A Parametric Study of Mixing in a Granular Flow a Bi-Axial Spherical Tumbler

We report on a computational parameter space study of mixing protocols for a half-full bi-axial spherical granular tumbler. The quality of mixing is quantified via the intensity of segregation (concentration variance) and computed as a function of three system parameters: angles of rotation about each tumbler axis and the flowing layer depth. Only the symmetric case is considered in which the flowing layer depth is the same for each rotation. We also consider the dependence on $\bar{R}$, which parametrizes the concentric spheroids ("shells") that comprise the volume of the tumbler. The intensity of segregation is computed over 100 periods of the mixing protocol for each choice of parameters. Each curve is classified via a time constant, $\tau$, and an asymptotic mixing value, $bias$. We find that most choices of angles and most shells throughout the tumbler volume mix well, with mixing near the center of the tumbler being consistently faster (small $\tau$) and more complete (small $bias$). We conclude with examples and discussion of the pathological mixing behaviors of the outliers in the so-called $\tau$-$bias$ scatterplots.