VELOCITY DISTRIBUTION FUNCTION FOR A DILUTE GRANULAR MATERIAL IN SHEAR-FLOW

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 collision s is large compared to particle-particle collisions. An asymptotic ana lysis is used in the small parameter epsilon, which is naL in two dime nsions and na(2)L in three dimensions, where; n is the number density of particles (per unit area in two dimensions and per unit volume in t hree 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 co nsidered. 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, O) 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 veloc ities of the walls of the channel. The effect of binary collisions is included using a self-consistent calculation, and the distribution fun ction is determined using the condition that the net collisional flux of particles at any point in velocity space is zero at steady state. C ertain approximations are made regarding the velocities of particles u ndergoing binary collisions :in order to obtain analytical results for the distribution function, and these approximations are justified ana lytically by showing that the error incurred decreases proportional to epsilon(1/2) in the limit epsilon --> 0. A numerical calculation of t he mean square of the difference between the exact flux and the approx imate flux confirms that the error decreases proportional to epsilon(1 /2) in the limit epsilon --> 0. The moments of the velocity distributi on function are evaluated, and it is found that [u(x)(2)] --> V-2, [u( y)(2)] similar to V-2 epsilon and -[u(x)u(y)] similar to V-2 epsilon l og(epsilon(-1)) in the limit epsilon --> 0. It is found that the distr ibution function and the scaling laws for the velocity moments are sim ilar for both two- and three-dimensional systems.