Evolution of a Mixing Zone in Granular Avalanches

A nonlinear first-order partial differential equation in two space variables and time describes the process of kinetic sieving in an avalanche, in which larger particles tend to rise to the surface while smaller particles descend, quickly leading to completely segregated layers. The interface between layers is a shock wave satisfying its own nonlinear equation. When the interface becomes vertical, it loses stability, and develops a mixing zone. The mixing zone is described explicitly under idealized initial conditions, and verified with numerical simulation. The problem and its solution are similar to twodimensional Riemann problems for scalar first-order conservation laws; the difference here is that the equation is not scale-invariant, due to shear in the avalanche, an essential ingredient of kinetic sieving.