Shear-Driven Particle Size Segregation: Models, Analysis, Numerical Solutions, and Experiments
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2009-12-04
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Abstract
Granular materials segregate by particle size when subject to shear, as in avalanches. Particles roll and slide across one another, and other particles fall into the voids that form, with smaller particles more likely to fit than larger particles. Small particles segregate to the bottom of the sample, and larger particles are levered upward. These processes are known as kinetic sieving and squeeze expulsion. The evolution of the volume fraction of small particles (ratio of the volume of small particles to the total volume of the system) corresponds to the evolution of segregation in a binary mixture of particles and can be modeled by a nonlinear first order partial differential equation, provided the velocity or
shear is a known function of position. In an avalanche, shear is approximately uniform in depth, however, in boundary driven shear, the velocity is nonlinear and a shear band forms
adjacent to the boundary.
We explore size segregation with a laboratory experiment and by analyzing a model. We classify solutions to a fundamental initial boundary value problem for avalanche flow in two space dimensions akin to a two dimensional Riemann problem. We describe three solution types; the initial condition determines which solution occurs. We also modify the partial differential equation to model segregation in a system that experiencing nonuniform shear. We measure a velocity profile from the experimental data from a Couette experiment, which provides parameters used to visualize the solution to the initial boundary value problem.
We experimentally investigate size segregation using an annular Couette cell, which is constructed of concentric cylinders and has a moving lower boundary that imparts shear to the system and an upper confining boundary that is free to move vertically to accommodate changes in the volume of the system. Initially, the Couette cell contains a layer of large particles below a layer of small particles. The system dilates as shear begins, then contracts as the sample mixes, and again expands as the sample resegregates; the height of the system is correlated to the amount of mixing or segregation. At the end of the experiment, we find a layer of small particles below a layer of large particles. The initial condition for the partial differential equation corresponds to the one dimensional initial configuration of the experiment. We solve two initial boundary value problems, one with a piecewise linear shear rate and one with an exponential shear rate, where the parameters for both cases are derived from the experimental data. In each case, we use the method of characteristics to solve the initial boundary value problem. In both cases, almost all pieces of the solution can be explicitly calculated, and those that cannot are calculated numerically. In the piecewise linear case, there is a material interface across which the characteristic speed jumps; in the exponential case, the characteristics are curved. We compare the model with the exponential shear rate to the experimental data. The model solution is the volume fraction of small particles at time t and location z. We cannot measure the volume fraction locally in the experiment; instead the height of the sample is an indirect measurement of the amount of mixing or segregation. We map the volume fraction to a theoretical height which we compare to the experimental height data. We conclude that the model captures qualitative features of the experimental data, but there are features of the experiment that we cannot capture with the model.
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Couette cell experiment, granular materials, partial differential equation model, size segregrgation
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Degree
PhD
Discipline
Applied Mathematics
