Shock Formation and Breaking in Granular Avalanches

In this paper, we explore properties of shock wave solutions of the Gray-Thornton model for particle size segregation in granular avalanches. The model equation is a nonlinear scalar conservation law expressing conservation of mass under shear for the concentration of small particles in a bidisperse mixture. Shock waves are weak solutions of the partial differential equation across which the concentration jumps. We give precise criteria on smooth initial conditions under which a shock wave forms in the interior of the avalanche in finite time. Shocks typically lose stability as they are sheared by the flow, giving way to a complex structure in which a two-dimensional rarefaction wave interacts dynamically with a pair of shocks. The rarefaction represents a mixing zone, in which small and large particles are mixed as they are transported up and down (respectively) through the zone. The mixing zone expands and twice changes its detailed structure before reaching the boundary.