The hard-particle model for a dense granular flow down an inclined plane

Abstract

Perfectly hard particles are those which experience an infinite repulsive force when they overlap, and no force when they do not overlap. In the hard‐particle model, the only static state is the isostatic state where the forces between particles are statically determinate. In the flowing state, the interactions between particles are instantaneous because the time of contact approaches zero in the limit of infinite particle stiffness. Here, we discuss the development of a hard particle model for a realistic granular flow down an inclined plane, and examine its utility for predicting the salient features both qualitatively and quantitatively. We first discuss Discrete Element simulations, that even very dense flows of sand or glass beads with volume fraction between 0.5 and 0.58 are in the rapid flow regime, due to the very high particle stiffness.An important length scale in the shear flow of inelastic particles is the ‘conduction length’ δ = (d/(1−e2)1/2), where d is the particle diameter and e is the coefficient of restitution. When the macroscopic scale h (height of the flowing layer) is larger than the conduction length, the rates of shear production and inelastic dissipation are nearly equal in the bulk of the flow, while the rate of conduction of energy is O((δ/h)2) smaller than the rate of dissipation of energy. Energy conduction is important in boundary layers of thickness δ at the top and bottom. The flow in the boundary layer at the top and bottom is examined using asymptotic analysis. We derive an exact relationship showing that the a boundary layer solution exists only if the volume fraction in the bulk decreases as the angle of inclination is increased. In the opposite case, where the volume fraction increases as the angle of inclination is increased, there is no boundary layer solution.The boundary layer theory also provides us with a way of understanding the cessation of flow when at a given angle of inclination when the height of the layer is decreased below a value hstop, which is a function of the angle of inclination. There is dissipation of energy due to particle collisions in the flow as well as due to particle collisions with the base, and the fraction of energy dissipation in the base increases as the thickness decreases. When the shear production in the flow cannot compensate for the additional energy drawn out of the flow due to the wall collisions, the temperature decreases to zero and the flow stops. Scaling relations can be derived for hstop as a function of angle of inclination.