  Flow behavior of a single-component yield stress fluid is addressed on the
hydrodynamic level. A basic ingredient of the model is a coupling between
fluctuations of density and velocity gradient via a Herschel-Bulkley-type
constitutive model. Focusing on the limit of low shear rates and high
densities, the model approximates well---but is not limited to---gently sheared
hard sphere colloidal glasses, where solvent effects are negligible. A detailed
analysis of the linearized hydrodynamic equations for fluctuations and the
resulting cubic dispersion relation reveals the existence of a range of
densities and shear rates with growing flow heterogeneity. In this regime,
after an initial transient, the velocity and density fields monotonically reach
a spatially inhomogeneous stationary profile, where regions of high shear rate
and low density coexist with regions of low shear rate and high density. The
steady state is thus maintained by a competition between shear-induced
enhancement of density inhomogeneities and relaxation via overdamped sound
waves. An analysis of the mechanical equilibrium condition provides a criterion
for the existence of steady state solutions. The dynamical evolution of the
system is discussed in detail for various boundary conditions, imposing either
a constant velocity, shear rate, or stress at the walls.
