Kinetic model for sheared granular flows in the high Knudsen number limit

The sheared granular flow of rough inelastic granular disks is analyzed in the high Knudsen number limit, where the frequency of particle-wall collisions is large compared with particle-particle collisions, using a kinetic theory approach. An asymptotic expansion is used in the small parameter epsilon =(nsigmaL), which is the ratio of the frequencies of particle-particle and particle-wall collisions, where n is the number of disks per unit area, sigma is the disk diameter, and L is the channel width. The collisions are specified using a normal coefficient of restitution e(n) and a tangential coefficient of restitution e(t). The analysis identifies two regions in the e(t) - e(n) parameter space, one where the final steady state is a static one in which the translational velocities of all particles decrease to zero, and the second where the final steady state is a dynamic one in which the mean square velocities scale as a power of epsilon in the limit epsilon --> 0. Both of these predictions are shown to be in quantitative agreement with computer simulations.