Avalanche mixing of granular solids in a rotating 2D drum: diffusion and fractionality

The dynamics of the avalanche mixing in a slowly rotated 2D upright drum is studied in the situation where the difference $\delta$ between the angle of marginal stability and the angle of repose of the granular material is finite. An analytical solution of the problem is found for a half filled drum, that is the most interesting case. The mixing is described by a simple linear difference equation. We show that the mixing looks like linear diffusion of fractions under consideration with the diffusion coefficient vanishing when $\delta$ is an integer part of $\pi$. The characteristic mixing time tends to infinity in these points. A full dependence of the mixing time on $\delta$ is calculated and predictions for an experiment are made.