Effects of temperature anisotropy in a hard sphere gas

We consider the effects of temperature anisotropy in a quiescent, hard sphere gas, as a model system for flowing granular matter. Utilising kinetic theory we obtain expressions for the streaming pressure, collisional pressure and the rate of transfer of kinetic energy between the different translational degrees of freedom, assuming a Maxwell-Boltzmann velocity distribution and an isotropic pair correlation function at contact. A theory for the collisional pressure with an anisotropic pair correlation function is also presented. Event driven simulations are carried out for frictionless, elastic, hard spheres, using a thermostat that maintains a different temperature in one of the coordinate directions. The pair correlation function, the velocity distribution function, streaming and collisional pressures, and the rate of energy transfer are computed in the simulations for different temperature anisotropy values. The pair correlation function becomes significantly anisotropic with increasing temperature anisotropy and solid fraction. The streaming pressure and the energy transfer rate are well described by the theory based on the isotropic pair correlation function. The predictions of this theory for the collisional pressure match well with the results of the simulations for a solid fraction of 0.1, but increasing deviations are obtained for higher solid fractions. The theory incorporating the anisotropy in the pair correlation function gives good predictions of the collisional pressure for all the solid fractions considered. The work shows the importance of incorporating the anisotropy in the pair correlation function in kinetic theories for systems with temperature anisotropy.