Transport phenomena in flows of granular materials
Physics of Condensed Matter and Complex Systems Group and Center for Nonlinear Studies
Los Alamos National Laboratory
Flowing granular materials are an example of a heterogeneous complex system away from equilibrium. As a result, their dynamics are still poorly understood. One canonical example is granular flow in a slowly-rotating container. Under some mild assumptions, the kinematics of the flow can be modeled and scalar mixing studied with the advection-diffusion equation paradigm. The shape of the container can induce chaotic trajectories, while the properties of the individual particles can lead to self-organization (demixing). The balance between these two effects leads to intricate persistent mixing patterns, which we show correspond to eigenmodes of an appropriate operator (Christov, Ottino & Lueptow, Phys. Fluids, 2011). However, granular materials do not perform thermally driven Brownian motion, so diffusion is observed in such systems because agitation (flow) causes inelastic collisions between particles. In a variation of the previous experiment, it has been suggested that axial diffusion of granular matter in a rotating drum might be "anomalous" in the sense that the mean squared displacement of particles follows a power law in time with exponent less than unity. Further numerical and experimental studies have been unable to definitively confirm or disprove whether a fractional diffusion equation describes this process. We can show that such a "paradox" can be resolved using Barenblatt's theory of self-similar intermediate asymptotics (Christov & Stone, Proc. Natl Acad. Sci. USA, 2012). Specifically, by incorporating concentration-dependent diffusivity into the model, we show the existence of a crossover from an anomalous scaling (consistent with experimental observations) to a normal diffusive scaling at very long times. Finally, time permitting, a third canonical granular flow will be discussed: chute flow down an incline. In this case, a model can be constructed for Taylor--Aris dispersion due to interaction between diffusion and shear, showing an enhancement for dispersion of granular materials over dispersion of a molecular solute in a fluid (Christov & Stone, Granular Matter, 2013). This research was supported by NSF Grant DMS-1104047.