Does the Chapman--Enskog expansion for sheared granular gases converge?

TL;DR

This paper investigates the convergence of the Chapman--Enskog expansion for sheared granular gases, demonstrating that unlike elastic gases, the series converges, with the radius of convergence increasing as inelasticity grows.

Contribution

It shows, using a kinetic model, that the Chapman--Enskog expansion converges for inelastic hard spheres, contrasting with the elastic case, and explains this through the evolution of shear rate.

Findings

The series converges for inelastic granular gases.

The radius of convergence increases with inelasticity.

The convergence is explained via shear flow dynamics.

Abstract

The fundamental question addressed in this paper is whether the partial Chapman--Enskog expansion $P_{xy}=-\sum_{k=0}^\infty \eta_k ({\partial u_x}/{\partial y})^{2k+1}$ of the shear stress converges or not for a gas of inelastic hard spheres. By using a simple kinetic model it is shown that, in contrast to the elastic case, the above series does converge, the radius of convergence increasing with inelasticity. It is argued that this paradoxical conclusion is not an artifact of the kinetic model and can be understood in terms of the time evolution of the scaled shear rate in the uniform shear flow.