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 deviator P, heat flux q, and rate of energy loss Γ for rapid flows of smooth inelastic 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 paradox for all wave 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 purely elastic collisions, thus it is linearly stable for all wave numbers. It agrees with the solution of the Boltzmann equation up to the Burnett order via the Chapman–Enskog expansion.