Hydrodynamic modes of a sheared granular flow from the Boltzmann and Navier–Stokes equations

Abstract

The initial growth rates for the hydrodynamic modes of the shear flow of a three-dimensional collection of inelastic spheres is analyzed using two models. The first is the generalized Navier–Stokes equations, derived for the shear flow of inelastic spheres using the Chapman–Enskog procedure, where the energy equation has an additional dissipation term due to inelastic collisions. The second is the solution of the linearized Boltzmann equation, where the distribution function in the base state is determined using a Hermite polynomial expansion in the velocity moments. For perturbations with variations in the velocity and gradient directions, it is found that the solutions obtained by two procedures are qualitatively similar, though there are quantitative differences. For perturbations with variations in the vorticity direction, it is found that there are qualitative differences in the predictions for the initial growth rate of the perturbations.