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Modeling and analysis of thin-film incline flow: bidensity suspensions and surface tension effects

Abstract

For flow of suspensions down an incline, particles are driven by shear-induced migration towards the surface, leading to separation of particle and fluid phases or aggregation at the leading edge. Reduced models are interesting from a mathematical standpoint as they take the form of PDEs for the evolution of the fluid and particle phases of hyperbolic/parabolic type. By assuming a separation of time scales, one can reduce the model to two simpler com- ponents: an equilibrium equation for the particle distribution and a set of one-dimensional PDEs for the evolution of the film. This approach is used here explore several more compli- cated problems concerning gravity-driven flow on an incline. We study bidensity suspensions (i.e. with two particle species of different densities) and develop the corresponding equilib- rium and dynamic theory. A system of hyperbolic conservation laws are derived and shock and rarefaction solutions are constructed to describe the separation of particle and fluid phases. Surface tension is also introduced into the model, which can lead to unusual behav- ior by allowing particles to aggregate to interior points in the fluid. We derive the thin-film system for the evolution of the fronts, develop a numerical method to handle the complicated fluxes and study solutions through simulations.

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