A Van der Waals-Cahn-Hilliard regularization of granular instability via dissipation potentials

This talk deals with the viscoplastic Hadamard (short-wavelength) instability of the so-called μ(I) model of dense rapidly-sheared granular flow, as reported recently by Barker et al. (2015, J. Fluid Mech, 779, 794-818). As explained in a subsequent paper by Goddard & Lee (2017, J. Fluid Mech., in press), one achieves stabilizing effects from higher velocity gradients by means of an enhanced-continuum model based on the dissipative analog of the Van der Waals-Cahn-Hilliard equation of equilibrium thermodynamics. This model involves a dissipative hyper-stress, with surface viscosity arising as counterpart of elastic surface tension. This allows for a description of diffuse shear bands as the rough analog of the diffuse interfaces of equilibrium thermodynamics. The later paper also presents a more comprehensive linear stability analysis, including convective (Kelvin) wave-vector stretching by the base flow that leads to asymptotic stabilization of the non-convective instability found by Barker et al. This suggests a theoretical connection between their non-convective instability and the loss of generalized ellipticity in the quasi-static field equations. Apart from the theoretical interest, the present work may suggest stratagems for the otherwise difficult numerical simulation of continuum field equations involving Hadamard-unstable viscoplasticity.