Granular materials segregate by size under shear, and the ability to quantitatively predict the time required to achieve complete segregation is a key test of our understanding of the segregation process. In this paper, we apply the Gray-Thornton model of segregation (developed for linear shear profiles) to a granular flow with an exponential profile, and evaluate its ability to describe the observed segregation dynamics. Our experiment is conducted in an annular Couette cell with a moving lower boundary. The granular material is initially prepared in an unstable configuration with a layer of small particles above a layer of large particles. Under shear, the sample mixes and then re-segregates so that the large particles are located in the top half of the system in the final state. During this segregation process, we measure the velocity profile and use the resulting exponential fit as input parameters to the model. To make a direct comparison between the continuum model and the observed segregation dynamics, we locally map the measured height of the experimental sample (which indicates the degree of segregation) to the local packing density. We observe that the model successfully captures the presence of a fast mixing process and relatively slower re-segregation process, but the model predicts a finite re-segregation time, while in the experiment re-segregation occurs only exponentially in time.
Granular materials will segregate by particle size when subjected to shear, as occurs, for example, in avalanches. The evolution of a bidisperse mixture of particles can be modeled by a nonlinear first order partial differential equation, provided the shear (or velocity) is a known function of position. While avalanche-driven shear is approximately uniform in depth, boundary-driven shear typically creates a shear band with a nonlinear velocity profile. In this paper, we measure a velocity profile from experimental data and solve initial value problems that mimic the segregation observed in the experiment, thereby verifying the value of the continuum model. To simplify the analysis, we consider only one-dimensional configurations, in which a layer of small particles is placed above a layer of large particles within an annular shear cell and is sheared for arbitrarily long times. We fit the measured velocity profile to both an exponential function of depth and a piecewise linear function which separates the shear band from the rest of the material. Each solution of the initial value problem is non-standard, involving curved characteristics in the exponential case, and a material interface with a jump in characteristic speed in the piecewise linear case.
Abstract. 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.
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