Subcritical Pattern Formation in Granular Flow

We study a minimal model for the flow of granular material on a conveyor belt consisting of a staircase-like array of K vertically vibrated compartments. Applying a steady inflow rate Q to the top compartment, we determine the maximum value Qcr(K) for which a continuous flow through the system is possible. BeyondQcr(K), which depends on the vibration strength and the dimensions of the system, a dense cluster forms in one of the first compartments and obstructs the flow. We find that the formation of this cluster is already announced below Qcr(K) by the appearance of an oscillatory density profile along the entire length of the conveyor belt, with a distinct two-compartment wavelength. These model predictions concerning the breakdown of the granular flow admit an elegant explanation in terms of bifurcation theory. In particular, the subcritical oscillatory pattern is shown to be a side effect of the period doubling bifurcation by which the uniform density profile (associated with a smooth particle flow) becomes unstable. The effect turns out to be robust enough to survive the presence of a reasonable amount of noise and even certain qualitative modifications to the flux model. The density oscillations may therefore well be of practical value and provide a warning signal for imminent clustering on actual conveyor belts.