Partial Regularisation of the Incompressible μ(I)-Rheology for Granular Flow

In recent years considerable progress has been made in the continuum modelling of granular flows, in particular the μ(I)-rheology, which links the local viscosity in a flow to the strain rate and pressure through the non-dimensional inertial number I. This formulation greatly benefits from its similarity to the incompressible Navier–Stokes equations as it allows many existing numerical methods to be used. Unfortunately, this system of equations is ill posed when the inertial number is too high or too low. The consequence of ill posedness is that the growth rate of small perturbations tends to infinity in the high wavenumber limit. Due to this, numerical solutions are grid dependent and cannot be taken as being physically realistic. Here we modify the μ(I)-rheology to make the equations well-posed for all inertial numbers below a large maximal limit. This allows us, for the first time, to reliably simulate complex two-dimensional flows such as decelerating flows on inclined planes and granular rollwaves.