283250 Intermediate Asymptotics of Axial Diffusion of Tumbled Granular Materials
Though granular materials consist of macroscopic particles that do not perform Brownian motion, "self-diffusion" can arise from agitation processes that cause non-equilibrium velocity fluctuations. Contrary to previous results , it has recently  been suggested that the axial diffusion process of granular matter in a long rotating cylinder might be "anomalous," i.e., the mean-squared-displacement of particles follows a power law in time with exponent less than unity. Further numerical  and experimental studies  were unable to settle this apparent paradox. We show two possible resolutions to the observations in  without the need to appeal to anomalous diffusion: (a) by considering the evolution of arbitrary initial data towards the self-similar intermediate asymptotics of diffusion, and (b) by accounting for the concentration-dependent diffusivity in bidisperse mixtures. Furthermore, since granular flow in the axial cross-section of a long rotating drum is composed of a thin surface shear layer over a fixed bed in static equilibrium, the cross-sectionally-averaged concentration can be separated into a diffusing and a non-diffusing component. We extend the standard axial diffusion model by using first-order kinetics to account for the continuous exchange of particles between the flowing layer and the fixed bed. The resulting transport equation is identical to the one governing the diffusion of momentum in a viscoelastic Jeffreys (also known as Oldroyd-B) fluid above an impulsively-moved plate. Results will be presented on the intermediate asymptotics of this new axial diffusion model equation with both constant and concentration-dependent diffusivities.
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