Kinetics of the shear banding instability in startup flows

Motivated by recent light scattering experiments on semidilute wormlike micelles, we study the early stages of the shear banding instability using the nonlocal Johnson-Segalman model with a “two-fluid” coupling of flow to micellar concentration. We perform a linear stability analysis for coupled fluctuations in shear rate $\dot{\gamma },$ micellar strain $\underset{¯}{\underset{¯}{W}},$ and concentration $\phi$ about an initially homogeneous state. This resembles the Cahn-Hilliard (CH) analysis of fluid-fluid demixing (although we discuss important differences). First, assuming the initial state to lie on the intrinsic constitutive curve, we calculate the “spinodal” onset of instability in sweeps along this curve. We then consider start-up “quenches” into the unstable region. Here the instability in general occurs before the intrinsic constitutive curve can be attained, so we analyze the fluctuations with respect to the time-dependent start-up flow. We calculate the selected length and time scales at which inhomogeneity first emerges. When the coupling between flow and concentration is switched off, fluctuations in the “mechanical variables” $\dot{\gamma }$ and $\underset{¯}{\underset{¯}{W}}$ are independent of those in $\phi ,$ and are unstable when the intrinsic constitutive curve has negative slope; but no length scale is selected. Coupling to the concentration enhances this instability at short length scales, thereby selecting a length scale, consistent with the recent light scattering experiments. The spinodal region is then broadened by an extent that increases with proximity to an underlying (zero-shear) CH fluid-fluid $\left(\phi \right)$ demixing instability. Far from demixing, the broadening is slight and the instability is still mechanically dominated (by $\delta \dot{\gamma }$ and $\delta \underset{¯}{\underset{¯}{W}})$ with only small $\delta \phi .$ Close to demixing, instability sets in at a very low shear rate, where it is dominated instead by $\delta \phi .$ In this way, the model captures a smooth crossover from shear banding instabilities that are perturbed by concentration coupling to demixing instabilities that are induced by shear.