Longitudinal Viscous Flow in Granular Gases

The flow characterized by a linear longitudinal velocity field $u_x(x,t)=a(t)x$, where $a(t)={a_0}/({1+a_0t})$, a uniform density $n(t)\propto a(t)$, and a uniform temperature $T(t)$ is analyzed for dilute granular gases by means of a BGK-like model kinetic equation in $d$ dimensions. For a given value of the coefficient of normal restitution $\alpha$, the relevant control parameter of the problem is the reduced deformation rate $a^*(t)=a(t)/\nu(t)$ (which plays the role of the Knudsen number), where $\nu(t)\propto n(t)\sqrt{T(t)}$ is an effective collision frequency. The relevant response parameter is a nonlinear viscosity function $\eta^*(a^*)$ defined from the difference between the normal stress $P_{xx}(t)$ and the hydrostatic pressure $p(t)=n(t)T(t)$. The main results of the paper are: (a) an exact first-order ordinary differential equation for $\eta^*(a^*)$ is derived from the kinetic model; (b) a recursion relation for the coefficients of the Chapman--Enskog expansion of $\eta^*(a^*)$ in powers of $a^*$ is obtained; (c) the Chapman--Enskog expansion is shown to diverge for elastic collisions ($\alpha=1$) and converge for inelastic collisions ($\alpha<1$), in the latter case with a radius of convergence that increases with inelasticity; (d) a simple approximate analytical solution for $\eta^*(a^*)$, hardly distinguishable from the numerical solution of the differential equation, is constructed.