Does the Chapman-Enskog expansion for viscous granular flows converge?

TL;DR

This paper investigates the convergence of the Chapman-Enskog series for viscous granular flows, revealing that, contrary to expectations, the series converge for inelastic granular gases, with convergence improving as inelasticity increases.

Contribution

It demonstrates the convergence of the Chapman-Enskog series for inelastic granular gases using an exact solution of a kinetic model, challenging previous assumptions about divergence.

Findings

Series converge for inelastic granular gases

Convergence radii increase with inelasticity

Paradoxical convergence behavior explained by steady states

Abstract

This paper deals with the convergence/divergence issue of the Chapman-Enskog series expansion of the shear and normal stresses for a granular gas of inelastic hard spheres. From the exact solution of a simple kinetic model in the uniform shear and longitudinal flows, it is shown that (except in the elastic limit) both series converge and their respective radii of convergence increase with inelasticity. This paradoxical result can be understood in terms of the time evolution of the Knudsen number and the existence of a nonequilibrium steady state.