A Bhatnagar–Gross–Krook-like Model Kinetic Equation for a Granular Gas of Inelastic Rough Hard Spheres

Abstract

The Boltzmann collision operator for a dilute granular gas of inelastic rough hard spheres is much more intricate than its counterpart for inelastic smooth spheres. Now the one-body distribution function depends not only on the translational velocity of the center of mass but also on the angular velocity of the particle. Moreover, the collision rules couple both velocities, involving not only the coefficient of normal restitution but also the coefficient of tangential restitution. The aim of this paper is to propose an extension to inelastic rough particles of a Bhatnagar-Gross-Krook-like kinetic model previously proposed for inelastic smooth particles. The Boltzmann collision operator is replaced by the sum of three terms representing: (i) the relaxation to a two-temperature local equilibrium distribution, (ii) the action of a nonconservative drag force proportional to the peculiar velocity, and (iii) the action of a nonconservative torque equal to a linear combination of the angular velocity and its mean value. The three coefficients in the force and torque are fixed to reproduce the Boltzmann collisional rates of change of the mean angular velocity and of the two granular temperatures (translational and rotational). A simpler version of the model is also constructed in the form of two coupled kinetic equations for the translational and rotational velocity distributions. The kinetic model is applied to the simple shear flow steady state and the combined influence of the two coefficients of restitution on the shear and normal stresses and on the translational velocity distribution function is analyzed.