Transverse stability of particle-laden flow on an inclined plane

We investigate a mixture of liquid and particles flowing down an incline. While many models have successfully described the dynamics of the particle-liquid mixture along the incline, studies focusing on the dynamics in the transverse direction, where interesting fingering instability occurs, are rare. To address the gap, it is essential to consider particle migration and fluid flow in the transverse direction. We extend the continuum model used to describe the mixture dynamics along the incline by incorporating these factors, treating the particle-fluid mixture as an incompressible Newtonian fluid with density and viscosity modified by the presence of particles. In the thin film limit, we derive a system of reduced equations governing the height of the free surface and particle volume fraction. We numerically simulate these reduced equations and, using linear stability analysis, investigate the dependence of the mixture front's instability on various physical parameters.