Longitudinal Viscous Flow in Granular Gases

TL;DR

This paper analyzes the nonlinear viscous flow in dilute granular gases using a kinetic model, deriving exact equations, exploring series expansions, and providing an approximate analytical solution for the viscosity function.

Contribution

It introduces an exact differential equation for the nonlinear viscosity function and examines the convergence of its series expansion based on inelasticity.

Findings

Exact differential equation for viscosity function derived

Chapman--Enskog expansion diverges for elastic collisions

Expansion converges for inelastic collisions with increasing radius

Abstract

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.