Granular flow of rough particles in the high-Knudsen-number limit

Abstract

The granular shear flow of rough inelastic particles driven by flat walls is considered in the high-Knudsen-number limit, where the frequency of particle collisions with the wall is large compared to the frequency of inter-particle collisions. An asymptotic analysis is used in the small parameter $\varepsilon = n d L$ in two dimensions and $\varepsilon = n d^2 L$ in three dimensions, where $n$ is the number of particles (per unit area in two dimensions and per unit volume in three dimensions), $d$ is the particle diameter and $L$ is the distance between the flat walls. In the collision model, the post-collisional velocity along the line joining the particle centres is $-e_n$ times the pre-collisional velocity, and the post-collisional velocity perpendicular to the line joining the particle centres is $-e_t$ times the pre-collisional value, where $e_n$ and $e_t$ are the normal and tangential coefficients of restitution. In the absence of binary collisions, a particle which has a random initial velocity tends to a final state where the translational velocities are zero, and the rotational velocity is equal to $(-2V_w/d)$, where $V_w$ is the wall velocity. When the effect of binary collisions is included, it is found that there are two possible final steady states, depending on the values of the tangential and normal coefficients of restitution. For certain parameter values, the final steady state is a stationary state, where the translational velocities of all the particles reduce to zero. For other parameter values, the final steady state is dynamic state where the translational velocity fluctuations are non-zero. In the dynamic state, the mean-square velocity has a power-law scaling with $\varepsilon$ in the limit $\varepsilon \rightarrow 0$. The exponents predicted by the theory are found to be in quantitative agreement with simulation results in two dimensions.