  We study shear banding flows in models of wormlike micelles or polymer
solutions, and explore the effects of different boundary conditions for the
viscoelastic stress. These are needed because the equations of motion are
inherently non-local and include ``diffusive'' or square-gradient terms. Using
the diffusive Johnson-Segalman model and a variant of the Rolie-Poly model for
entangled micelles or polymer solutions, we study the interplay between
different boundary conditions and the intrinsic stress gradient imposed by the
flow geometry. We consider prescribed gradient (Neumann) or value (Dirichlet)
of the viscoelastic stress tensor at the boundary, as well as mixed boundary
conditions in which an anchoring strength competes with the gradient
contribution to the stress dynamics. We find that hysteresis during shear rate
sweeps is suppressed if the boundary conditions favor the state that is induced
by the sweep. For example, if the boundaries favor the high shear rate phase
then hysteresis is suppressed at the low shear rate edges of the stress
plateau. If the boundaries favor the low shear rate state, then the high shear
rate band can lie in the center of the flow cell, leading to a three-band
configuration. Sufficiently strong stress gradients due to curved flow
geometries, such as that of cylindrical Couette flow, can convert this to a
two-band state by forcing the high shear rate phase against the wall of higher
stress, and can suppress the hysteresis loop observed during a shear rate
sweep.
