Transport coefficients of a granular gas of inelastic rough hard spheres

TL;DR

This paper derives explicit expressions for the transport coefficients of a dilute granular gas of inelastic rough spheres using the Boltzmann equation and Chapman-Enskog method, accounting for both translational and rotational degrees of freedom.

Contribution

It extends the kinetic theory of granular gases by providing a detailed analysis of transport coefficients for inelastic rough spheres, including their dependence on restitution coefficients and moment of inertia.

Findings

Transport coefficients expressed as nonlinear functions of restitution and inertia.

Recovery of known results for smooth and elastic spheres.

Explicit formulas using first Sonine approximation.

Abstract

The Boltzmann equation for inelastic and rough hard spheres is considered as a model of a dilute granular gas. In this model, the collisions are characterized by constant coefficients of normal and tangential restitution and hence the translational and rotational degrees of freedom are coupled. A normal solution to the Boltzmann equation is obtained by means of the Chapman-Enskog method for states near the homogeneous cooling state. The analysis is carried out to first order in the spatial gradients of the number density, the flow velocity, and the granular temperature. The constitutive equations for the momentum and heat fluxes and for the cooling rate are derived and the associated transport coefficients are expressed in terms of the solutions of linear integral equations. For practical purposes, a first Sonine approximation is used to obtain explicit expressions of the transport coefficients as nonlinear functions of both coefficients of restitution and the moment of inertia. Known results for purely smooth inelastic spheres and perfectly elastic and rough spheres are recovered in the appropriate limits.