Cahn-Hilliard Regularization of the "mu(I)" Rheology

Recently Barker et al. [J. Fluid Mech. 779 (2015) 794-818] have shown that the popular $\mu(I)$ model for the viscoplasticity of granular media is ill-posed, exhibiting short wave-length instabilities of the Hadamard variety. As one possible regularization of the model, we employ the dissipative analog of the classical Cahn-Hilliard (CH) model, with dissipation potential given by: $\psi(\nabla{\bf v}, \nabla\nabla{\bf v}) = \psi_0({\bf D}) + k||\nabla\nabla{\bf v}||^2$, with ${\bf D} = \rm{Sym}(\nabla{\bf v})$ and $ k>0$, with stress for the standard $\mu(I)$ model given by $\partial\psi_0/\partial{\bf D}$, and with hyperstress given by $\partial\psi/\partial\nabla\nabla{\bf v}$. Following the linear-stability analysis of Barker et al. of the momentum balance and continuity equation, we obtain a modification of their dispersion relation giving growth rate in terms of spatial wave number. It is found that the higher-gradient terms in the CH model lead to a large wave number cut-off of the instability, so that the model provides a possibly useful regularization of the $\mu (I)$ model.