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e a ct E Landau A E 7 Lifshitz E: º Mi | Fluid Mechanics | Third Revised English Edition j Course of Theoretical Physics | Tn Volume 6 [= O D 2) o pod) o, A A ergamon COURSE OF THEORETICAL PHYSICS Vol. 1 Mechanios Vol, 2 The Classical Theory of Fields Vol. 3 Quantum Mechanics —Non-Relativistic Theory Vol. 4 Relativistir Quantum Theory Vol. 5 Statistical Physics Vol. 7 Theory of Elasticity Vol. 8 Electrodymamies of Continuous Media Vol. 9 Physical Kinetics FLUID MECHANICS by L, D. LANDAU anp E. M. LIFSHITZ INSTITUTE OF PHYSICAL PROBLEMS, U.S.5.R, ACADEMY OF SCIENCES Volume 6 of Course of Theoretical Physics Translated from the Russian by J. B. SYKES anp W. H. REID PERGAMON PRESS Oxford - London - Edinburgh - New York Toronto - Sydney - Paris - Braunschweig 226. 529. CONTENTS Preface ta the English edition Notation I IDEAL FLUIDS The equation of continuity Euler's equation Hydrostatics The condition that convection is absent Bernoulli's equation The energy flux The momentum flux The conservation of circulation Potential flow Incompressible fluids The drag force in potential fow past a body Gravity waves Long gravity waves Waves in an incompressible fluid II. VISCOUS FLUIDS “The equations of motion of a viscous fluid Energy dissipation in an incompressible fuid Flow in a pipe Flow between rotating cylinders The law of similarity Stokes” formula The laminar wake The viscosity of suspensions Exact solutions of the equations of motion for a viscous fluid Oscillatory motion in a viscous fluid Damping of gravity waves It. TURBULENCE Stability of steady flow “The onset of turbulence Stability of flow between rotating cylinders Stability of flow in a pipe v Page xdi 102 103 107 ni vi 830. $31. 32. 333. 34. $3s. 836. 37. 538. 557. 558. $59. 560. S61. 562. Contents Instability of tangential discontinuities Fully developed turbulence Local turbulence The velocity correlation The turbulent region and the phenomenon of separation The turbulent jet “The turbulent wake Zhukovskii's theorem Isotropie turbulence IV. BOUNDARY LAYERS “The laminar boundary layer Flow near the line of separation Stability of flow in the laminar boundary layer The logarithmic velocity profile “Turbulent fow in pipes The turbulent boundary layer The drag crisis Flow past streamlined bodies Induced drag The lift of a thin wing V. THERMAL CONDUCTION IN FLUIDS “The general equation of heat transfer Thermal conduction in an incompressible fluid Thermal conduction in an infinite medium 'Thermal conduction in a finite medium The similarity law for heat transfer Heat transfer in a boundary layer Heating of a body in a moving fluid Free convection VI. DIFFUSION The equations of fluid dynamics for a mixture of fluids Coefficients of mass transfer and thermal diffusion Diffusion of particles suspended in a fluid VII SURFACE PHENOMENA Laplace's formula Capillary waves The effect of adsorbed films on the motion of a liquid Page 114 116 120 123 128 130 136 137 140 145 151 156 159 166 168 172 175 179 219 222 227 230 237 241 vii XI. NUR stoz. g103. SI04. S105, 8106. $107, s108. 8109. g110. SHI guia. gi13. gua. sits. gH6. su7. gs. g119. s120. ga. g122. g123. EIZA s125. $126. g127. Contents THE INTERSECTION OF SURFACES OF DISCONTINUITY Rarefaction waves 'The intersection of shock waves The intersection of shock waves with a solid surface Supersonic fow round an angle Flow past a conical obstacle XII, TWO-DIMENSIONAL GAS FLOW Potential flow of a gas Steady simple waves Chaplygin's equation: the general problem of steady two- dimensional gas fow Characteristics in steady two-dimensional flow The Euler-Tricomi equation. Transonic flow Solutions of the Euler-Tricomi equation near non-singular points of the sonic surface Flow at the velocity of sound “The intersection of discontinuities with the transition line XIII. FLOW PAST FINITE BODIES *The formation of shock waves in supersonic flow past bodies Supersonic fow past a pointed body Subsonic flow past a thin wing Supersonic flow past a wing The law of transonic similarity The law of hypersonic similarity XIV. FLUID DYNAMICS OF COMBUSTION Slow combustion Detonation The propagation of a detonation wave The relation between the different modes of combustion Condensation discontinuities Xv. RELATIVISTIC FLUID DYNAMICS The energy-momentum tensor 'The equations of relativistic fluid dynamics Relativistic equations for dissipative processes Page 399 405 410 413 418 422. 425 430 433 436 441 451 457 460 464 466 469 472 a74 480 487 496 499 505 Contents ix XVI. DYNAMICS OF SUPERFLUIDS Page 3128. Principal properties of superfluids 507 $129. “The thermo-mechanical effect 509 $130. The equations of superfiuid dynamics 510 $131. “The propagation of sound in a superfluid 517 XVI. FLUCTUATIONS IN FLUID DYNAMICS $132. “The general theory of fluctuations in fluid dynamics 523 $133. Fluctuations in an infinite medium 526 Index 530 e 8aunso Box aes NOTATION density pressure temperature entropy per unit mass internal energy per unit mass <+2p/p heat function (enthalpy) pico ratio of specific heats at constant pressure and constant volume dynamic viscosity np kinematic viscosity thermal conductivity «/pcp thermometric conductivity Reynolds number velocity of sound ratio of fluid velocity to velocity of sound CHAPTER 1 IDEAL FLUIDS $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 particie 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 mole- cule, 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,y2,t) and of any two thermodynamic quantities pertaining to the fluid, for instance the pressure (x, y, 2, £) and the density p(x, y, 2; £). 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 p and the density p, the state of the moving fluid is completely determined. All these quantities are, in general, functions of the co-ordinates x, y, 2 and of the time t. We emphasise that v(x, y, 2, t) is the velocity of the fluid at a given point (x, y, 2) 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 p 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 Vy of space. The mass of fluid in this volume is Jp dV, where p is the fluid density, and the integration is taken over the volume Vo. The mass of fluid flowing in unit time through an element df of the surface bounding this volume is pv » 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 pv » df is positive if the fluid is flowing out of the volume, and negative if the flow 1 EV Euler's equation 3 of the pressure, taken over the surface bounding the volume. Transforming it to a volume integral, we have - pat = — [grado dr. Hence we see that the fluid surrounding any volume element dV exerts on that element a force —dV gradp. 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 Auid by equating the force — grad p to the product of the mass per unit volume (p) and the acceleration dv/dt: pdvidt = —gradp. (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 df 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 di. "The first part is (ov/ot)dt, where the derivative Bv/ôt is taken for constant x,y, %, i.e. at the given point in space. The second part is ev ow ev da— + dy—+ de— = (de-grad)v. àx ay [a Thus dy = (ov/at)dt+(dr-grad)v, or, dividing both sides by dz, dv J êv a 22) qa grad)v. (2. Substituting this in (2.1), we find ow 1 x + tvesrad)y = —-gradp. (2.3) p 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 funda- mental equations of fluid dynamics. Tf the fluid is in a gravitationa! field, an additional force pg, where g is the acceleration due to gravity, acts on any unit volume. This force 4 Ideal Fluids s2 must be added to the right-hand side of equation (2.1), so that equation (2.3) takes the form dy gradp % +(vgrad)v= ->—— tg. (2.4) p 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 motion as dsjdt = 0, (2.5) where the total derivative with respect to time denotes, as in (2.1), the rate of change of entropy for a given fluid particle as it moves about. This condition can aiso be written êsfor+v-grads = 0, (2.6) This is the general equation describing adiabatic motion of an ideal fluid. Using (1.2), we can write it as an “equation of continuity” for entropy: dps)/ot-+ div (psv) = 0. (2.7) The product psv is the “entropy flux density”. Tt must be borne in mind that the adiabatic equation usually takes a much simpler form. If, as usually happens, the entropy is constant throughout the volume of the fluid at some initial instant, it retains everywhere the same constant value at all times and for any subsequent motion of the fluid. In this case we can write the adiabatic equation simply as s= constant, (2.8) and we shall usually do so in what follows. Such a motion is said to be isentropic. We may use the fact that the motion is isentropic to put the equation of motion (2.3) in a somewhat different form. To do so, we employ the familiar thermodynamic relation dy = Tds+Vdp, where w is the heat function per unit mass of fluid (enthalpy), W = 1/p 6 Ideal Fluids 3 SoLuriox. In these variables the co-ordinato 4 of any fluid particle at any instant is re- garded as a function of £ and its co-ordinate a at the initial instant: x = a(a, 4). The condition of conservation of mass during the motion of a fluid element (the equation of continuity) is accordingly written p dx: = py da, or 2x elx) = Py, da Ji where po(a) is a given initial density distribution. The velocity of a fluid particle is, by definition, v = (0x/60)a, and the derivative (20/8t)a gives the rate of change of the velocity of the particle during its motion. Euler's equation becomes (5) (aa): (2s/ô8)a = 0, and the adiabatic equation is $3. Hydrostatics For a fluid at rest in a uniform gravitational field, Euler's equation (2.4) takes the form grado = pg. (41) “This equation describes the mechanical equilibrium of the fluid. (If there is no external force, the equation of equilibrium is simply grad 2 = 0, Le. p = constant; the pressure is the same at every point in the fluid.) Equation (3.1) can be integrated immediately if the density of the fluid may be supposed constant throughout its volume, i.e. if there is no signi- ficant compression of the fluid under the action of the external force. Taking the z-axis vertically upward, we have ôpjôx =pjoy =0, Opjêz= —pg. Henece b= —pgz+constant. Jthe fluid at rest has a free surface at height À, to which an external pressure bo, the same at every point, is applied, this surface must be the horizontal plane z = k. From the condition p = po for z = h, we find that the constant is £o+ pgh, so that p= potemth-a). (82) For large masses of liquid, and for a gas, the density p cannot in general be supposed constant; this applies especially to gases (for example, the atmosphere). Let us suppose that the fluid is not only in mechanical equilibrium but also in thermal equilibrium. Then the temperature is the 8 Hydrostatics 7 same at every point, and equation (3.1) may be integrated as follows. We use the familiar thermodynamic relation db = -sdT+Vdp, where is the thermodynamic potential per unit mass. For constant tem- perature do = Vdp = dpfp Hence we see that the expression (grad )/p can be written in this case as grad db, so that the equation of equilibrium (3.1) takes the form gradO = g. For a constant vector g directed along the negative s-axis we have g= —grad(g). Thus grad(D+g2) = 0, whence we find that throughout the fluid P+gz = constant; (3.3) gz is the potential energy of unit mass of fluid in the gravitational field. The condition (3.3) is known from statistical physics to be the condition for thermodynamic equilibrium of a system in an external field. We may mention here another simple consequence of equation (3.1). K a fluid (such as the atmosphere) is in mechanical equilibrium in a gravi- tational field, the pressure in it can be a function only of the altitude x (since, if the pressure were different at different points with the same alti- tude, motion would result). It then follows from (3.1) that the density 1dp ed (3.4) is also a function of z only. The pressure and density together determine the temperature, which is therefore again a function of z only. Thus, in mechanical equilibrium in a gravitational field, the pressure, density and temperature distributions depend only on the altitude. If, for example, the temperature is different at different points with the same altitude, then mechanical equilibrium is impossible. Finally, let us derive the equation of equilibrium for a very large mass of fluid, whose separate parts are held together by gravitational attraction— a star. Let & be the Newtonian gravitational potential of the field due to the fluid. Tt satisfies the differential equation Ad = 47Gp, (3.5) 
