Simple model for heterogeneous flows of yield stress fluids

Various experiments evidence spatial heterogeneities in sheared yield stress fluids. To account for heterogeneities in the velocity gradient direction, we use a simple model corresponding to a nonmonotonic local flow curve and study a simple shear geometry. Different types of boundary conditions are considered. Under controlled macroscopic shear stress Sigma, we find homogeneous flow in the bulk and a hysteretic macroscopic stress-shear-rate curve. Under controlled macroscopic shear rate Gamma, shear banding is predicted within a range of values of Gamma. For small shear rates, stick-slip can also be observed. These qualitative behaviors are robust to changes in the boundary conditions.