Landau / Lifshitz Fluid Mechanics

Volume 6
1. Auflage 2013
ISBN: 978-1-4831-4050-6
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 558 Seiten

Course of Theoretical Physics, Volume 6: Fluid Mechanics discusses several areas of concerns regarding fluid mechanics. The book provides a discussion on the phenomenon in fluid mechanics and their intercorrelations, such as heat transfer, diffusion in fluids, acoustics, theory of combustion, dynamics of superfluids, and relativistic fluid dynamics. The text will be of great interest to researchers whose work involves or concerns fluid mechanics.

Autoren/Hrsg.

Landau, L D

Lifshitz, E. M.

Inhaltsverzeichnis

1;Front Cover;1
2;Fluid Mechanics;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface to the English edition;12
6;Notation;13
7;CHAPTER I. IDEAL FLUIDS;14
7.1;§1. The equation of continuity;14
7.2;§2. Euler's equation;15
7.3;§3. Hydrostatics;19
7.4;§4. The condition that convection is absent;21
7.5;§5. Bernoulli's equation;22
7.6;§6. The energy flux;23
7.7;§7. The momentum flux;25
7.8;§8 The conservation of circulation;27
7.9;§9. Potential flow;29
7.10;§10. Incompressible fluids;33
7.11;§11. The drag force in potential flow past a body;44
7.12;§12. Gravity waves;49
7.13;§13. Long gravity waves;55
7.14;§14. Waves in an incompressible fluid;57
8;CHAPTER II. VISCOUS FLUIDS;60
8.1;§15. The equations of motion of a viscous fluid;60
8.2;§16. Energy dissipation in an incompressible fluid;66
8.3;§17. Flow in a pipe;68
8.4;§18. Flow between rotating cylinders;73
8.5;§19. The law of similarity;74
8.6;§20. Stokes' formula;76
8.7;§21. The laminar wake;84
8.8;§22. The viscosity of suspensions;89
8.9;§23. Exact solutions of the equations of motion for a viscous fluid;92
8.10;§24. Oscillatory motion in a viscous fluid;101
8.11;§25. Damping of gravity waves;111
9;CHAPTER III. TURBULENCE;115
9.1;§26. Stability of steady flow;115
9.2;§27. The onset of turbulence;116
9.3;§28. Stability of flow between rotating cylinders;120
9.4;§29. Stability of flow in a pipe;124
9.5;§30. Instability of tangential discontinuities;127
9.6;§31. Fully developed turbulence;129
9.7;§32. Local turbulence;133
9.8;§33. The velocity correlation;136
9.9;§34. The turbulent region and the phenomenon of separation;141
9.10;§35. The turbulent jet;143
9.11;§36. The turbulent wake;149
9.12;§37. Zhukovskii's theorem;150
9.13;§38. Isotropie turbulence;153
10;CHAPTER IV. BOUNDARY LAYERS;158
10.1;§39. The laminar boundary layer;158
10.2;§40. Flow near the line of separation;164
10.3;§41. Stability of flow in the laminar boundary layer;169
10.4;§42. The logarithmic velocity profile;172
10.5;§43. Turbulent flow in pipes;176
10.6;§44. The turbulent boundary layer;179
10.7;§45. The drag crisis;181
10.8;§46. Flow past streamlined bodies;185
10.9;§47. Induced drag;188
10.10;§48. The lift of a thin wing;192
11;CHAPTER V. THERMAL CONDUCTION IN FLUIDS;196
11.1;§49. The general equation of heat transfer;196
11.2;§50. Thermal conduction in an incompressible fluid;201
11.3;§51. Thermal conduction in an infinite medium;205
11.4;§52. Thermal conduction in a finite medium;209
11.5;§53. The similarity law for heat transfer;215
11.6;§54. Heat transfer in a boundary layer;218
11.7;§55. Heating of a body in a moving fluid;222
11.8;§56. Free convection;225
12;CHAPTER VI. DIFFUSION;232
12.1;§57. The equations of fluid dynamics for a mixture of fluids;232
12.2;§58. Coefficients of mass transfer and thermal diffusion;235
12.3;§59. Diffusion of particles suspended in a fluid;240
13;CHAPTER VII. SURFACE PHENOMENA;243
13.1;§60. Laplace's formula;243
13.2;§61. Capillary waves;250
13.3;§62. The effect of adsorbed films on the motion of a liquid;254
14;CHAPTER VIII. SOUND;258
14.1;§63. Sound waves;258
14.2;§64. The energy and momentum of sound waves;262
14.3;§65. Reflection and refraction of sound waves;266
14.4;§66. Geometrical acoustics;269
14.5;§67. Propagation of sound in a moving medium;272
14.6;§68. Characteristic vibrations;275
14.7;§69. Spherical waves;278
14.8;§70. Cylindrical waves;281
14.9;§71. The general solution of the wave equation;283
14.10;§72. The lateral wave;286
14.11;§73. The emission of sound;292
14.12;§74. The reciprocity principle;301
14.13;§75. Propagation of sound in a tube;304
14.14;§76. Scattering of sound;307
14.15;§77. Absorption of sound;311
14.16;§78. Second viscosity;317
15;CHAPTER IX. SHOCK WAVES;323
15.1;§79. Propagation of disturbances in a moving gas;323
15.2;§80. Steady flow of a gas;326
15.3;§81. Surfaces of discontinuity;330
15.4;§82. The shock adiabatic ;332
15.5;§83. Weak shock waves;335
15.6;§84. The direction of variation of quantities in a shock wave;338
15.7;§85. Shock waves in a perfect gas;342
15.8;§86. Oblique shock waves;346
15.9;§87. The thickness of shock waves;350
15.10;§88. The isothermal discontinuity;355
15.11;§89. Weak discontinuities;357
16;CHAPTER X. ONE-DIMENSIONAL GAS FLOW;360
16.1;§90. Flow of gas through a nozzle;360
16.2;§91. Flow of a viscous gas in a pipe;363
16.3;§92. One-dimensional similarity flow;366
16.4;§93. Discontinuities in the initial conditions;373
16.5;§94. One-dimensional travelling waves;379
16.6;§95. Formation of discontinuities in a sound wave;385
16.7;§96. Characteristics;391
16.8;§97. Riemann invariants;394
16.9;§98. Arbitrary one-dimensional gas flow;399
16.10;§99. The propagation of strong shock waves;405
16.11;§100. Shallow-water theory;409
17;CHAPTER XI. THE INTERSECTION OF SURFACES OF DISCONTINUITY;412
17.1;§101. Rarefaction waves;412
17.2;§102. The intersection of shock waves;418
17.3;§103. The intersection of shock waves with a solid surface;423
17.4;§104. Supersonic flow round an angle;426
17.5;§105. Flow past a conical obstacle;431
18;CHAPTER XII. TWO-DIMENSIONAL GAS FLOW;435
18.1;§106. Potential flow of a gas;435
18.2;§107. Steady simple waves;438
18.3;§108. Chaplygin's equation: the general problem of steady two-dimensional gas flow;443
18.4;§109. Characteristics in steady two-dimensional flow;446
18.5;§110. The Euler–Tricomi equation Transonic flow;449
18.6;§111. Solutions of the Euler–Tricomi equation near non-singular points of the sonic surface;454
18.7;§112. Flow at the velocity of sound;459
18.8;§113. The intersection of discontinuities with the transition line;464
19;CHAPTER XIII. FLOW PAST FINITE BODIES;470
19.1;§114. The formation of shock waves in supersonic flow past bodies;470
19.2;§115. Supersonic flow past a pointed body;473
19.3;§116. Subsonic flow past a thin wing;477
19.4;§117. Supersonic flow past a wing;479
19.5;§118. The law of transonic similarity;482
19.6;§119. The law of hypersonic similarity;485
20;CHAPTER XIV. FLUID DYNAMICS OF COMBUSTION;487
20.1;§120. Slow combustion;487
20.2;§121. Detonation;493
20.3;§122. The propagation of a detonation wave;499
20.4;§123. The relation between the different modes of combustion;506
20.5;§124. Condensation discontinuities;509
21;CHAPTER XV. RELATIVISTIC FLUID DYNAMICS;512
21.1;§125. The energy-momentum tensor;512
21.2;§126. The equations of relativistic fluid dynamics;513
21.3;§127. Relativistic equations for dissipative processes;518
22;CHAPTER XVI. DYNAMICS OF SUPERFLUIDS;520
22.1;§128. Principal properties of superfluids;520
22.2;§129. The thermo-mechanical effect;522
22.3;§130. The equations of superfluid dynamics;523
22.4;§131. The propagation of sound in a superfluid;530
23;CHAPTER XVII. FLUCTUATIONS IN FLUID DYNAMICS;536
23.1;§132. The general theory of fluctuations in fluid dynamics;536
23.2;§133. Fluctuations in an infinite medium;539
24;INDEX;543

Leseproben

CHAPTER I

IDEAL FLUIDS

Publisher Summary

This chapter discusses the study of motion of fluids. Because 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. The mathematical description of the state of a moving fluid is affected by means of functions that give the distribution of the fluid velocity and of any two thermodynamic quantities pertaining to the fluid. All the thermodynamic quantities are determined by the values of any two of them, together with the equation of state. Therefore, if the three components of the velocity, the pressure, and the density are given, the state of the moving fluid is completely determined. At a boundary between two immiscible fluids, the condition is that the pressure and the velocity component normal to the surface of separation must be the same for the two fluids, and each of these velocity components must be equal to the corresponding component of the velocity of the surface. The properties of the flow in this boundary layer decide the choice of one out of the infinity of solutions of the equations of motion for an ideal 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 ?(). As is well known, 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 co-ordinates and of the time We emphasise 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 fixed 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 ? ? d, 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 d f 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??dV.

Equating the two expressions, we have

??t??dV=-??v·df. (1.1)

(1.1)

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

? v·df=?div (?v)dV.

Thus

[???t+div (?v)]dV=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 df

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

?pdf=-?grad  p dv.

Hence we see that the fluid surrounding any volume element d exerts on that element a force –dgrad . 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 :

dv/dt=-grad p. (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 d r is the distance moved by the given fluid particle during the time The first part is (?v/?t)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/?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 p. (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 forcemust 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 p?+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...