Kinetic theory of binary particles with unequal mean velocities and non-equipartition energies

TL;DR

This paper develops a three-dimensional kinetic theory for binary granular mixtures with unequal velocities and energies, deriving hydrodynamic equations and analyzing how mass, temperature, and velocity ratios influence stress and energy transfer.

Contribution

It introduces a kinetic theory framework for binary granular mixtures considering non-equipartition energies and different mean velocities, extending previous models.

Findings

Collision frequency increases with granular temperatures.

Solid viscosity is maximized at maximum granular temperatures.

Energy transfer depends strongly on relative velocities.

Abstract

The hydrodynamic conservation equations and constitutive relations for a binary granular mixture composed of smooth, nearly elastic, hard spheres with non-equipartition energies and different mean velocities are derived. This research is aimed to build three-dimensional kinetic theory to characterize the behaviors of two species of particles suffering different forces. The standard Enskog method is employed assuming a Maxwell velocity distribution for each species of particles. The collision components of the stress tensor and the other parameters are calculated from the zeroth- and first-order approximation. Our results demonstrate that three factors, namely the ratios between two granular masses, temperatures and mean velocities all play important roles in the stress-strain relation of the binary mixture. The collision frequency and the solid viscosity escalate with increasing of two granular temperatures and are maximized when both of two granular temperatures reach maximums. The first part of the energy source varies greatly with the mean velocities of spheres of two species, and further, it reaches maximum at the maximum of relative velocity between two mean velocities of spheres of two species.