In micro-structural fluids, the homogeneous shear flow breaks into alternate regions of low and high shear rates (i.e., shear localization), respectively, when the applied shear rate exceeds a critical value and this is known as gradient banding. On the other hand, if the applied shear stress exceeds a critical value, the homogeneous flow separates into bands of different shear stresses (having the same shear rate) along the vorticity (spanwise) direction, leading to stress localization, and the resulting pattern is dubbed vorticity banding. Here we provide a brief overview of our recent work on nonlinear order-parameter theory to describe various pattern formation scenario in a sheared granular fluid, with a specific focus on the vorticity-banding phenomena. The analysis holds for any general constitutive model, but the results are presented for a kinetic-theory constitutive model that holds for rapid granular flows. Our theory predicts that the vorticity banding can occur via supercritical/subcritical pitchfork and subcritical Hopf bifurcations in dilute and dense flows, respectively, resulting in an inhomogeneous state of shear stress and pressure.
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