Kinetic theory and boundary conditions for flows of high inelastic spheres: Application to gravity driven granular flows down bumpy inclines

Abstract

In this paper, we employ formal methods of statistical averaging to calculate the rates at which momentum and energy are transferred from anisotropically fluctuating bumpy boundaries to dense granular assemblies. The assemblies consist of identical, smooth, nearly elastic spheres that are thermalized by repeated collisions with the boundaries, but experience no mean motion as a consequence of these collisions. The boundaries vibrate with velocities that are governed by a tri-axial Gaussian distribution function that depends on both the normal and tangential mean square fluctuation speeds of the boundaries. Using the transfer rates calculated, we write down conditions that ensure that momentum and energy are balanced at such boundaries, and employ these conditions with a corresponding kinetic constitutive theory to analyze steady, gravity-free, thermalized states of granular assemblies between parallel, vibrating, bumpy boundaries. We find that, as the boundaries become bumpier, vibrations that are tangent the boundaries become more effective and vibrations that are normal to the boundaries become less effective at transferring energy to the assemblies.