12 Towards hydrodynamic equations J Nonlinear Dynamics II: Continuum Systems Lecture 12 Spring 2015

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1 18.354J Nonlinear Dynaics II: Continuu Systes Lecture 12 Spring Towards hydrodynaic equations The previous classes focussed on the continuu description of static (tie-independent) elastic systes. We becoe ore abitious now and look to derive the hydrodynaic equations for a syste of particles obeying Newton s laws. There are two ways we can go about this: by going fro the icroscopic to the acroscopic scale, or by adopting a continuu approxiation and deriving the acro-equations fro general considerations. The general principle underlying this subject is that the acroscopic variables are quantities that are icroscopically conserved. The reason for this is siply that the entire notion of a acroscopic equation relies on scale separation with the icroscopic scale. Any quantity which is not conserved icroscopically necessarily varies on a acroscopic scale. The only quantities that are candidate hydrodynaic variables are therefore those which are 56

2 conserved. This siple arguent shows that what we really need to do is figure out how to generalise our rando walkers to soething that conserves oentu and energy. To rigorously justify the acroscopic equations of otion for a fluid (i.e., a collection of particles interacting with each other by Newton s laws) it is necessary to find a way of passing in detail fro the icroscopic (quantu) echanical description, to the acroscopic description. The ideas behind this are highly related to what we have already done for the rando walker, albeit with another level of coplexity Euler equations Instead of obtaining acroscopic equations of fluid otion fro icroscopic principles (this will be done in Sec below), we shall first derive the equations of inviscid (frictionless) hydrodynaics purely fro acro-considerations alone. This requires us to adopt a continuu approxiation, which assues a acroscopic scale large copared with the distance between olecules. We assue that the fluid is continuous in structure, and physical quantities such as the ass and oentu are spread uniforly over sall volue eleents. The validity of the continuu hypothesis under everyday conditions is clear, as two of the ore coon fluids, air and water, are so obviously continuous and soothly varying that no different hypothesis would see natural. One or two nubers readily show the great difference between the lengthscale representative of the fluid as a whole and that representative of the particle structure. For ost laboratory experients, a characteristic linear diension of the region occupied by the fluid is at least 1c, and very little variation of the physical and dynaical properties of the fluid occurs over a distance of 10 3 c. Thus an instruent with a sensitive volue of 10 9 c 3 would give a local easureent. Sall though this volue is, it contains about olecules of air at noral teperature and pressure (and an even larger nuber of olecules of water), which is large enough for an average of all the olecules to be independent of their nuber. The continuu hypothesis iplies that it is possible to attach a definite eaning to the notion of value at a point of the various fluid properties such as density, velocity and teperature, and that in general values of these quantities are continuous functions of position and tie. There is aple observational evidence that coon real fluids ove as if they were continuous, under noral conditions and indeed for considerable departures fro noral conditions. However, soe of the properties of the equivalent continuous edia need to be deterined epirically, and cannot be derived directly fro icroscopic principles The continuity equation Let s suppose the fluid density is described by a function ρ(r, t). The total ass enclosed in a fixed volue V is ρdv. (297) V The ass flux leaving this volue through the bounding surface S = V is ρu nds, (298) S 57

3 where u(x, t) is the velocity of the fluid and n is the outward noral. Hence we have ρ dv = ρu nds = (ρu)dv. (299) V S V This ust hold for any arbitrary fluid eleent dv, thus ρ + (ρu) = 0. (300) This is called the continuity equation. For fluids like water, the density does not change very uch and we will often be tepted to neglect the density variations. If we ake this approxiation the continuity equation reduces to the incopressibility condition u = 0. (301) Like all approxiations, this one is soeties very good and soeties not so good. We will have to figure out where it fails Moentu equations So far we have ore unknowns than equations (three velocity coponents but only one equation). We now consider the conservation of linear oentu and, adopting an alternative viewpoint to that used in deriving the continuity equation, consider Newton s laws for a particular oving eleent of fluid: d ρudv = pnds + fdv, (302) dt V (t) S(t) V (t) where V (t) is the volue of the eleent enclosed by the surface S(t), f is the density of body forces, such as gravity ρg, and p is a pressure force. The pressure force is a noral force per unit area (usually copressive) exerted across the surface of a fluid eleent, and is related to both interolecular forces and oentu transfer across an interface. For any volue, the pressure force is pnds = pdv. (303) Both V (t) and S(t) are being defored by the otion of the fluid, so if we want to take the d/dt inside the integral sign we ust take account of this. The Reynolds transport theore does so, and it can be shown that for a deforing, incopressible fluid eleent d dt V (t) ρudv = V (t) ρ Du dv (304) Dt Here D Dt = + (u ) (305) 58

4 is called the convective derivative, and we shall discuss it s significance in a oent. Hence, assuing that f is solely given by gravity, f = ρg, we find Du dv = ( p + ρg)dv (306) Dt ρ V (t) V (t) Since this ust hold for any arbitrary fluid eleent we arrive at Du Dt = p + g. (307) ρ This, cobined with the the continuity equation (300), constitutes the Euler equations. Things can be tidied up a little if we realise that the gravitational force, being conservative, can be written as the gradient of a scalar potential ψ. It is therefore usual to redefine pressure as p + ψ p. This iplies that gravity siply odifies the pressure distribution in the fluid and does nothing to change the velocity. However, we cannot do this if ρ is not constant or if we have a free surface (as we shall see later with water waves). Assuing the density is constant eans we now have four equations in four unknowns: three coponents of u and p. Note that if we do not deand constant density then the equations (continuity+oentu) only close with another relation, an equation of state p(ρ) Fro Newton s laws to hydrodynaic equations To copleent the purely acroscopic considerations fro the previous section, we will now discuss how one can obtain hydrodynaic equations fro the icroscopic dynaics. To this end, we consider a any-particle syste governed by Newton s equations dx i dt = v i, dv i = F i, (308) dt assuing that all particles have the sae ass, and that the forces F i can be split into an external contribution G and pair interactions H(r) = H( r) F (x 1,..., x n ) = G(x i ) + H(x i x j ) = xi Φ(x 1,..., x n ) (309) j i We define the fine-grained phase-space density N f(t, x, v) = δ(x x i (t))δ(v v i (t)) (310) i=1 where δ(x x i ) = δ(x x i )δ(y y i )δ(z z i ) in three diensions. Intuitively, the density f counts the nuber of particles that at tie t are in the sall volue [x, x + dx] while having velocities in [v, v + dv]. By chain and product rule N f = d [δ(x xi )δ(v v i )] dt i=1 N = {δ(v v i ) xi δ(x x i ) ẋ i + δ(x x i ) vi δ(v v i ) v i} i N N F i = x δ(v v i )δ(x x i ) v i v δ(x x i )δ(v v i ) i=1 i=1 (311) 59

5 where, in the last step, we inserted Newton s equations and used that δ(x x i ) = δ(x x i ) (312) x i x Furtherore, aking use of the defining properties of the delta-function N N F i f = v x δ(v v i )δ(x x i ) v δ(x x i )δ(v v i ) i=1 i=1 = v x f 1 N v δ(x x i )δ(v v i ) F i. (313) i=1 Writing = x and inserting (309) for the forces, we ay rewrite ( ) N + v f = v δ(x x i )δ(v v i ) G(x i ) + H(x i x j ) i=1 j i N = v δ(x x i )δ(v v i ) G(x) + H(x x j ) x j x i=1 = G(x) + H(x x j ) v f (314) x j x In the second line, we have again exploited the properties of the delta function which allow us to replace x i by x. Also note the appearance of the convective derivative on the lhs.; the above derivation shows that it results fro Newton s first equation. To obtain the hydrodynaic equations fro (314), two additional reductions will be necessary: We need to replace the fine-grained density f(t, x, v), which still depends iplicitly on the (unknown) solutions x j (t), by a coarse-grained density f(t, x, v). We have to construct the relevant field variables, the ass density ρ(t, r) and velocity field u, fro the coarse-grained density f. To otivate the coarse-graining procedure, let us recall that the Newton equations (308) for a syste of deterinistic ODEs whose solutions are {x 1 (t),..., x N (t)} are uniquely deterined by the initial conditions {x 1 (0),..., x N (0); v 1 (0),..., v N (0)} =: Γ 0. However, for any experiental realization of a acroscopic syste (say, a glass of water), it is practically ipossible to deterine the initial conditions exactly. To account for this lack of knowledge, we ay assue that the initial conditions are drawn fro soe probability distribution P(Γ 0 ). Without specifying the exact details of this distribution at this point, we ay define the coarse-grained density f by averaging the fine-grained density f with respect to P(Γ 0 ), forally expressed as f(t, x, v) = dp(γ 0 ) f(t, x, v). (315) 60

6 Averaging Eq. (314) and using the fact that integration over initial conditions coutes with the partial differentiations, we have ( ) + v f = v [G(x) f + C] (316) where the collision-ter C(t, x, v) := H(x x j )f(t, x, v) (317) x j x represents the average effect of the pair interactions on a fluid particle at position x. We now define the ass density ρ, the velocity field u, and the specific kinetic energy tensor Σ by ρ(t, x) = d 3 v f(t, x, v), (318a) ρ(t, x) u(t, x) = d 3 v f(t, x, v) v. (318b) ρ(t, x) Σ(t, x) = d 3 v f(t, x, v) vv. (318c) The tensor Σ is, by construction, syetric as can be seen fro the definition of its individual coponents ρ(t, x) Σ ij (t, x) = d 3 v f(t, x, v) v i v j, and the trace of Σ defines the local kinetic energy density 1 ɛ(t, x) := 2 Tr(ρΣ) = d 3 v f(t, x, v) v 2. (319) 2 Integrating Eq. (316) over v, we get ρ + (ρu) = dv 3 v [G(x) f + C], (320) but the rhs. can be transfored into a surface integral (in velocity space) that vanishes since for physically reasonable interactions [G(x) f + C] 0 as v. We thus recover the ass conservation equation ρ + (ρu) = 0. (321) To obtain the oentu conservation law, lets ultiply (316) by v and subsequently integrate over v, ( ) dv 3 + v f v = dv 3 v v [G(x) f + C]. (322) 61

7 The lhs. can be rewritten as ( ) dv 3 + v f v = (ρu) + dv 3 f vv = (ρu) + (ρσ) = (ρu) + (ρuu) + [ρ(σ uu)] = ρ u + u ρ + u (ρu) + ρu u + [ρ(σ uu)] ( ) (321) = ρ + u u + [ρ(σ uu)] (323) The rhs. of (322) can be coputed by partial integration, yielding dv 3 3 v v [G(x) f + C] = dv [G(x) f + C] = ρg + c(t, x), (324) where g(x) := G(x)/ is the force per unit ass (acceleration) and the last ter c(t, x) = dv 3 C = dv 3 H(x x j )f(t, x, v) (325) x j = x encodes the ean pair interactions. Cobining (323) and (324), we find ( ) ρ + u u = [ρ(σ uu)] + ρg(x) + c(t, x). (326) The syetric tensor Π := Σ uu (327) easures the covariance of the local velocity fluctuations of the olecules which can be related to their teperature. To estiate c, let us assue that the pair interaction force H can be derived fro a pair potential ϕ, which eans that H(r) = r ϕ(r). Assuing further that H(0) = 0, we ay write c(t, x) = dv 3 [ x ϕ(x x j )]f(t, x, v) (328) x j (t) Replacing for soe function ζ(x) the su over all particles by the integral 1 ζ(x j ) d 3 y ρ(t, y) ζ(y) (329) we have x j c(t, x) 1 dv 3 1 = dv 3 d 3 y ρ(t, y) [ x ϕ(x y)]f(t, x, v) d 3 y ρ(t, y) [ y ϕ(x y)]f(t, x, v) = 1 dv 3 d 3 y [ ρ(t, y)] ϕ(x y)f(t, x, v) (330) 62

8 In general, it is ipossible to siplify this further without soe explicit assuptions about initial distribution P that deterines the average. There is however one exception, naely, the case when interactions are very short-range so that we can approxiate the potential by a delta-function, ϕ(r) = φ 0 a 3 δ(r), (331) where ϕ 0 is the interaction energy and a 3 the effective particle volue. In this case, ϕ0a 3 c(t, x) = dv 3 d 3 y [ ρ(t, y)] δ(x y)f(t, x, v) ϕ 0 a 3 = [ ρ(t, x)] dv 3 f(t, x, v) = ϕ 0a 3 [ ρ(t, x)]ρ(t, x) 2 ϕ 0 a 3 = ρ(t, x) 2 (332) 2 2 Inserting this into (326), we have thus derived the following hydrodynaic equations where ρ + (ρu) = 0 (333a) ( ) ρ + u u = Ξ + ρg(x), (333b) Ξ := [ρ(σ uu) + ϕ 0a 3 ] ρ 2 I 2 2 (333c) is the stress tensor with I denoting unit atrix. Note that Eqs. (333) do not yet for a closed syste, due to the appearance of the second-oent tensor Σ. This is a anifestation of the well-known hierarchy proble, encountered in all 14 attepts to derive hydrodynaic equations fro icroscopic odels. More precisely, the hierarchy proble eans that the tie evolution of the nth-oent depends on that of the higher oents. The standard approach to overcoing this obstacle is to postulate (guess) reasonable ad-hoc closure conditions, which essentially eans that one tries to express higher oents, such as Σ, in ters of the lower oents. For exaple, a coonly adopted closure condition is the ideal isotropic gas approxiation kt Σ uu = I, (334) where T is the teperature and k the Boltzann constant. For this closure condition, Eqs. (333a) and (333b) becoe to a closed syste for ρ and u. Traditionally, and in ost practical applications, one does not bother with icroscopic derivations of Ξ; instead one erely postulates that 14 Except, perhaps for very trivial exaples. µ Ξ = pi + µ( u + u 2 ) ( u), (335) 3 63

9 where p(t, x) is the pressure field and µ the dynaic viscosity, which can be a function of pressure, teperature etc. depending on the fluid. Equations (333a) and (333b) cobined with the epirical ansatz (335) are the faous Navier-Stokes equations. The second suand in Eq. (335) contains the rate-of-strain tensor 1 E = ( u + u ) (336) 2 and ( u) is the rate-of-expansion of the flow. For incopressible flow, defined by ρ = const., the Navier-Stokes equations siplify to In this case, one has to solve for (p, u). u = 0 (337a) ( ) ρ + u u = p + µ 2 u + ρg. (337b) 64

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