Modeling polydisperse particle-laden flow down an incline

Particle-laden flows are fundamental to geology, mining, and food science. We propose a new approach to model polydisperse particle-laden flow down an incline. We consider two cases: finitely many particle species of different sizes and infinitely many particle species, the sizes of which we represent as a continuous distribution. We model the flow after the particles have reached equilibrium in the direction normal to the incline, which motivates using the diffusive flux approach. The equilibrium dynamics are governed by a system of ordinary differential equations for the finite case and an integro-differential equation for the continuous case, which we derive by applying the lubrication approximation to the Stokes equation and particle conservation equations. We also obtain and numerically study the associated transport equations. Extending this analysis to the dilute limit results in simplified expressions for the model equations. We validate our model by conducting systematic polydisperse experiments to measure the front positions. We present experimental results characterized by distinct fronts for the liquid, small, and large particles. Notably, at high particle concentrations, the larger particles outpace the smaller ones, resulting in a ridge rich in large particles.