Several applications of a model for dense granular flows
This dissertation describes efforts to evaluate a recently proposed continuum model for the dense flow of dry granular materials (Jop, Forterre & Pouliquen, 2006, Nature, 441, 167-192). The model, based upon a generalisation of Coulomb sliding friction, is known to perform well when modelling certain simple free surface flows. We extend the application of this model to a wide range of flow configurations, beginning with six simple flows studied in detailed experiments (GDR MiDi, 2004, Eur. Phys. J. E, 14, 341-366). Two-dimensional shearing flows and problems of linear stability are also addressed. These examples are used to underpin a thorough discussion of the strengths and weaknesses of the model. In order to calculate the behaviour of granular material in more complicated configurations, it is necessary to undertake a numerical solution. We discuss several computational techniques appropriate to the model, with careful attention paid to the evolution of any shear-free regions that may arise. In addition, we develop a numerical scheme, based upon a marker-and-cell method, that is capable of modelling two-dimensional granular flow with a moving free surface. A detailed discussion of our unsuccessful attempt to construct a scheme based upon Lagrangian finite elements is presented in an appendix. We apply the marker-and-cell code to the key problem of granular slumping (Balmforth & Kerswell, 2005, J. Fluid Mech., 538, 399-428), which has hitherto resisted explanation by modelling approaches based on various reduced (shallow water) models. With our numerical scheme, we are able to lift the assumptions required for other models, and make predictions in good qualitative agreement with the experimental data. An additional chapter describes the largely unrelated problem of contact between two objects separated by a viscous fluid. Although classical lubrication theory suggests that two locally smooth objects converging under gravity will make contact only after infinite time, we discuss several physical effects that may promote contact in finite time. Detailed calculations are presented to illustrate how the presence of a sharp asperity can modify the approach to contact.