Gravity-Driven Particle-Laden Flow on an Inclined Plane with Varying Topography

We develop a dynamic model to investigate mixtures of particles and viscous fluid on an inclined plane with varying topography. Our model accounts for several important factors, including gravity-induced settling of particles, particle resuspension due to shear-induced migration, liquid surface tension, and topography profile. In the thin film limit, we derive a reduced model consisting of hyperbolic-parabolic equations for the liquid layer thickness and particle volume fraction. The resulting equation depends only on the direction parallel to the inclined plane, making it easier to solve compared to the governing equation while still capturing its essential features. A noteworthy aspect of our study is the incorporation of additional terms in the reduced model, which consider the influence of the normal component of gravity, surface tension, and variations in the topography. This inclusion allows us to gain a deeper understanding of how these factors influence the dynamics of the system. We analyze several cases of topographical variations, such as sinusoids, Gaussians, and sharp steps, using both theoretical and numerical approaches. These analyses hold significant practical applications, including the study of mudslides, microchip manufacturing, and material processing.