A conservation law model for bidensity suspensions on an incline

We study bidensity suspensions of a viscous fluid on an incline. The particles migrate within the fluid due to a combination of gravity-induced settling and shear induced migration. We propose an extension of a recent model (Murisic et al., 2013) for monodisperse suspensions to two species of particles, resulting in a hyperbolic system of three conservation laws for the height and particle concentrations. We analyze the Riemann problem and show that the system exhibits three-shock solutions representing distinct fronts of particles and liquid traveling at different speeds as well as singular shock solutions for sufficiently large concentrations, for which the mechanism is essentially the same as the single-species case. We also consider initial conditions describing a fixed volume of fluid, where solutions are rarefaction-shock pairs, and present a comparison to recent experimental results. The long-time behavior of solutions is identified for settled mono- and bidisperse suspensions and some leading-order asymptotics are derived in the single-species case for moderate concentrations.