Characterization of invariant patterns in a slowly rotated granular tumbler

We report experimental results of the pattern developed by a mixture of two types of grains in a triangular rotating tumbler operating in the avalanche regime. At the centroid of the triangular tumbler an invariant zone appears where the grains do not move relative to the tumbler. We characterize this invariant zone by its normalized area, $A_i$, and its circularity index as a function of the normalized filling area $A$. We find a critical filling area so that only for $A>A_c$ invariant zones are obtained. These zones scale as $A_i\sim (A-A_c)^2$ near $A_c$. We have obtained a maximum in the circularity index for $A\approx 0.8$, for which the shape of the invariant zone is closer to a circular one. The experimental results are reproduced by a simple model which, based on the surface position, accounts for all the possible straight lines within the triangle that satisfy the condition of constant $A$. We have obtained an analytic expression for the contour of the invariant zone.