Velocity distribution function for a dilute granular material in shear flow

Abstract

The velocity distribution function for the steady shear flow of disks (in two dimensions) and spheres (in three dimensions) in a channel is determined in the limit where the frequency of particle–wall collisions is large compared to particle–particle collisions. An asymptotic analysis is used in the small parameter ε, which is naL in two dimensions and n²L in three dimensions, where n is the number density of particles (per unit area in two dimensions and per unit volume in three dimensions), L is the separation of the walls of the channel and a is the particle diameter. The particle–wall collisions are inelastic, and are described by simple relations which involve coefficients of restitution e_t and e_n in the tangential and normal directions, and both elastic and inelastic binary collisions between particles are considered. In the absence of binary collisions between particles, it is found that the particle velocities converge to two constant values (u_x, u_y) =(±V, 0) after repeated collisions with the wall, where u_x and u_y are the velocities tangential and normal to the wall, V=(1−e_t) V_w/(1+e_t), and V_w and −V_w are the tangential velocities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution function is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. Certain approximations are made regarding the velocities of particles undergoing binary collisions in order to obtain analytical results for the distribution function, and these approximations are justified analytically by showing that the error incurred decreases proportional to ε^1/2 in the limit ε→0. A numerical calculation of the mean square of the difference between the exact flux and the approximate flux confirms that the error decreases proportional to ε^1/2 in the limit ε→0. The moments of the velocity distribution function are evaluated, and it is found that 〈u²_x〉→V², 〈u²_y〉 ∼V²ε and − 〈u_xu_y〉 ∼ V²εlog(ε⁻¹) in the limit ε→0. It is found that the distribution function and the scaling laws for the velocity moments are similar for both two- and three-dimensional systems.