Title:

Wellposed continuum modelling of granular flows

Inertial granular flows lie in a region of parameter space between quasistatic and collisional regimes. In each of these phases the mechanisms of energy dissipation are often taken to be the defining features. Frictional contacts between grains and the transmission of energy through cooperative force chains dominate slowly sheared flows. In the opposite extreme infrequent highenergy collisions are responsible for dissipation in socalled gaseous granular flows. Borrowing from each of these extremes, it is postulated that during liquidlike flow, grain energy is transferred through frequent frictional interactions as the particles rearrange. This thesis focuses on the μ(I)rheology which generalises the simple Coulomb picture, where greater normal forces lead to greater tangential friction, by including dependence on the inertial number I, which reflects the frequency of grain rearrangements. The equations resulting from this rheology, assuming that the material is incompressible, are first examined with a maximalorder linear stability analysis. It is found that the equations are linearly wellposed when the inertial number is not too high or too low. For inertial numbers in which the equations are instead illposed numerical solutions are found to be griddependent with perturbations growing unboundedly as their wavelength is decreased. Interestingly, experimental results also diverge away from the original μ(I) curve in the illposed regions. A generalised wellposedness analysis is used alongside the experimental findings to suggest a new functional form for the curve. This is shown to regularise numerical computations for a selection of inclined plane flows. As the incompressibility assumption is known to break down more drastically in the highI and lowI limits, compressible μ(I) equations are also considered. When the closure of these equations takes the form suggested by critical state soil mechanics, it is found that the resultant system is wellposed regardless of the details of the deformation. Wellposed equations can also be formed by depthaveraging the μ(I)rheology. For threedimensional chute flows experimental measurements are captured well by the depthaveraged model when the flows are shallow. Furthermore, numerical computations are much less expensive than those with the full μ(I) system.
