Gaussian Kinetic Model for Granular Gases

TL;DR

This paper introduces a Gaussian kinetic model for dilute granular gases that accurately reproduces key properties of the Boltzmann equation, including the homogeneous cooling state and transport coefficients, facilitating studies of complex granular flows.

Contribution

A new Gaussian kinetic model for granular gases that closely matches Boltzmann equation results and simplifies analysis of inhomogeneous states.

Findings

Model reproduces qualitative features of the homogeneous cooling state.

Transport coefficients from the model agree well with Boltzmann equation.

Specialized Maxwell Model variant shows consistent results.

Abstract

A kinetic model for the Boltzmann equation is proposed and explored as a practical means to investigate the properties of a dilute granular gas. It is shown that all spatially homogeneous initial distributions approach a universal "homogeneous cooling solution" after a few collisions. The homogeneous cooling solution (HCS) is studied in some detail and the exact solution is compared with known results for the hard sphere Boltzmann equation. It is shown that all qualitative features of the HCS, including the nature of over population at large velocities, are reproduced semi-quantitatively by the kinetic model. It is also shown that all the transport coefficients are in excellent agreement with those from the Boltzmann equation. Also, the model is specialized to one having a velocity independent collision frequency and the resulting HCS and transport coefficients are compared to known results for the Maxwell Model. The potential of the model for the study of more complex spatially inhomogeneous states is discussed.