Constitutive equations for a simple shear flow of a disk shaped granular mixture

Abstract

Abstract Constitutive equations are derived for a granular flow of disk shaped solids in a 2-dimensional, simple shear field. Binary collisions are assumed to be the major mechanism for momentum transfer. Stresses are computed as the average rate of momentum transfer across a surface due to the interparticle collisions. Stresses are formulated explicitly in terms of the shearing rate, concentration of solids, and the physical properties of the solid constituent. The anisotropic collision distribution on the circumference of the disks due to the mean flow gradient is explicitly quantified. The frictional impact during collisions is treated in sufficient detail so that the singularity which existed in previous constitutive equations [1, 2] is removed. This detailed analysis of collision geometry and its effect upon frictional dissipation includes important considerations that have not been included in the numerical simulation models of Campbell and Brennen [3] where particles, after impact, are considered to not have any relative tangential velocity. This assumption by Campbell and Brennen will most likely be the technique by which frictional effects will be commonly handled in the future. Hence the present study will provide an important reference for assessing the effects of using simplifying assumptions when determining frictional effects during particle collisions. The constitutive relationships developed in this investigation are compared to data obtained from computer simulated experiments [3]. The result of the 2-dimensional analysis has practical applications in the study of ice flows on a river surface as reported in [4] and in the conveyance of bulk solids in chutes. Except for geometric complications, the procedure developed in the analysis can be extended to 3-dimensional flows of spherical particles.