Characterization of invariant patterns in a slowly rotated granular tumbler

• A simple geometric model explains the invariant patterns experimentally obtained.
• The finite width of avalanches has quantifiable consequences.
• There is a special filling area for which the invariant patterns are closer to a circle.
• We have obtained an explicit expression for the contour of the invariant patterns.

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, AiAi, and its circularity index as a function of the normalized filling area AA. We find a critical filling area so that only for A>AcA>Ac invariant zones are obtained. These zones scale as Ai∼(A−Ac)2Ai∼(A−Ac)2 near AcAc. We have obtained a maximum in the circularity index for A≈0.8A≈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 AA. We have obtained an analytic expression for the contour of the invariant zone. Experimentally, we obtained a displacement in AcAc that we explain in terms of a finite width of the avalanche region. This displacement is needed only to correct the size of the invariant zone, not its shape.