 |
Center for Nonlinear Studies |
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, 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 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.
Host: Eli Ben-Naim, T-4