Time of avalanche mixing of granular materials in a half filled rotated drum

The avalanche mixing of granular solids in a slowly rotated 2D upright drum is studied. We demonstrate that the account of the difference $\delta$ between the angle of marginal stability and the angle of repose of the granular material leads to a restricted value of the mixing time $\tau$ for a half filled drum. The process of mixing is described by a linear discrete difference equation. We show that the mixing looks like linear diffusion of fractions with the diffusion coefficient vanishing when $\delta$ is an integer part of $\pi$. Introduction of fluctuations of $\delta$ supresses the singularities of $\tau(\delta)$ and smoothes the dependence $\tau(\delta)$.