A generalized lever rule for shear banding of yield stress fluids.

We study the local dynamics of a yield stress fluid that shows a pronounced non-monotonic flow curve. Such a mechanically unstable behavior generally leads to the coexistence of a flowing band with an arrested region below a critical shear rate $\dot \gamma_c$. Combining ultrasound velocimetry with standard rheometry, we discover an original shear banding scenario in the decreasing branch of the flow curve, in which the flow profile of the flowing band is set by the applied shear rate $\dot \gamma$ instead of $\dot \gamma_c$. As a consequence, the material slips at the walls with a velocity that shows a non-trivial dependence on $\dot \gamma$. To capture our experimental observations, we propose a differential version of the so-called lever rule that describes the extent of the flowing band and the evolution of wall slip velocity with shear rate. Our approach therefore provides a generalized lever rule where constraints imposed in the classical shear banding scenario can be relaxed, while wall slip plays the role of an additional degree of freedom.