Velocity distribution for a two-dimensional sheared granular flow

Abstract

The velocity distribution for a two-dimensional collection of disks of number density n per unit area and radius a in a channel of width L is studied. The particle-particle collisions are considered to be inelastic with a coefficient of restitution e, while the particle-wall coefficients of restitution are inelastic with a tangential and normal coefficients of restitution, e_t and e_n, respectively. The Knudsen number, which is the ratio of the channel width and the mean free path of the particles, is varied from K_n « 1 to K_n » 1. In the limit of high Knudsen number, the distribution function for the streamwise velocity is bimodal, as predicted by theory [V. Kumaran, J. Fluid Mech., 340, 3l9 (1997)], and the scalings of the moments of the velocity distribution with the Knudsen number are in agreement with the theory. In the low Knudsen number limit, the distribution function for the streamwise velocity is a Gaussian if the coefficient of restitution is close to 1, but assumes the form of a "composite Gaussian" if the coefficient of restitution is not close to 1. The distribution function has a complex structure in the intermediate regime, where there are three maxima in the distribution function near the wall, while the distribution function is bimodal at the center. The granular temperature is accurately predicted by kinetic theory at the center of the channel, but there is dissipation at the wall due to inelastic particle-wall collisions, which results in a significant decrease in the temperature even when the coefficient of restitution is 0.9; this finding is in agreement with previous results with bumpy wall boundary conditions and with specular reflection conditions. The slip velocity at the wall has a power law dependence on the Knudsen number, and the exponent in this power law depends on the coefficients of restitution.