Gradient and Vorticity Banding Phenomena in a Sheared Granular Fluid

In many complex fluids, including granular systems, 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 shear rate 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.'' Here we outline a Landau-type nonlinear order-parameter theory for both gradient and vorticity banding phenomena in a sheared granular fluid. Our analysis holds for any general constitutive model, but the specific results will be presented for a kinetic-theory constitutive model that holds for rapid granular flows. Our theory predicts that while the vorticity banding [1] can occur via supercritical/subcritical pitchfork and subcritical Hopf bifurcations in dilute and dense flows, respectively, the gradient banding [2] occurs only via pitchfork bifurcations, both resulting in inhomogeneous states.\\[4pt] [1] P. Shukla and M. Alam, (2012, submitted).\\[0pt] [2] J.~Fluid Mech. {\bf 666}, 203 (2011); Phys.~Rev.~Lett. {\bf 100}, 068001 (2009).