Geometric aspects of mixing and segregation in granular tumblers

Recently, much attention has been given to granular systems due to their widespread use in industrial processes and their often baffling physical behavior. In the midst of the emerging field of complex systems, granular materials are quickly becoming a prototypical system exhibiting spontaneous organization. Flowing or vibrated mixtures of granular materials, differing in physical properties such as size or density, spontaneously segregate. Constitutive equations describing the behavior of granular materials, analogous to the Navier-Stokes equations, do not exist. This field, as compared to fluids, is poorly understood.; Mixing in fluids may be enhanced by the addition of time-periodic perturbations to the flow, generating chaos. This same concept is applied to granular flow in a tumbler. A simple two-dimensional continuum model developed by Khakhar et al. 1997b describing continuous flow in a circular cross section of a tumbler is modified to describe flow in noncircular geometries. Time-periodic flow in a square is chaotic and sensitive to fill level. When bi-disperse materials are tumbled in a square, there is direct competition between mixing and segregation, and pattern formation is nontrivial-segregation resembles invariant structures of the underlying flow. This model is extended to describe flow in a sphere and a cube to shed light on the interplay between segregation and chaotic flow in three-dimensions.; The competition between granular mixing and segregation is also investigated in a novel spherical tumbler capable of undergoing independent programmed motions in two orthogonal directions. The primary mode of operation is a combination of rotating and rocking of the axes. Space-time plots are used to compare experimental results with surface Poincaré sections obtained using a continuum model of the flow. A phase plot showing modes of segregation—band formation/no axial bands—in the frequency/amplitude domain is used to organize the experimental results; segregated bands are remarkably robust and survive rocking amplitudes of as much as 60 degrees. Details differ, but the phenomenon occurs both in dry materials and under slurry conditions.