A three-phase mixture theory for particle size segregation in shallow granular free-surface flows

Particle-size segregation within granular materials is of great technological significance yet it is still very poorly understood. There are several causes of segregation, but this paper focuses on kinetic sieving which is the dominant mechanism in dense gravity-driven shallow free-surface flows, or, granular avalanches. The segregation model is derived from a three-phase mixture theory composed of large particles, small particles and a passive interstitial fluid. Steady-state solutions are constructed for a normally graded inflow in a steady uniform flow field. This problem is of fundamental interest, because it shows how an unstably stratified layer readjusts into a stable configuration. Expansion fans and concentration shocks are generated and sufficiently far downstream inversely graded segregated layers form, with the larger particles overlying the finer ones. This a good approximation for segregation in flows with weak diffusive remixing. The distance for complete segregation to occur is shown to increase with rising fluid density and tends to infinity as its density approaches that of the grains. If the particles are buoyant then the initial configuration is stable. An exact time-dependent two-dimensional solution is constructed for plug flow, which exploits the uncoupling of material columns of grains in the absence of shear. This yields insight into the nature of more complex numerical solutions for strong shear, which are computed with a high-resolution shock-capturing numerical scheme.