The Gray-Thornton Model of Granular Segregation

In this paper, we explore properties of the Gray-Thornton model for particle size segregation in granular avalanches. The model equation is a single conservation law expressing conservation of mass under shear for the concentration of the smaller of two types of particle in a bidisperse mixture. Sharp interfaces across which the concentration jumps are shock wave solutions of the partial differential equation. We show that they can form internally from smooth data, as well as propagate in from boundaries of the domain. We prove a general stability result that expresses the physically reasonable notion that an interface should be stable only if the concentration of small particles is larger below the interface than above. Once shocks form, they are sheared by the flow, leading to loss of stability when an interface becomes vertical. The subsequent evolution of a mixing zone, a two-dimensional rarefaction solution of the equation that replaces the unstable part of the shock can be tracked explicitly for a short time. We conducted experiments to test the continuum model against real flow in a Couette geometry, in which a bidisperse mixture is confined in the annular region between concentric vertical cylinders. Initially, the material is placed in the annulus with a layer of large particles below a layer of small particles. The sample is then sheared by rotating the bottom confining plate, while a heavy top plate is allowed to move vertically to accommodate Reynolds dilatancy. Comparison to predictions of the model show reasonable agreement with the rate at which the sample mixes, and with the rate of the subsequent resegregation. However, the model naturally fails to capture short-time dilatancy, finite size effects, or three-dimensional effects.