Velocity Distribution and Cumulants in the Unsteady Uniform Longitudinal Flow of a Granular Gas

Abstract

The uniform longitudinal flow is characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$ is the strain rate, a uniform density $n(t)\propto a(t)$, and a uniform granular temperature $T(t)$. Direct simulation Monte Carlo solutions of the Boltzmann equation for inelastic hard spheres are presented for three (one positive and two negative) representative values of the initial strain rate $a_0$. Starting from different initial conditions, the temporal evolution of the reduced strain rate $a^*\propto a_0/\sqrt{T}$, the non-Newtonian viscosity, the second and third velocity cumulants, and three independent marginal distribution functions has been recorded. Elimination of time in favor of the reduced strain rate $a^*$ shows that, after a few collisions per particle, different initial states are attracted to common "hydrodynamic" curves. Strong deviations from Maxwellian properties are observed from the analysis of the cumulants and the marginal distributions.