  A general phenomenological reaction-diffusion model for flow-induced phase
transitions in complex fluids is presented. The model consists of an equation
of motion for a nonconserved composition variable, coupled to a Newtonian
stress relations for the reactant and product species. Multivalued reaction
terms allow for different homogeneous phases to coexist with each other,
resulting in banded composition and shear rate profiles. The one-dimensional
equation of motion is evolved from a random initial state to its final
steady-state. We find that the system chooses banded states over homogeneous
states, depending on the shape of the stress constitutive curve and the
magnitude of the diffusion coefficient. Banding in the flow gradient direction
under shear rate control is observed for shear-thinning transitions, while
banding in the vorticity direction under stress control is observed for
shear-thickening transitions.
