Short-Wave Limit of Hydrodynamics: A Soluble Example.

Abstract

A derivation of hydrodynamics from the Boltzmann kinetic equation is the classical problem of physical kinetics. The Chapman-Enskog (CE) method [1] gives, in principle, a possibility to compute a solution as a formal series in powers of Knudsen number e (where e is a ratio between the mean free path of a particle and the scale of variations of hydrodynamic quantities, density, mean flux, and temperature). The CE solution leads to a formal expansion of stress tensor and of heat flux vector in balance equations for density, momentum, and energy. Retaining the first order term (e) in the latter expansions, we come to the NavierStokes equations, while further corrections are known as the Burnett (e 2 ) and the super-Burnett (e 3 ) corrections [1]. However, as demonstrated by Bobylev [2], even in the simplest regime (one-dimensional linear deviation from global equilibria), the Burnett and super-Burnett hydrodynamics violate the basic physics behind the Boltzmann equation. Namely, sufficiently short acoustic waves are increasing with time instead of decaying. This contradicts the H theorem, since all near-equilibrium perturbations must decay. It should also be noted that the instability of equilibria just mentioned is not a feature of the NavierStokes approximation where waves of arbitrary length are decaying, though this approximation is formally not valid in a short-wave domain. A possible root of this violation is poor convergency properties of CE series, and this, in particular, creates serious difficulties for an extension of hydrodynamics, as derived from a microscopic description, into a highly nonequilibrium domain. The latter problem remains one of the central open problems of the Boltzmann equation theory, in particular, and of the physical kinetics, in general. In this Letter we consider the CE procedure for a simple model of nonhydrodynamic description (one-dimensional linearized 10-moment Grad equations [3]). The CE series, which is due to a nonlinear procedure even here and which also suffers the Bobylev instability in low-order approximations, is summed up in a closed form. This result leads to a quantitative discussion of the CE solution in a shortwave domain in frames of the model, and to a preliminary discussion of what can be expected in more realistic models. Exact results on the CE method for other Grad moment systems will be reported elsewhere. Throughout the Letter, p and u are dimensionless deviations of pressure and of mean flux from their equilibrium values, respectively (see Ref. [4] for relations of these variables to dimensional quantities). The point of departure is the set of linearized Grad equations [4] for p, u, and s, where s is a dimensionless xx component of stress tensor,