Fluid mechanics class notes

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Fluid Mechanics a short course for physicists Lyon - Moscow, 2010 Gregory Falkovichiii Preface Why study fluid mechanics? The primary reason is not even technical, it is cultural: a physicist is defined as one who looks around and under- stands at least part of the material world. One of the goals of this book is to let you understand how the wind blows and how the water flows so that swimming or flying you may appreciate what is actually going on. The secondary reason is to do with applications: whether you are to engage with astrophysics or biophysics theory or to build an appara- tus for condensed matter research, you need the ability to make correct fluid-mechanics estimates; some of the art for doing this will be taught in the book. Yet another reason is conceptual: mechanics is the basis of the whole of physics in terms of intuition and mathematical methods. Concepts introduced in the mechanics of particles were subsequently applied to optics, electromagnetism, quantum mechanics etc; here you will see the ideas and methods developed for the mechanics of fluids, which are used to analyze other systems with many degrees of freedom in statistical physics and quantum field theory. And last but not least: at present, fluid mechanics is one of the most actively developing fields of physics, mathematics and engineering so you may wish to participate in this exciting development. Even for physicists who are not using fluid mechanics in their work takingaone-semestercourseonthesubjectwouldbewellworththeiref- fort.Thisisonesuchcourse.Itpresumesnoprioracquaintancewiththe subject and requires only basic knowledge of vector calculus and analy- sis. On the other hand, applied mathematicians and engineers working on fluid mechanics may find in this book several new insights presented from a physicist’s perspective. In choosing from the enormous wealth of materialproducedbythelastfourcenturiesofever-acceleratingresearch, preference was given to the ideas and concepts that teach lessons whose importance transcends the confines of one specific subject as they prove useful time and again across the whole spectrum of modern physics. To much delight, it turned out to be possible to weave the subjects into a single coherent narrative so that the book is a novel rather than a collection of short stories.Contents 1 Basic equations and steady flows page 3 1.1 Definitions and basic equations 3 1.1.1 Definitions 3 1.1.2 Equations of motion for an ideal fluid 5 1.1.3 Hydrostatics 8 1.1.4 Isentropic motion 11 1.2 Conservation laws and potential flows 14 1.2.1 Kinematics 14 1.2.2 Kelvin’s theorem 15 1.2.3 Energy and momentum fluxes 17 1.2.4 Irrotational and incompressible flows 19 1.3 Flow past a body 24 1.3.1 Incompressible potential flow past a body 25 1.3.2 Moving sphere 26 1.3.3 Moving body of an arbitrary shape 27 1.3.4 Quasi-momentum and induced mass 29 1.4 Viscosity 34 1.4.1 Reversibility paradox 34 1.4.2 Viscous stress tensor 35 1.4.3 Navier-Stokes equation 37 1.4.4 Law of similarity 40 1.5 Stokes flow and wake 41 1.5.1 Slow motion 42 1.5.2 Boundary layer and separation phenomenon 45 1.5.3 Flow transformations 48 1.5.4 Drag and lift with a wake 49 Exercises 542 Contents Prologue ”The water’s language was a wondrous one, some narrative on a recurrent subject...” 1 A. Tarkovsky There are two protagonists in this story: inertia and friction. One meets them first in the mechanics of particles and solids where their interplay is not very complicated: inertia tries to keep the motion while friction tries to stop it. Going from a finite to an infinite number of degrees of freedom is always a game-changer. We will see in this book how an infinitesimal viscous friction makes fluid motion infinitely more complicated than inertia alone would ever manage to produce. Without friction, most incompressible flows would stay potential i.e. essentially trivial.Atsolidsurfaces,frictionproducesvorticitywhichiscarriedaway byinertiaandchangestheflowinthebulk.Instabilitiesthenbringabout turbulence, and statistics emerges from dynamics. Vorticity penetrating the bulk makes life interesting in ideal fluids though in a way different from superfluids and superconductors. On the other hand, compressibil- ity makes even potential flows non-trivial as it allows inertia to develop a finite-time singularity (shock), which friction manages to stop. On a formal level, inertia of a continuous medium is described by a nonlinear term in the equation of motion. Friction is described by a linear term which, however, have the highest spatial derivatives so that thelimitofzerofrictionissingular.Frictionisnotonlysingularbutalso asymmetry-breakingperturbation,whichleadstoananomalywhenthe effect of symmetry breaking remains finite even in the limit of vanishing viscosity. The first chapter introduces basic notions and describes stationary flows, inviscid and viscous. Time starts to run in the second chapter where we discuss instabilities, turbulence and sound. This is a short version (about one half), the full version is to be published by the Cam- bridge Academic Press.1 Basic equations and steady ows In this Chapter, we define the subject, derive the equations of motion and describe their fundamental symmetries. We start from hydrostatics where all forces are normal. We then try to consider flows this way as well, neglecting friction. That allows us to understand some features of inertia, most important induced mass, but the overall result is a failure to describe a fluid flow past a body. We then are forced to introduce friction and learn how it interacts with inertia producing real flows. We briefly describe an Aristotelean world where friction dominates. In an opposite limit we discover that the world with a little friction is very much different from the world with no friction at all. 1.1 De nitions and basic equations Continuous media. Absence of oblique stresses in equilibrium. Pressure and density as thermodynamic quantities. Continuous motion. Continu- ity equation and Euler’s equation. Boundary conditions. Entropy equa- tion.Isentropicflows.Steadyflows.Bernoulliequation.Limitingvelocity for the efflux into vacuum. Vena contracta. 1.1.1 Definitions We deal with continuous media where matter may be treated as homo- geneousinstructuredownto thesmallestportions.Term uidembraces both liquids and gases and relates to the fact that even though any fluid may resist deformations, that resistance cannot prevent deforma- tionfromhappening.Thereasonisthattheresistingforcevanisheswith the rate of deformation. Whether one treats the matter as a fluid or a4 Basic equations and steady ows solid may depend on the time available for observation. As prophetess Deborah sang, “The mountains flowed before the Lord” (Judges 5:5). The ratio of the relaxation time to the observation time is called the 1 Deborah number . The smaller the number the more fluid the material. Afluidcanbeinequilibriumonlyifallthemutualforcesbetweentwo adjacent parts are normal to the common surface. That experimental observation is the basis of Hydrostatics. If one applies a force parallel (tangential) to the common surface then the fluid layer on one side of the surface start sliding over the layer on the other side. Such sliding motion will lead to a friction between layers. For example, if you cease tostirteainaglassitcouldcometorestonlybecauseofsuchtangential forces i.e. friction. Indeed, if the mutual action between the portions on the same radius was wholly normal i.e. radial, then the conservation of the moment of momentum about the rotation axis would cause the fluid to rotate forever. Since tangential forces are absent at rest or for a uniform flow, it is naturaltoconsiderfirsttheflowswheresuchforcesaresmallandcanbe neglected. Therefore, a natural first step out of hydrostatics into hydro- dynamics is to restrict ourselves with a purely normal forces, assuming velocity gradients small (whether such step makes sense at all and how long such approximation may last is to be seen). Moreover, the intensity of a normal force per unit area does not depend on the direction in a fluid, the statement called the Pascal law (see Exercise 1.1). We thus characterize the internal force (or stress) in a fluid by a single scalar function p(r,t) called pressure which is the force per unit area. From the viewpoint of the internal state of the matter, pressure is a macro- scopic (thermodynamic) variable. To describe completely the internal state of the fluid, one needs the second thermodynamical quantity, e.g. the density ρ(r,t), in addition to the pressure. What analytic properties of the velocity field v(r,t) we need to pre- sume? We suppose the velocity to be finite and a continuous function of r. In addition, we suppose the first spatial derivatives to be everywhere finite. That makes the motion continuous, i.e. trajectories of the fluid particlesdonotcross.Theequationforthedistanceδrbetweentwoclose fluid particles is dδr/dt =δv so, mathematically speaking, finiteness of ∇v is Lipschitz condition for this equation to have a unique solution a simple example of non-unique solutions for non-Lipschitz equation is 1− 1= dx/dt=x with two solutions, x(t)=(αt) and x(t)=0 starting from zero forα0. For a continuous motion, any surface moving with the fluid completely separates matter on the two sides of it. We don’t1.1 De nitions and basic equations 5 yet know when exactly the continuity assumption is consistent with the equations of the fluid motion. Whether velocity derivatives may turn into infinity after a finite time is a subject of active research for an in- compressible viscous fluid (and a subject of the one-million-dollar Clay prize). We shall see below that a compressible inviscid flow generally develops discontinuities called shocks. 1.1.2 Equations of motion for an ideal fluid The Euler equation. The force acting on any fluid volume is equal to H thepressureintegraloverthesurface:− pdf.Thesurfaceareaelement df is a vector directed as outward normal: df H Let us transform the surface integral into the volume one:− pdf = ∫ − ∇pdV. The force acting on a unit volume is thus−∇p and it must be equal to the product of the mass ρ and the acceleration dv/dt. The latter is 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. One uses the chain rule differentiation to express this (substantial or material) derivative in terms of quantities referring to points fixed in space. During the timedt the fluid particle changes its velocity by dv which is composed of two parts, temporal and spatial: ∂v ∂v ∂v ∂v ∂v dv =dt +(dr·∇)v =dt +dx +dy +dz . (1.1) ∂t ∂t ∂x ∂y ∂z It is the change in the fixed point plus the difference at two points dr apart wheredr=vdt is the distance moved by the fluid particle during dt. Dividing (1.1) by dt we obtain the substantial derivative as local derivative plus convective derivative: dv ∂v = +(v·∇)v . dt ∂t Any function F(r(t),t) varies for a moving particle in the same way according to the chain rule differentiation: dF ∂F = +(v·∇)F . dt ∂t6 Basic equations and steady ows Writing now the second law of Newton for a unit mass of a fluid, we come to the equation derived by Euler (Berlin, 1757; Petersburg, 1759): ∂v ∇p +(v·∇)v =− . (1.2) ∂t ρ BeforeEuler,theaccelerationofafluidhadbeenconsideredasduetothe differenceofthepressureexertedbytheenclosingwalls.Eulerintroduced the pressure field inside the fluid. We see that even when the flow is steady, ∂v/∂t = 0, the acceleration is nonzero as long as (v·∇)v̸= 0, that is if the velocity field changes in space along itself. For example, for a steadily rotating fluid shown in Figure 1.1, the vector (v·∇)v 2 has a nonzero radial componentv /r. The radial acceleration times the 2 density must be given by the radial pressure gradient: dp/dr =ρv /r. v p p Figure 1.1 Pressure p is normal to circular surfaces and cannot change the moment of momentum of the fluid inside or outside the surface; the radial pressure gradient changes the direction of velocity v but does not change its modulus. We can also add an external body force per unit mass (for gravity f =g): ∂v ∇p +(v·∇)v =− +f . (1.3) ∂t ρ The term (v·∇)v describes inertia and makes the equation (1.3) non- linear. Continuity equation expresses conservation of mass. If Q is the vol- ume of a moving element then dρQ/dt=0 that is dρ dQ Q +ρ =0 . (1.4) dt dt The volume change can be expressed via v(r,t).1.1 De nitions and basic equations 7 δ y Q B A δ x The horizontal velocity of the point B relative to the point A is δx∂v /∂x. After the time interval dt, the length of the AB edge is x δx(1+dt∂v /∂x). Overall, after dt, one has the volume change x ( ) dQ ∂v ∂v ∂v x y z dQ=dt =δxδyδzdt + + =Qdtdivv . dt ∂x ∂y ∂z Substituting that into (1.4) and canceling (arbitrary) Q we obtain the continuity equation dρ ∂ρ ∂ρ +ρdivv = +(v·∇)ρ+ρdivv = +div(ρv)=0 . (1.5) dt ∂t ∂t The last equation is almost obvious since for any xed volume of space ∫ the decrease of the total mass inside,− (∂ρ/∂t)dV, is equal to the flux H ∫ ρv·df = div(ρv)dV. Entropyequation. Wehavenowfourequations(1.3,1.5)forfivequan- titiesp,ρ,v ,v ,v , so we need one extra equation. In deriving (1.3,1.5) x y z we have taken no account of energy dissipation neglecting thus internal friction (viscosity) and heat exchange. Fluid without viscosity and ther- mal conductivity is called ideal. The motion of an ideal fluid is adiabatic that is the entropy of any fluid particle remains constant: ds/dt = 0, where s is the entropy per unit mass. We can turn this equation into a continuity equation for the entropy density in space ∂(ρs) +div(ρsv)=0 . (1.6) ∂t Attheboundariesofthefluid,thecontinuityequation(1.5)isreplaced by the boundary conditions: 1) On a fixed boundary, v =0; n 2) On a moving boundary between two immiscible fluids, p =p and v =v . 1 2 n1 n2 Theseareparticularcasesofthegeneralsurfacecondition.LetF(r,t)=8 Basic equations and steady ows 0 be the equation of the bounding surface. Absence of any fluid flow across the surface requires dF ∂F = +(v·∇)F =0, dt ∂t which means, as we now know, the zero rate of F variation for a fluid particle. For a stationary boundary, ∂F/∂t=0 and v⊥∇F ⇒v =0. n Eulerian and Lagrangian descriptions. We thus encountered two alternative ways of description. The equations (1.3,1.6) use the coordi- natesystemfixedinspace,likefieldtheoriesdescribingelectromagnetism or gravity. That way of description is called Eulerian in fluid mechan- ics. Another approach is called Lagrangian, it is a generalization of the approach taken in particle mechanics. This way one follows fluid parti- 2 cles and treats their current coordinates, r(R,t), as functions of time andtheirinitialpositionsR=r(R,0).Thesubstantialderivativeisthus the Lagrangian derivative since it sticks to a given fluid particle, that is keeps R constant: d/dt = (∂/∂t) . Conservation laws written for a R unit-mass quantityA have a Lagrangian form: dA ∂A = +(v∇)A=0 . dt ∂t EveryLagrangianconservationlawtogetherwithmassconservationgen- erates an Eulerian conservation law for a unit-volume quantity ρA: ∂(ρA) ∂ρ ∂A +div(ρAv)=A +div(ρv) +ρ +(v∇)A =0 . ∂t ∂t ∂t On the contrary, if the Eulerian conservation law has the form ∂(ρB) +div(F)=0 ∂t and the flux is not equal to the density times velocity, F = ̸ ρBv, then the respective Lagrangian conservation law does not exist. That means thatfluidparticlescanexchangeB conservingthetotalspaceintegral— we shall see below that the conservation laws of energy and momentum have that form. 1.1.3 Hydrostatics A necessary and sufficient condition for fluid to be in a mechanical equi- librium follows from (1.3): ∇p=ρf . (1.7)1.1 De nitions and basic equations 9 Not any distribution ofρ(r) could be in equilibrium sinceρ(r)f(r) is not necessarily a gradient. If the force is potential, f = −∇ϕ, then taking curl of (1.7) we get ∇ρ×∇ϕ=0. That means that the gradients of ρ and ϕ are parallel and their level surfacescoincideinequilibrium.Thebest-knownexampleisgravitywith ϕ=gz and ∂p/∂z =−ρg. For an incompressible fluid, it gives p(z)=p(0)−ρgz . For an ideal gas under a homogeneous temperature, which has p = ρT/m, one gets dp pgm =− ⇒ p(z)=p(0)exp(−mgz/T) . dz T ◦ For air at 0 C, T/mg ≃ 8km. The Earth atmosphere is described by neitherlinearnorexponentiallawbecauseofaninhomogeneoustemper- ature. Assuming a linear temperature decay,T(z)=T −αz, one gets a 0 p isothermal (exponential) incompressible (linear) real atmosphere z Figure 1.2 Pressure-height dependence for an incompressible fluid (broken line), isothermal gas (dotted line) and the real atmosphere (solid line). better approximation: dp pmg =−ρg =− , dz T −αz 0 mg= p(z)=p(0)(1−αz/T ) , 0 ◦ which can be used not far from the surface with α≃6.5 /km. In a (locally) homogeneous gravity field, the density depends only on10 Basic equations and steady ows vertical coordinate in a mechanical equilibrium. According to dp/dz = −ρg, the pressure also depends only on z. Pressure and density deter- mine temperature, which then must also be independent of the horizon- tal coordinates. Different temperatures at the same height necessarily produce fluid motion, that is why winds blow in the atmosphere and currents flow in the ocean. Another source of atmospheric flows is ther- mal convection due to a negative vertical temperature gradient. Let us derive the stability criterium for a fluid with a vertical profile T(z). If a fluid element is shifted up adiabatically from z by dz, it keeps its en- ′ tropy s(z) but acquires the pressure p = p(z +dz) so its new density ′ is ρ(s,p). For stability, this density must exceed the density of the dis- placedairattheheightz+dz,whichhasthesamepressurebutdifferent ′ entropy s = s(z +dz). The condition for stability of the stratification is as follows: ( ) ∂ρ ds ′ ′ ′ ρ(p,s)ρ(p,s) ⇒ 0 . ∂s dz p Entropy usually increases under expansion, (∂ρ/∂s) 0, and for sta- p bility we must require ( ) ( ) ( ) ds ∂s dT ∂s dp c dT ∂V g p = + = − 0 . (1.8) dz ∂T dz ∂p dz T dz ∂T V p T p Here we used specific volume V = 1/ρ. For an ideal gas the coefficient of the thermal expansion is as follows: (∂V/∂T) =V/T and we end up p with dT g − . (1.9) dz c p 3 For the Earth atmosphere, c ∼ 10 J/kg·Kelvin, and the convection p ◦ ◦ threshold is 10 /km, not far from the average gradient 6.5 /km, so that 3 the atmosphere is often unstable with respect to thermal convection . 4 Human body always excites convection in a room-temperature air . The convection stability argument applied to an incompressible fluid rotating with the angular velocity Ω(r) gives the Rayleigh’s stability 2 2 criterium, d(r Ω) /dr0, which states that the angular momentum of 2 the fluid L =r Ω must increase with the distance r from the rotation 5 ′ axis .Indeed,ifafluidelementisshiftedfromr tor itkeepsitsangular ′ 2 ′ momentum L(r), so that the local pressure gradient dp/dr =ρr Ω (r ) ′ 2 4 ′4 must overcome the centrifugal force ρr (L r /r ).1.1 De nitions and basic equations 11 1.1.4 Isentropic motion Thesimplestmotioncorrespondstos=constandallowsforasubstantial simplification of the Euler equation. Indeed, it would be convenient to represent ∇p/ρ as a gradient of some function. For this end, we need a function which depends on p,s, so that at s =const its differential is expressed solely via dp. There exists the thermodynamic potential called enthalpy defined asW =E+pV per unit mass (E is the internal energy of the fluid). For our purposes, it is enough to remember from thermodynamics the single relation dE = Tds−pdV so that dW = Tds+Vdp one can also show thatW =∂(Eρ)/∂ρ). Sinces=const for −1 an isentropic motion andV =ρ for a unit mass thendW =dp/ρ and without body forces one has ∂v +(v·∇)v =−∇W . (1.10) ∂t Such a gradient form will be used extensively for obtaining conservation laws, integral relations etc. For example, representing 2 (v·∇)v =∇v /2−v×(∇×v), we get ∂v 2 =v×(∇×v)−∇(W +v /2) . (1.11) ∂t The first term in the right-hand side is perpendicular to the veloc- ity. To project (1.11) along the velocity and get rid of this term, we define streamlines as the lines whose tangent is everywhere parallel to the instantaneous velocity. The streamlines are then determined by the relations dx dy dz = = . v v v x y z Note that for time-dependent flows streamlines are different from par- ticle trajectories: tangents to streamlines give velocities at a given time while tangents to trajectories give velocities at subsequent times. One recordsstreamlinesexperimentallybyseedingfluidswithlight-scattering particles; each particle produces a short trace on a short-exposure pho- tograph, the length and orientation of the trace indicates the magnitude and direction of the velocity. Streamlines can intersect only at a point of zero velocity called stagnation point. Let us now consider a steady flow assuming ∂v/∂t = 0 and take the12 Basic equations and steady ows component of (1.11) along the velocity at a point: ∂ 2 (W +v /2)=0 . (1.12) ∂l 2 2 We see that W +v /2 = E +p/ρ+v /2 is constant along any given streamline, but may be different for different streamlines (Bernoulli, 1738). Why W rather than E enters the conservation law is discussed 2 after (1.16) below. In a gravity field, W +gz +v /2 =const. Let us consider several applications of this useful relation. Incompressible fluid. Under a constant temperature and a constant density and without external forces, the energy E is constant too. One can obtain, for instance, the limiting velocity with which such a liquid escapes from a large reservoir into vacuum: √ v = 2p /ρ . 0 3 −3 5 −2 For water (ρ = 10 kgm ) at atmospheric pressure (p = 10 Nm ) 0 √ one gets v = 200≈14m/s. Adiabatic gas flow. The adiabatic law, p/p = (ρ/ρ ) , gives the 0 0 enthalpy as follows: ∫ dp γp W = = . ρ (γ−1)ρ The limiting velocity for the escape into vacuum is √ 2γp 0 v = (γ−1)ρ √ thatis γ/(γ−1)timeslargerthanforanincompressiblefluid(because the internal energy of the gas decreases as it flows, thus increasing the kinetic energy). In particular, a meteorite-damaged spaceship looses the air from the cabin faster than the liquid fuel from the tank. We shall 2 see later that (∂P/∂ρ) = γP/ρ is the sound velocity squared, c , so s √ that v = c 2/(γ−1). For an ideal gas with n degrees of freedom, W =E+p/ρ=nT/2m+T/m so thatγ =(2+n)/n. For bi-atomic gas at not very high temperature, n=5.1.1 De nitions and basic equations 13 Efflux from a small orifice under the action of gravity. Supposing the external pressure to be the same at the horizonal surface and at the orifice, we apply the Bernoulli relation to the streamline which origi- nates at the upper surface with almost zero velocity and exits with the √ velocity v = 2gh (Torricelli, 1643). The Torricelli formula is not of much use practically to calculate the rate of discharge as the orifice area √ times 2gh (the fact known to wine merchants long before physicists). Indeed, streamlines converge from all sides towards the orifice so that thejetcontinuestoconvergeforawhileaftercomingout.Moreover,that convergingmotionmakesthepressureintheinteriorofthejetsomewhat greater that at the surface so that the velocity in the interior is some- √ whatlessthan 2gh.Theexperimentshowsthatcontractionceasesand p p Figure 1.3 Streamlines converge coming out of the orifice. the jet becomes cylindrical at a short distance beyond the orifice. That point is called “vena contracta” and the ratio of jet area there to the orifice area is called the coefficient of contraction. The estimate for the √ discharge rate is 2gh times the orifice area times the coefficient of con- traction. For a round hole in a thin wall, the coefficient of contraction is experimentally found to be 0.62. The Exercise 1.3 presents a particular case where the coefficient of contraction can be found exactly. Bernoulli relation is also used in different devices that measure the flow velocity. Probably, the simplest such device is the Pitot tube shown in Figure 1.4. It is open at both ends with the horizontal arm facing up- stream. Since the liquid does not move inside the tube than the velocity is zero at the point labelledB. On the one hand, the pressure difference at two pints on the same streamline can be expressed via the velocity at 2 A: P −P =ρv /2. On the other hand, it is expressed via the height B A h by which liquid rises above the surface in the vertical arm of the tube: 2 P −P =ρgh. That gives v =2gh. B A14 Basic equations and steady ows h v . . A B Figure 1.4 Pitot tube that determines the velocity v at the point A by measuring the height h. 1.2 Conservation laws and potential ows Kinematics: Strain and Rotation. Kelvin’s theorem of conservation of circulation. Energy and momentum fluxes. Irrotational flow as a poten- tial one. Incompressible fluid. Conditions of incompressibility. Potential flows in two dimensions. 1.2.1 Kinematics Therelativemotionnearapointisdeterminedbythevelocitydifference between neighbouring points: δv =r ∂v /∂x . i j i j It is convenient to analyze the tensor of the velocity derivatives by decomposing it into symmetric and antisymmetric parts: ∂v /∂x = i j S +A . The symmetric tensor S = (∂v /∂x +∂v /∂x )/2 is called ij ij ij i j j i strain, it can be always transformed into a diagonal form by an or- thogonal transformation (i.e. by the rotation of the axes). The diagonal componentsaretheratesofstretchingindifferentdirections.Indeed,the equation for the distance between two points along a principal direction has a form: r˙ = δv = r S (no summation over i). The solution is as i i i ii follows: ∫ t ′ ′ r (t)=r (0)exp S (t)dt . i i ii 0 For a permanent strain, the growth/decay is exponential in time. One recognizes that a purely straining motion converts a spherical material element into an ellipsoid with the principal diameters that grow (or1.2 Conservation laws and potential ows 15 decay) in time, the diameters do not rotate. Indeed, consider a circle of √ 2 2 the radius R at t = 0. The point that starts at x ,y = R −x goes 0 0 0 into S t 11 x(t)=e x , 0 √ √ S t S t S t 22 22 2 22 2 2 2 −2S t 11 y(t)=e y =e R −x =e R −x (t)e , 0 0 2 −2S t 2 −2S t 2 11 22 x (t)e +y (t)e =R . (1.13) The equation (1.13) describes how the initial fluid circle turns into the ellipse whose eccentricity increases exponentially with the rate S − 11 S . 22 Thesumofthestraindiagonalcomponentsisdivv =S whichdeter- ii −1 −1 minestherateofthevolumechange:Q dQ/dt=−ρ dρ/dt=divv = S . ii exp(S t) xx t exp(S t) yy Figure 1.5 Deformation of a fluid element by a permanent strain. The antisymmetric part A = (∂v /∂x −∂v /∂x )/2 has only three ij i j j i independent components so it could be represented via some vector ω: A = −ϵ ω /2. The coefficient −1/2 is introduced to simplify the ij ijk k relation between v and ω: ω =∇×v . The vector ω is called vorticity as it describes the rotation of the fluid element:δv =ω×r/2. It has a meaning of twice the effective local an- gular velocity of the fluid. Plane shearing motion likev (y) corresponds x to strain and vorticity being equal in magnitude. 1.2.2 Kelvin’s theorem That theorem describes the conservation of velocity circulation for isen- tropic flows. For a rotating cylinder of a fluid, the momentum of mo- mentum is proportional to the velocity circulation around the cylinder circumference. The momentum of momentum and circulation are both conserved when there are only normal forces, as was already mentioned16 Basic equations and steady ows strain shear t shear vorticity Figure 1.6 Deformation and rotation of a fluid element in a shear flow. Shearing motion is decomposed into a straining motion and rotation. at the beginning of Sect. 1.1.1. Let us show that this is also true for every ”fluid” contour which is made of fluid particles. As fluid moves, both the velocity and the contour shape change: I I I d v·dl= v(dl/dt)+ (dv/dt)·dl=0 . dt The first term here disappears because it is a contour integral of the H H 2 complete differential: since dl/dt = δv then v(dl/dt) = δ(v /2) = 0. In the second term we substitute the Euler equation for isentropic motion, dv/dt = −∇W, and use the Stokes formula which tells that the circulation of a vector around the closed contour is equal to the flux H of the curl through any surface bounded by the contour: ∇W ·dl = ∫ ∇×∇Wdf =0. H ∫ Stokesformulaalsotellsusthat vdl= ω·df.Therefore,theconser- vation of the velocity circulation means the conservation of the vorticity flux.Tobetterappreciatethis,consideranalternativederivation.Taking curl of (1.11) we get ∂ω =∇×(v×ω) . (1.14) ∂t This is the same equation that describes the magnetic field in a perfect conductor: substituting the condition for the absence of the electric field in the frame moving with the velocity v, cE +v×H = 0, into the Maxwellequation∂H/∂t=−c∇×E,onegets∂H/∂t=∇×(v×H).The magnetic flux is conserved in a perfect conductor and so is the vorticity flux in an isentropic flow. One can visualize vector field introducing field lines which give the direction of the field at any point while their density is proportional to the magnitude of the field. Kelvin’s theorem means that vortex lines move with material elements in an inviscid fluid exactly like magnetic lines are frozen into a perfect conductor. One way to prove that is to show that ω/ρ (and H/ρ) satisfy the same equation1.2 Conservation laws and potential ows 17 as the distance r between two fluid particles: dr/dt = (r·∇)v. This is done using dρ/dt=−ρdivv and applying the general relation ∇×(A×B)=A(∇·B)−B(∇·A)+(B·∇)A−(A·∇)B (1.15) to∇×(v×ω)=(ω·∇)v−(v·∇)ω−ωdivv. We then obtain d ω 1dω ω dρ 1 ∂ω divv = − = +(v·∇)ω + 2 dtρ ρ dt ρ dt ρ ∂t ρ ( ) 1 divv ω = (ω·∇)v−(v·∇)ω−ωdivv+(v·∇)ω+ = ·∇ v . ρ ρ ρ Sincer andω/ρ move together, then any two close fluid particles chosen on the vorticity line always stay on it. Consequently any fluid particle stays on the same vorticity line so that any fluid contour never crosses vorticity lines and the flux is indeed conserved. 1.2.3 Energy and momentum fluxes Let us now derive the equation that expresses the conservation law of 2 energy.Theenergydensity(perunitvolume)intheflowisρ(E+v /2). For isentropic flows, one can use ∂ρE/∂ρ = W and calculate the time derivative ( ) 2 ( ) ∂ ρv ∂ρ ∂v 2 2 ρE+ = W +v /2 +ρv· =−divρv(W +v /2) . ∂t 2 ∂t ∂t Since the right-hand side is a total derivative then the integral of the energy density over the whole space is conserved. The same Eulerian conservation law in the form of a continuity equation can be obtained in a general (non-isentropic) case as well. It is straightforward to calculate the time derivative of the kinetic energy: 2 2 ∂ ρv v =− divρv−v·∇p−ρv·(v∇)v ∂t 2 2 2 v 2 =− divρv−v(ρ∇W−ρT∇s)−ρv·∇v /2 . 2 −2 For calculating ∂(ρE)/∂t we use dE = Tds−pdV = Tds+pρ dρ so that d(ρE)=Edρ+ρdE =Wdρ+ρTds and ∂(ρE) ∂ρ ∂s =W +ρT =−Wdivρv−ρTv·∇s . ∂t ∂t ∂t Adding everything together one gets ( ) 2 ∂ ρv 2 ρE+ =−divρv(W +v /2) . (1.16) ∂t 218 Basic equations and steady ows As usual, the rhs is the divergence of the flux, indeed: ∫ ( ) I 2 ∂ ρv 2 ρE+ dV =− ρ(W +v /2)v·df . ∂t 2 Note the remarkable fact that the energy flux is 2 2 ρv(W +v /2)=ρv(E+v /2)+pv which is not equal to the energy density times v but contains an extra pressure term which describes the work done by pressure forces on the fluid. In other terms, any unit mass of the fluid carries an amount of 2 2 energyW+v /2ratherthanE+v /2.Thatmeans,inparticular,thatfor energythereisno(Lagrangian)conservationlawforunitmassd(·)/dt= 0 that is valid for passively transported quantities like entropy. This is natural because different fluid elements exchange energy by doing work. Momentum is also exchanged between different parts of fluid so that theconservationlawmusthavetheformofacontinuityequationwritten for the momentum density. The momentum of the unit volume is the vector ρv whose every component is conserved so it should satisfy the equation of the form ∂ρv ∂Π i ik + =0 . ∂t ∂x k Let us find the momentum flux Π — the flux of the i-th component ik of the momentum across the surface with the normal along k. Substi- tute the mass continuity equation ∂ρ/∂t =−∂(ρv )/∂x and the Euler k k −1 equation ∂v /∂t=−v ∂v /∂x −ρ ∂p/∂x into i k i k i ∂ρv ∂v ∂ρ ∂p ∂ i i =ρ +v =− − ρv v , i i k ∂t ∂t ∂t ∂x ∂x i k that is Π =pδ +ρv v . (1.17) ik ik i k Plainly speaking, along v there is only the flux of parallel momentum 2 p+ρv while perpendicular to v the momentum component is zero at the given point and the flux is p. For example, if we direct the x-axis 2 along velocity at a given point then Π =p+v , Π = Π =p and xx yy zz all the off-diagonal components are zero. We have finished the formulations of the equations and their general properties and will discuss now the simplest case which allows for an analytic study. This involves several assumptions.

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