System of elastic hard spheres which mimics the transport properties of a granular gas

TL;DR

This paper develops an elastic hard sphere model with an effective drag force and adjusted collision rate to accurately mimic the transport properties of inelastic hard sphere granular gases, simplifying their kinetic description.

Contribution

It introduces a novel elastic hard sphere model that reproduces key transport properties of inelastic granular gases using simple modifications.

Findings

EHS model reproduces self-diffusion coefficient accurately

Heat flux transport coefficients are reasonably matched

Shear viscosity deviation is less than 14% for certain inelasticities

Abstract

The prototype model of a fluidized granular system is a gas of inelastic hard spheres (IHS) with a constant coefficient of normal restitution $\alpha$. Using a kinetic theory description we investigate the two basic ingredients that a model of elastic hard spheres (EHS) must have in order to mimic the most relevant transport properties of the underlying IHS gas. First, the EHS gas is assumed to be subject to the action of an effective drag force with a friction constant equal to half the cooling rate of the IHS gas, the latter being evaluated in the local equilibrium approximation for simplicity. Second, the collision rate of the EHS gas is reduced by a factor $(1+\alpha)/2$, relative to that of the IHS gas. Comparison between the respective Navier-Stokes transport coefficients shows that the EHS model reproduces almost perfectly the self-diffusion coefficient and reasonably well the two transport coefficients defining the heat flux, the shear viscosity being reproduced within a deviation less than 14% (for $\alpha\geq 0.5$). Moreover, the EHS model is seen to agree with the fundamental collision integrals of inelastic mixtures and dense gases. The approximate equivalence between IHS and EHS is used to propose kinetic models for inelastic collisions as simple extensions of known kinetic models for elastic collisions