Universal Unfolding of Pitchfork Bifurcations and Shear-Band Formation in Rapid Granular Couette Flow

A numerical bifurcation analysis is carried out to understand the role of gravity on the shear-band formation in rapid granular Couette flow. At {\it low} shear rates, there is a unique solution with a {\it plug} near the bottom wall and a {\it shear-layer} near the top-wall; this solution mirrors typical shearbanding-type profiles in earth-bound shear-cell experiments. Interestingly, a {\it stable} plug near the top-wall is also a solution of these equations at {\it high} shear rates; there is a multitude of other plugged states, with the plugs being located in an ordered fashion within the Couette gap. The origin of such shearbanding solutions is tied to the spontaneous symmetry-breaking {\it shearbanding} instabilities of the gravity-free uniform shear flow, leading to both subcritical and supercritical pitchfork bifurcations. In the language of singularity theory, we have established that this bifurcation problem admits {\it universal unfolding} of pitchfork bifurcations.