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We study shear banding flows in models of wormlike micelles or polymersolutions, and explore the effects of different boundary conditions for theviscoelastic stress. These are needed because the equations of motion areinherently non-local and include ``diffusive'' or square-gradient terms. Usingthe diffusive Johnson-Segalman model and a variant of the Rolie-Poly model forentangled micelles or polymer solutions, we study the interplay betweendifferent boundary conditions and the intrinsic stress gradient imposed by theflow geometry. We consider prescribed gradient (Neumann) or value (Dirichlet)of the viscoelastic stress tensor at the boundary, as well as mixed boundaryconditions in which an anchoring strength competes with the gradientcontribution to the stress dynamics. We find that hysteresis during shear ratesweeps is suppressed if the boundary conditions favor the state that is inducedby the sweep. For example, if the boundaries favor the high shear rate phasethen hysteresis is suppressed at the low shear rate edges of the stressplateau. If the boundaries favor the low shear rate state, then the high shearrate band can lie in the center of the flow cell, leading to a three-bandconfiguration. Sufficiently strong stress gradients due to curved flowgeometries, such as that of cylindrical Couette flow, can convert this to atwo-band state by forcing the high shear rate phase against the wall of higherstress, and can suppress the hysteresis loop observed during a shear ratesweep.

The interplay between boundary conditions and flow geometries in shear banding: hysteresis, band configurations, and surface transitions