  This article deals with the Hadamard instability of the so-called $\mu(I)$
model of dense rapidly-sheared granular flow, as reported recently by Barker et
al. (2015,this journal, ${\bf 779}$, 794-818). The present paper presents a
more comprehensive study of the linear stability of planar simple shearing and
pure shearing flows, with account taken of convective Kelvin wave-vector
stretching by the base flow. We provide a closed form solution for the linear
stability problem and show that wave-vector stretching leads to asymptotic
stabilization of the non-convective instability found by Barker et al. We also
explore the stabilizing effects of higher velocity gradients achieved by an
enhanced-continuum model based on a dissipative analog of the van der
Waals-Cahn-Hilliard equation of equilibrium thermodynamics. This model involves
a dissipative hyper-stress, as the analog of a special Korteweg stress, with
surface viscosity representing the counterpart of elastic surface tension.
Based on the enhanced continuum model, we also present a model of steady shear
bands and their non-linear stability against parallel shearing. Finally, we
propose a theoretical connection between the non-convective instability of
Barker et al. 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 numerical simulation of continuum field equations involving
the $\mu(I)$ rheology and variants thereof.
