# Moment theories for a d-dimensional dilute granular gas of Maxwell molecules

## TL;DR

This paper derives and analyzes moment equations for a d-dimensional dilute granular gas of Maxwell molecules, providing insights into the Navier-Stokes relations and stability of the homogeneous cooling state.

## Contribution

It introduces a comprehensive set of moment equations for granular gases and demonstrates their sufficiency for Navier-Stokes relations and stability analysis.

## Key findings

- Moment equations derived up to (d+3)(d^2+6d+2)/6 moments.
- Navier-Stokes constitutive relations depend only on certain moments.
- Critical system size for instability matches simulation results.

## Abstract

Various systems of moment equations---consisting of up to (d+3)(d^2+6d+2)/6 moments---in a general dimension d for a dilute granular gas composed of Maxwell molecules are derived from the inelastic Boltzmann equation by employing the Grad moment method. The Navier--Stokes-level constitutive relations for the stress and heat flux appearing in the system of mass, momentum and energy balance equations are determined from the derived moment equations. It has been shown that the moment equations only for the hydrodynamic field variables (density, velocity and granular temperature), stress and heat flux---along with the time-independent value of the fourth cumulant---are sufficient for determining the Navier--Stokes-level constitutive relations in the case of inelastic Maxwell molecules, and that the other higher-order moment equations do not play any role in this case. The homogeneous cooling state of a freely cooling granular gas is investigated with the system of the Grad (d+3)(d^2+6d+2)/6-moment equations and its various subsystems. By performing a linear stability analysis in the vicinity of the homogeneous cooling state, the critical system size for the onset of instability is estimated through the considered Grad moment systems. The results on critical system size from the presented moment theories are found to be in reasonably good agreement with those from simulations.