Granular flow in rotating cylinders with noncircular cross sections

An experimental and theoretical study is carried out of the flow of granular material in cylinders with different cross-sectional shapes rotated about their axes. The flow of particles in such geometries is confined to a shallow layer at the free surface. The length and thickness of the layer shrink and expand periodically with rotation of the cylinder, resulting in chaotic advection and improved mixing of passive tracers. Experimental results obtained by flow visualization are reported for quasi-two-dimensional mixers half filled with glass beads. A depth-averaged flow model to predict the time-varying layer thickness profile is presented, along with a perturbation solution in terms of a small parameter k, which is the ratio of the maximum layer thickness to the half length of the layer (L), at the cross-section orientation when the length is minimum. To the lowest order [O(k0)], the model predicts that the layer profiles scaled with L(θ) at different mixer orientation angles (θ) are identical and the same as that for a circle. The measured layer thickness profiles averaged over different orientations of noncircular mixers match reasonably well with the theory, but the standard deviations are larger for the noncircular cylinders compared to the circle. The O(k) perturbation solution and the full theory both predict that the scaled layer thickness varies periodically; the deviations are proportional to the rate of change of the length with orientation. The perturbation solution gives results close to those from the numerical solution except at cylinder orientations when the length of the flowing layer changes sharply. The measured variation of the scaled midlayer thickness with orientation for all geometries is well predicted by the theory

Some Remarks on Avalanches Modelling: An Introduction to Shallow Flows Models

International audienceThe main goal of these notes is to present several depth-averaged models with application in granular avalanches. We begin by recalling the classical Saint-Venant or Shallow Water equations and present some extensions like the Saint-Venant-Exner model for bedload sediment transport. The first part is devoted to the derivation of several avalanche models of Savage-Hutter type, using a depth-averaging procedure of the 3D momentum and mass equations. First, the Savage-Hutter model for aerial avalanches is presented. Two other models for partially fluidized avalanches are then described: one in which the velocities of both the fluid and the solid phases are assumed to be equal, and another one in which both velocities are unknowns of the system. Finally, a Savage-Hutter model for submarine avalanches is derived. The second part is devoted to non-newtonian models, namely viscoplastic fluids. Indeed, a one-phase viscoplastic model can also be used to simulate fluidized avalanches. A brief introduction to Rheology and plasticity is presented in order to explain the Herschel-Bulkley constitutive law. We finally present the derivation of a shallow Herschel-Bulkley model