Dense shearing flows of inelastic disks

We introduce a simple phenomenological modification to the hydrodynamic equations for dense flows of identical, frictionless, inelastic disks and show that the resulting theory describes the area fraction dependence of quantities that are measured in numerical simulations of steady, homogeneous shearing flows and steady, fully developed flows down inclines. The modification involves the incorporation of a length scale other than the particle diameter in the expression for the rate of collisional dissipation. The idea is that enduring contacts between grains forced by the shearing reduce the collisional rate of dissipation while continuing to transmit momentum and force. The length and orientation of the chains of particles in contact are determined by a simple algebraic equation. When the resulting expression for the rate of dissipation is incorporated into the theory, numerical solutions of the boundary-value problem for steady, fully developed flow of circular disks down a bumpy incline exhibit a core with a uniform area fraction that decreases with increasing angles of inclination. When the height at which an inclined flow stops is assumed to be proportional to this chain length, a scaling between the average velocity, flow height, and stopping height similar to that seen in experiments and numerical simulations is obtained from the balance of fluctuation energy.