Modeling size segregation of granular materials: the roles of segregation, advection and diffusion

Predicting segregation of granular materials composed of different-sized particles is a challenging problem. In this paper, we develop and implement a theoretical model that captures the interplay between advection, segregation, and diffusion in size bidisperse granular materials. The fluxes associated with these three driving factors depend on the underlying kinematics, whose characteristics play key roles in determining particle segregation configurations. Unlike previous models for segregation, our model uses parameters based on kinematic measures from discrete element method simulations instead of arbitrarily adjustable fitting parameters, and it achieves excellent quantitative agreement with both experimental and simulation results when applied to quasi-two-dimensional bounded heaps. The model yields two dimensionless control parameters, both of which are only functions of physically control parameters (feed rate, particle sizes, and system size) and kinematic parameters (diffusion coefficient, flowing layer depth, and percolation velocity). The P\'eclet number, $Pe$, captures the interplay of advection and diffusion, and the second dimensionless parameter, $\Lambda$, describes the interplay between segregation and advection. A parametric study of $\Lambda$ and $Pe$ demonstrates how the particle segregation configuration depends on the interplay of advection, segregation, and diffusion. The model can be readily adapted to other flow geometries.