Particle Size Segregation In Granular Avalanches: A Study In Shocks

In this thesis, we explore properties of shock wave solutions of the Gray-Thornton model for particle size segregation in granular avalanches. In these avalanches, particles segregate by size when subject to shear. As the particles roll across each other, other particles fall into the gaps that form, with smaller particles more likely to fit. These small particles fall to the bottom of the avalanche and force the larger particles upward. These processes are called kinetic sieving and squeeze expulsion. The Gray-Thornton model is a nonlinear scalar conservation law expressing conservation of mass under shear for the concentration of small particles in a bidisperse mixture. In this model, the velocity (and thus, shear) is a function of the height of the avalanche. We first discuss characteristic surfaces of the model, which are used in combination with shock waves to construct and analyze solutions of the model. .;Shock waves are weak solutions of the partial differential equation across which the concentration of small particles jumps. For a linear velocity profile, we give criteria on smooth initial conditions under which a shock wave forms in the interior of the avalanche in finite time. Additionally, numerical simulations show how and when these shocks form, verifying our analysis.;Shocks will often lose stability as they are sheared by the flow of the avalanche. Upon the loss of stability a complex structure develops in which a two-dimensional rarefaction wave interacts dynamically with a pair of shocks. This rarefaction represents a mixing zone in which small and large particles are mixed as they are transported up and down (respectively) through the zone. Under a linear velocity profile, the structure of this region twice changes over time before reaching the boundary of the avalanche. We also present a special case where the structure of the mixing region does not change over time. By introducing a scaling, we can find a similarity solution for this case.;Linear velocity profiles are not always present in granular materials, especially in the case of boundary driven shear. Thus, we analyze shock formation from smooth initial data under a general increasing velocity profile. Additionally, we analyze the short time solution of the mixing zone under an increasing velocity profile. Here, we present several cases, with each case more general than the previous one. For each case, we analyze the structure of the mixing zone as much as possible, and discuss limitations to the more general cases. Numerical simulations show how the mixing region evolves for each case.;We look at the evolution of an avalanche that is uniform in the downslope direction. Analysis of this solution is important because it appears in the most general version of the mixing zone problem.