Tuesday, January 15, 2013 at 4:00 PM to 5:00 PM in SAS 4201
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 slowlyrotating container. Under some mild assumptions, the kinematics of the flow can be modeled and scalar mixing studied with the advectiondiffusion equation paradigm. The shape of the container can induce chaotic trajectories, while the properties of the individual particles can lead to selforganization (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 Barenblatts theory of selfsimilar intermediate asymptotics (Christov & Stone, Proc. Natl Acad. Sci. USA, 2012). Specifically, we find an analytical expression for the instantaneous scaling exponent of a macroscopic concentration profile, as a function of the initial distribution. Then, by incorporating concentrationdependent 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.
