Fundamental aspects of continuum modeling of granular diffusion and dispersion in tumbler flows
Granular materials do not perform Brownian motion, yet diffusion can be observed in such systems when agitation causes inelastic collisions between particles. It has been suggested that axial diffusion of granular matter in a rotating drum might be “anomalous”. I will discuss several mathematical results, grounded in continuum modeling of granular diffusion, that can justify observation of “anomalous” diffusion without resorting to non-local effects. First, the evolution of arbitrary (non-point-source) initial data towards the self-similar intermediate asymptotics of diffusion can exhibit an instantaneous collapse exponent that appears anomalous. Second, concentration-dependent diffusivity in bidisperse mixtures, for which an exact self-similar analytical solution does not exist, leads to two intermediate asymptotic regimes: one with an anomalous scaling, and another with a “normal” diffusive scaling at longer times. Third, accounting for localization of shear in the flow to a thin surface layer in the cross-section leads to an exactly-solvable non-Fickian macroscopic model of axial diffusion. This model is a member of the general class of linear constitutive relations with memory, and suggests new ideas for interpreting the results of experimental measurements. Finally, I will conclude with some comments on the interaction of shear with diffusion in granular flows. I will formulate and solve a model problem of shear dispersion of dense granular materials in rapid flow down an incline. The effective dispersivity of the depth-averaged concentration of the dispersing powder is shown to vary as the Péclet number squared, as in classical Taylor–Aris dispersion of molecular solutes.
Ivan Christov received his Ph.D. in Engineering Sciences and Applied Mathematics from Northwestern University. Subsequently, he was awarded an NSF Mathematical Sciences Postdoctoral Research Fellowship and spent two years with the Complex Fluids Group at Princeton University, working on interfacial instabilities and fluid–structure interactions at low Reynolds number. Following that, he spent two and a half years as the Richard P. Feynman Distinguished Postdoctoral Fellow in Theory and Computing at the Center for Nonlinear Studies at Los Alamos National Laboratory, working on problems of granular materials and porous media flow related to geophysics and unconventional energy utilization. Previously, he has interned at the U.S. Naval Research Laboratory and the ExxonMobil Upstream Research Company. His research interests are primarily in the area of modeling and numerical simulation of transport phenomena with an emphasis on complex and nonlinear systems. He is now an Assistant Professor of Mechanical Engineering at Purdue University directing the Transport: Modeling, Numerics & Theory laboratory, where advanced mathematical concepts and experimental results are combined with physical intuition towards the modeling of flowing materials in order to make progress on fundamental questions in mechanics; specifically, regarding transport as a means of effecting mixing or for mitigating separation.