Flow instabilities and shear banding in complex fluids

Shear banding instabilities are seen in complex fluids such as wormlike micellar surfactant solutions, which separate into coexisting bands of differing viscosity when subject to shear. This effect can be viewed as a driven, non-equilibrium phase transition. Indeed, much of the phenomenology mirrors that of conventional, equilibrium phase separation (oil and water demixing, for example). In steady state, this analogy can be pursued to construct flow phase diagrams. Likewise, the kinetics of shear band formation show unstable and metastable regimes strongly resembling those of conventional phase separation. We pursue this analogy to study theoretically the early stages of band formation in the unstable regime, by analogy with the Cahn-Hilliard theory for equilibrium fluid-fluid demixing. By coupling a simple shear banding instability to a ''two-fluid model'' for fluctuations in surfactant concentration, we predict an initial length scale at which banding inhomogeneity first emerges in unstable startup flows. We also demonstrate a smooth crossover in types of instability, from ''mechanical shear banding instabilities, perturbed by concentration-coupling'' to ''shear induced demixing instabilities''. These are distinguished according to the nature of the most unstable ''eigenvector'', which is measurable experimentally.