Theory for the continuation of singular shocks in systems of conservation laws for particle laden flow

We consider the problem of particle laden flow on an incline in the lubrication limit. Such problems arise in many industrial applications including particle separation in the mining industry, food manufacture, and cosmetics. We review the derivation of conservation law models for such problems and note that in the high volume fraction and high inclination angle setting, that singular shocks arise in these models. These shocks exhibit seemingly unphysical behavior of mass concentration. However another aspect of the shocks is the finite volume fraction of particles at the maximum packing fraction. We show that there is a way to interpret this behavior in the physical setting where particles acculumate at the leading front in a particle rich ridge. The experiment exhibits mass shedding from the front in which clumps of particles take on a solid-like behavior, breaking off from the front and sliding down the incline at a much faster speed than the particle front. We propose a theoretical continuation of the singular shock in which statistical mass shedding balances acculumation of particles in the leading edge of the front. This model has interesting properties including statistical stationarity and universality in the mass shedding dynamics and an equilibrium balance between the fluid dynamics and the solid-like mass shedding in the singular shock.