Modeling dynamic flows of grain-fluid mixtures by coupling the mixture theory with a dilatancy law

A depth-averaged two-velocity grain-fluid mixture model is proposed to describe flows of grain-fluid mixtures. Motivated by the experimental observations, the proposed model considers that the granular and the fluid phases are moving with different velocities, and the velocity difference between the granular phase and the fluid phase is coupled with the granular dilatancy that is described by a granular dilatancy law. The characteristics of flows allow to formulate a simpler depth-averaged PDE system. To scrutinize the proposed equations, an analysis for steady flows in rectangular channels is performed, which reproduces the cross-stream velocity profiles commonly observed in fields. Additionally, a uniform flow is investigated to illustrate the effects of the granular dilatancy on the velocities, flow depth, and volume fractions.