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A Continuum Framework for the Description of Mixing and Segregation in Three-Dimensional Granular Flows

Steven W. Meier and Julio M. Ottino. Department of Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Road, Tech E136, Evanston, IL 60208

Granular mixing in three-dimensional tumblers is complicated, as flow induces segregation by particle size or density. Flow in three-dimensional systems appears complicated; however, all of the dynamics takes place in a thin flowing surface layer. This observation, coupled with two key experimental results, leads to a framework for modeling mixing and segregation in three-dimensional granular flows. Particle tracking velocimetry measurements indicate that the streamwise velocity decreases linearly with depth in the surface layer. Measurements across the spanwise direction of the free surface shows that streamwise velocity at the midpoint of the flowing layer is a linear function of local flowing layer length (the distance in the direction of flow between the tumbler walls for a given spanwise position). These observations enter as assumptions into a continuum model of the flow. Segregation is modeled via a constitutive model involving two interpenetrating continua representing the different particle types (large and small or heavy and light). The model captures experimental results in time periodic flow in quasi-two-dimensional tumblers. Here we extend results for three-dimensional tumblers (such as spheres and cubes) rotated about one or more axes of rotation. We also present a symmetry analysis technique that requires only basic information about the nature of the flow. There appears to be a strong relationship between the symmetries of the underlying flow and the segregation patterns seen in experiments. An understanding of the symmetries of the underlying flow may yield insights into improved mixing of granular material. [Supported by NSF and DOE.]