A Continuum Model for a Chute Flow of Grains

Abstract

This paper is motivated by a problem from the food-sorting industry and is concerned with the development of a continuum model for a chute flow of grains, based on analogies with the Lighthill–Whitham model of traffic flow. A fundamental relationship between the density and the flux is proposed which is multivalued due to the fact that grains can move leftwards and rightwards across the chute. The subsequent implications are discussed in detail. Results are presented first by solving with a method of characteristics and second by solving analytically and numerically after including a viscous dissipation term to smooth out discontinuities. The fundamental diagram is revisited in view of the viscous dissipation term and a substantial modification at the cusps is found necessary to allow physically sensible solutions to emerge. If the viscous dissipation is small, then in general the viscous effects also are small. Exceptions, however, are shocks, where the viscosity has a smoothing effect, as is well known, and two new features/observations: crossovers, which allow two-way solutions, and steady states, which can exist only in the viscous case and appear over a long time scale.