Higher-order effects on orientational correlation and relaxation dynamics in homogeneous cooling of a rough granular gas

Abstract

The orientational or angular correlation between the directions of the translational and rotational motions is analyzed theoretically for the homogeneous cooling state of a rough granular gas. The dynamical equations are derived using an approximate form of the single-particle distribution function that incorporates angular correlations. The goal is to assess the effects of higher-order angular corrections for which both quadratic- and quartic-order terms (in translational and rotational velocities of particles) are retained in the perturbation expansion of the distribution function. We show that higher-order corrections can markedly affect the steady-state orientational correlation when the normal restitution coefficient is moderate or small, and this effect is more prominent for nearly smooth particles. The transient evolution of orientational correlation is found to be significantly affected by higher-order terms. In particular the higher-order orientational correlations can dominate over the leading-order contribution during short times even in the quasi-elastic limit, although the steady correlation remains unaffected by such corrections in the same limit. The buildup of correlations during the transient stage seems to be closely tied to the evolution of the ratio between the rotational and translational temperatures. It is demonstrated that the transient dynamics of the temperature ratio and its steady state remain insensitive to higher-order angular correlation.