Regularization of the Burnett equations for rapid granular flows via relaxation

In [J. Fluid Mech. 361 (1998) 41] Sela and Goldhirsch have used the Chapman -Enskog expansion to derive constitutive relations for the pressure deviato r P, heat Bur q, and rate of energy loss R for rapid flows of smooth inelas tic spheres. Unfortunately as in the classical Chapman-Enskog expansion for elastic spheres any truncation of the expansion beyond Navier-Stokes order (n = 1) will possess unphysical instabilities. In this paper we propose a visco-elastic relaxation approximation that eliminates the instability para dox for all Nave numbers, and provide a system of local equations allowing robust numerical approximations of gas dynamics valid to the Burnett order. This system is weakly parabolic, has a linearly hyperbolic convection part , and is endowed with a generalized entropy inequality in the case of purel y elastic collisions, thus it is linearly stable for all wave numbers. It a grees with the solution of the Boltzmann equation up to the Burnett order v ia the Chapman-Enskog expansion. (C) 2001 Elsevier Science B.V. All rights reserved.