Lecture 5: Kinetic theory of fluids

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1 Lecture 5: Kinetic theory of fluids September 21, Goal 2 From atoms to probabilities Fluid dynamics descrines fluids as continnum media (fields); however under conditions of strong inhomogeneities (shocks, boundary layers), the continuum picture breaks down and the granular nature of matter is exposed. However, describing strongly inhomogenous fluids in terms of atoms or molecules is simply unviable. It is then very convenient to take an intermediate point of view, a 1

2 probabilistic one. What is the probability of finding a molecule at position r in space at time t with velocity v? This is precisely the Boltzmann probability distribution function (pdf): N = f( r, v; t) r v where N is the number of particles (atoms/molecules) in a volume r v of phase-space. 3 The Boltzmann equation The BE is a continuity equation in phase space: df dt = tf + v i xi f + a i vi f = C G L The lhs (Streaming) is a mirror of Newtonian mechanics, the rhs describes the interparticle collisions (Collide). The Stream-Collide paradigm leads to Advection-Diffusion in teh macrosciopic limit, but it is more general. 4 Global and local equilibria The local equilibrium is defined by teh condition C = 0 namely a dynamic balance between gain and loss terms, G = L. Inspection of the Boltzmann collision operator shows, after making the molecular chaos assumption f(1, 2) = f(1) f(2) shows that the logarithm of the product f(1)f(2) is conserved in the collision process. This means that it must be depend on the five collision invariants only f eq = A + B i v i + Cv 2 By imposing the conservation of density, momentum and energy, one readily derives the local maxwellian distribution M(v, r; t) = An(r; t)e ξ2 (r;t)/2 (1) where ξ i (r; t) = (v i u(r; t))/v th(r;t) is the peculiar speed in units of the thernal speed, and A = (2πvth 3 is a normalization constant. To be noted that the space-time dependence of the local equilibrium comes exclusively throught the macroscopic fields. 2

3 This dependence is arbitrary, except that it must be slow in the scale of the mean free path. Global equilibria (thermodynamics) are attained when u = 0 (no flow) and v th = const. The transient dynamics from local to global equilibria defines transport phenomena, which are key to pratical applications. 5 From Boltzmann to Navier-Stokes Kinetic moments: they describe the statistic in velocity space Number density n(r; t) = f(r, v; t)dv (2) Mass density ρ(r; t) = m f(r, v; t)dv = nm (3) Mass current density: J i (r; t) = m f(r, v; t)v i dv ρu i (4) Energy density: e(r; t) = m f(r, v; t)v 2 /2dv (5) Temperature Energy density: nk B T (r; t) = m f(r, v; t)(v u) 2 /2 dv (6) Momentum flux: P ij = m f(r, v; t)v i v j dv (7) Energy flux: Q i = m f(r, v; t)(v 2 /2)v i dv (8) Note that these quantities are statistical probes of velocity space. It s happens that close enough to local equilibrium (low Knudsen) a few fields. density, velocity and temperature, capture the full story (hydrodynamics). Project upon zeroth order basis (integrate over...dv): t ρ + i J i = 0 (9) t J i + j P ij = 0 (10) 3

4 t P ij + k Q ijk = ω P (P ij P eq ij ) (11) where Q ijk = fv i v j v k dv. Enslaving assumption: neglect the time derivative and replace the triple tesnors with its equilibrium expressions. This delivers: P ij P eq ij By computing the equilibrium tensors + τ P k Q eq ijk (12) and P eq ij = ρv2 thδ ij + ρu i u j Q eq ijk = J iv 2 thδ jk + P erm 6 Chapman-Enskog and higher order expansions Formal solution of model Boltzmann equation (BGK): Formal series expansion in k τ = τ t + vτ x (Knudsen number operator) Order 0 (Euler): f = f eq f = 1 τ (f eq f) (13) f = (1 + τ ) 1 f eq Order 1 (Navier-Stokes):f = f eq kf eq Order 2 (Burnett):f = f eq kf eq + k 2 f eq Order 3 (Super-Burnett):f = f eq kf eq + k 2 + k 3 f eq Order (Boltzmann):. This expansion does not converge term by term: it is the series 1/(1 + z): each term is a growing power, alternating sign, all terms must be included to obtain a finite result. Mathematically: the small parameter upfronts the space derivatives, if k << 1 all is well, otherwise no convergence. Physically: boundary layers where k = O(1) require all terms of the summation! Going beyond the first order expansion (Navier-Stokes) typically leads to ill-posed equations. That is why transport phenomena far from equilibrium are hard to handle. 4

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Fluid Equations for Rarefied Gases

1 Fluid Equations for Rarefied Gases Jean-Luc Thiffeault Department of Applied Physics and Applied Mathematics Columbia University http://plasma.ap.columbia.edu/~jeanluc 23 March 2001 with E. A. Spiegel