Landau / Lifshitz Fluid Mechanics

Landau and Lifshitz: Course of Theoretical Physics, Volume 6
2. Auflage 2013
ISBN: 978-1-4831-6104-4
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 554 Seiten

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.

Autoren/Hrsg.

Landau, L D

Lifshitz, E. M.

Inhaltsverzeichnis

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

Leseproben

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

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 and 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() and of any two thermodynamic quantities pertaining to the fluid, for instance the pressure () and the density ?(). 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 and the density ?, the state of the moving fluid is completely determined.

All these quantities are, in general, functions of the coordinates and of the time We emphasize that v() is the velocity of the fluid at a given point () in space and at a given time , 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 ? and

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 0 of space. The mass of fluid in this volume is , where ? is the fluid density, and the integration is taken over the volume 0. The mass of fluid flowing in unit time through an element df of the surface bounding this volume is ?v · 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 ?v · 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 0 in unit time is therefore

?v·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 0 can be written

??t??d V·

Equating the two expressions, we have

?t??d V=-??v·df. (1.1)

(1.1)

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

?v·df=?div(?v)d V.

Thus

[???t+div(?v)]d V=0.

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

?/?t+div(?v)=0. (1.2)

(1.2)

This is the Expanding the expression div (?v), we can also write (1.2) as

?/?t+? div v+v· grad ?=0. (1.3)

(1.3)

The vector

=?v (1.4)

(1.4)

is called the 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 d exerts on that element a force - d grad In other words, we can say that a force - grad 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 to the product of the mass per unit volume (?) and the acceleration dv/d:

d v/dt=- grad ?. (2.1)

(2.1)

The derivative dv/d 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 d is composed of two parts, namely the change during d 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 d. The first part is (?v/?)d, where the derivative ?v/? is taken for constant , 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 d,†

vdt=?v?t+(v· grad )v. (2.2)

(2.2)

Substituting this in (2.1), we find

v?t+(v· grad )v=-?1 grad ?. (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 and is one of the fundamental equations of fluid dynamics.

If the fluid is in a gravitational field, an additional force ?g, 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 ??+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

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 the entropy per unit mass, we can express the condition for adiabatic...