travelling waves 395
§102. Formation of discontinuities in a sound wave 402
§103. Characteristics 408
§104. Riemann invariants 411
§105. Arbitrary one-dimensional gas flow 414
§106. A strong explosion 420
§107. An imploding spherical shock wave 423
§108. Shallow-water theory 428
CHAPTER XI. THE INTERSECTION OF SURFACES OF DISCONTINUITY 431
§109. Rarefaction waves 431
§110. Classification of intersections of surfaces of discontinuity 436
§111. The intersection of shock waves with a solid surface 442
§112. Supersonic flow round an angle 444
§113. Flow past a conical obstacle 449
CHAPTER XII. TWO-DIMENSIONAL GAS FLOW 452
§114. Potential flow of a gas 452
§115. Steady simple waves 455
§116. Chaplygin's equation: the general problem of steady two-dimensional gas flow 459
§117. Characteristics in steady two-dimensional flow 462
§118. The Euler–Tricomi equation. Transonic flow 464
§119. Solutions of the Euler–Tricomi equation near non-singular points of the sonic surface 469
§120. Flow at the velocity of sound 473
§121. The reflection of a weak discontinuity from the sonic line 478
CHAPTER XIII. FLOW PAST FINITE BODIES 484
§122. The formation of shock waves in supersonic flow past bodies 484
§123. Supersonic flow past a pointed body 487
§124. Subsonic flow past a thin wing 491
§125. Supersonic flow past a wing 493
§126. The law of transonic similarity 496
§127. The law of hypersonic similarity 498
CHAPTER XIV. FLUID DYNAMICS OF COMBUSTION 501
§128. Slow combustion 501
§129. Detonation 506
§130. The propagation of a detonation wave 511
§131. The relation between the different modes of combustion 517
§132. Condensation discontinuities 520
CHAPTER XV. RELATIVISTIC FLUID DYNAMICS 522
§133. The energy-momentum tensor 522
§134. The equations of relativistic fluid dynamics 523
§135. Shock waves in relativistic fluid dynamics 527
§136. Relativistic equations for flow with viscosity and thermal conduction 529
CHAPTER XVI. DYNAMICS OF SUPERFLUIDS 532
§137. Principal properties of superfluids 532
§138. The thermo-mechanical effect 534
§139. The equations of superfluid dynamics 535
§140. Dissipative processes in superfluids 540
§141. The propagation of sound in superfluids 543
Index 550

Textauszug

CHAPTER I

IDEAL FLUIDS

Publisher Summary

This chapter discusses the ideal fluids or fluids in motions in which thermal conductivity and the viscosity of the fluids are unimportant. Fluid dynamics concerns itself with the study of the motion of fluids, such as liquids and gases. As the phenomena considered in fluid dynamics are macroscopic, a fluid is regarded as a continuous medium. The mathematical description of the state of a moving fluid is effected by means of functions that give the distribution of the fluid velocity and of any two thermodynamic quantities pertaining to the fluid. Such quantities are, in general, functions of the coordinates and of time. The equations of fluid dynamics are simplified in the case of steady flow, which is one where the velocity is constant in time at any point occupied by fluid.

§1 The equation of continuity

Fluid dynamics concerns itself with the study of the motion of fluids (liquids and gases). Since the phenomena considered in fluid dynamics are macroscopic, a fluid is regarded as a continuous medium. This means that any small volume element in the fluid is always supposed so large that it still contains a very great number of molecules. Accordingly, when we speak of infinitely small elements of volume, we shall always mean those which are “physically” infinitely small, i.e. very small compared with the volume of the body under consideration, but large compared with the distances between the molecules. The expressions fluid particle and point in a fluid are to be understood in a similar sense. If, for example, we speak of the displacement of some fluid particle, we mean not the displacement of an individual molecule, but that of a volume element containing many molecules, though still regarded as a point.

The mathematical description of the state of a moving fluid is effected by means of functions which give the distribution of the fluid velocity v = v(x, y, z, t) and of any two thermodynamic quantities pertaining to the fluid, for instance the pressure p(x, y, z, t) and the density ρ(x, y, z, t). All the thermodynamic quantities are determined by the values of any two of them, together with the equation of state; hence, if we are given five quantities, namely the three components of the velocity v, the pressure p and the density ρ, the state of the moving fluid is completely determined.

All these quantities are, in general, functions of the coordinates x, y, z and of the time t. We emphasize that v(x, y, z, t) is the velocity of the fluid at a given point (x, y, z) in space and at a given time t, i.e. it refers to fixed points in space and not to specific particles of the fluid; in the course of time, the latter move about in space. The same remarks apply to ρ and p.

We shall now derive the fundamental equations of fluid dynamics. Let us begin with the equation which expresses the conservation of matter. We consider some volume V0 of space. The mass of fluid in this volume is , where ρ is the fluid density, and the integration is taken over the volume V0. The mass of fluid flowing in unit time through an element df of the surface bounding this volume is ρv · df; the magnitude of the vector df is equal to the area of the surface element, and its direction is along the normal. By convention, we take df along the outward normal. Then ρv · df is positive if the fluid is flowing out of the volume, and negative if the flow is into the volume. The total mass of fluid flowing out of the volume V0 in unit time is therefore

ρv⋅df,

where the integration is taken over the whole of the closed surface surrounding the volume in question.

Next, the decrease per unit time in the mass of fluid in the volume V0 can be written

∂∂t∫ρd V⋅

Equating the two expressions, we have

∂t∫ρd V=−∮ρv⋅df. (1.1)

(1.1)

The surface integral can be transformed by Green’s formula to a volume integral:

ρv⋅df=∫div(ρv)d V.

Thus

[∂ρ∂t+div(ρv)]d V=0.

Since this equation must hold for any volume, the integrand must vanish, i.e.

ρ/∂t+div(ρv)=0. (1.2)

(1.2)

This is the equation of continuity. Expanding the expression div (ρv), we can also write (1.2) as

ρ/∂t+ρ div v+v⋅ grad ρ=0. (1.3)

(1.3)

The vector

=ρv (1.4)

(1.4)

is called the mass flux density. Its direction is that of the motion of the fluid, while its magnitude equals the mass of fluid flowing in unit time through unit area perpendicular to the velocity.

§2 Euler’s equation

Let us consider some volume in the fluid. The total force acting on this volume is equal to the integral

∮p d f

of the pressure, taken over the surface bounding the volume. Transforming it to a volume integral, we have

∮pdf=−∫ grad p d V.

Hence we see that the fluid surrounding any volume element dV exerts on that element a force - d V grad p. In other words, we can say that a force - grad p acts on unit volume of the fluid.

We can now write down the equation of motion of a volume element in the fluid by equating the force - grad p to the product of the mass per unit volume (ρ) and the acceleration dv/dt:

d v/dt=− grad ρ. (2.1)

(2.1)

The derivative dv/dt which appears here denotes not the rate of change of the fluid velocity at a fixed point in space, but the rate of change of the velocity of a given fluid particle as it moves about in space. This derivative has to be expressed in terms of quantities referring to points fixed in space. To do so, we notice that the change dv in the velocity of the given fluid particle during the time dt is composed of two parts, namely the change during dt in the velocity at a point fixed in space, and the difference between the velocities (at the same instant) at two points dr apart, where dr is the distance moved by the given fluid particle during the time dt. The first part is (∂v/∂t)dt, where the derivative ∂v/∂t is taken for constant x, y, z, i.e. at the given point in space. The second part is

x∂v∂x+dy∂v∂y+dz∂v∂z=(dr⋅ grad )v.

Thus

 V=(∂v∂t)dt+(dr⋅ grad )v,

or, dividing both sides by dt,†

vdt=∂v∂t+(v⋅ grad )v. (2.2)

(2.2)

Substituting this in (2.1), we find

v∂t+(v⋅ grad )v=−ρ1 grad ρ. (2.3)

(2.3)

This is the required equation of motion of the fluid; it was first obtained by L. Euler in 1755. It is called Euler’s equation and is one of the fundamental equations of fluid dynamics.

If the fluid is in a gravitational field, an additional force ρg, where g is the acceleration due to gravity, acts on any unit volume. This force must be added to the right-hand side of equation (2.1), so that equation (2.3) takes the form

v∂t+(v⋅ grad )v=− grad ρρ+g. (2.4)

(2.4)

In deriving the equations of motion we have taken no account of processes of energy dissipation, which may occur in a moving fluid in consequence of internal friction (viscosity) in the fluid and heat exchange between different parts of it. The whole of the discussion in this and subsequent sections of this chapter therefore holds good only for motions of fluids in which thermal conductivity and viscosity are unimportant; such fluids are said to be ideal.

The absence of heat exchange between different parts of the fluid (and also, of course, between the fluid and bodies adjoining it) means that the motion is adiabatic throughout the fluid. Thus the motion of an ideal fluid must necessarily be supposed adiabatic.

In adiabatic motion the entropy of any particle of fluid remains constant as that particle moves about in space. Denoting by s the entropy per unit mass, we can express the condition for adiabatic...

Produkt-Details

• Erscheint lt. Verlag: 3.9.2013
• Sprache: englisch
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Strömungsmechanik
• Themenwelt: Technik ► Maschinenbau
• ISBN-10: 1-4831-6104-8 / 1483161048
• ISBN-13: 978-1-4831-6104-4 / 9781483161044