On the basis of the Saint-Venant equations for flowing granular matter, we study the various travelling waveforms that are encountered in chute flow for growing Froude number. Generally, for $Fr<2/3$ one finds either a uniform flow of constant thickness or a monoclinal flood wave, i.e. a shock structure monotonically connecting a thick region upstream to a shallower region downstream. For $Fr>2/3$ both the uniform flow and the monoclinal wave cease to be stable; the flow now organizes itself in the form of a train of roll waves. From the governing Saint-Venant equations we derive a dynamical system that elucidates the transition from monoclinal waves to roll waves. It is found that this transition involves several intermediate stages, including an undular bore that had hitherto not been reported for granular flows.

We study granular chute flow using the classic Saint-Venant approach, with the shear stresses within the granular sheet being incorporated via a friction law due to Pouliquen & Forterre (J. Fluid Mech., vol. 453, 2002, pp. 113–151) and with the in-plane stresses (which are ignored in the traditional formulation for normal fluids) being represented by a viscous-like term recently derived by Gray & Edwards (J. Fluid Mech., vol. 755, 2014, pp. 503–534). On the basis of this model, we predict that the granular sheet is able to sustain monoclinal waves, i.e. travelling shock structures that monotonically connect a thick region of uniform flow to a thinner one. We examine the balance of forces that determine the shape of this particular waveform and give the precise window of system parameters for which monoclinal waves are expected to appear in experiments.

Abstract

A granular cluster, placed on a staircase setup, is brought into motion by vertical shaking. Molecular dynamics simulations show that the system goes through three phases. After a rapid initial breakdown of the cluster, the particle stream organizes itself in the form of a shock wave moving down the steps of the staircase. As this wave becomes diluted, it transforms into a more symmetric flow, in which the particles move not only downwards but also toward the top of the staircase. This series of events is accurately reproduced by a dynamical model in which the particle flow from step to step is modeled by a flux function. To explain the observed scaling behavior during the three stages, we study the continuum version of this model ͑a nonlinear partial differential equation͒ in three successive limiting cases. ͑i͒ The first limit gives the correct t −1/3 decay law during the rapid initial phase, ͑ii͒ the second limit reveals that the transient shock wave is of the Burgers type, with the density of the wave front decreasing as t −1/2 , and ͑iii͒ the third limit shows that the eventual symmetric flow is a slow diffusive process for which the density falls off as t −1/3 again. For any finite number of compartments, the system finally reaches an equilibrium distribution with a bias toward the lower compartments. For an unbounded staircase, however, the t −1/3 decay goes on forever and the distribution becomes increasingly more symmetric as the dilution progresses.

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 Q cr (K) for which a continuous flow through the system is possible. Beyond Q cr (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 belowQ cr (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.