Stagnation point flow of a viscoplastic fluid

Stagnation points occur in many configurations, such as flow around blunt objects, flow through a T-junction, and squeeze flow between plates. For viscoplastic fluids, vanishing strain rate at a stagnation point results in regions of stagnant unyielded fluid, or "plugs". We explore the planar flow of a Bingham fluid in the neighbourhood of a stagnation point in a general flow configuration. When the Bingham number is small, this local problem reduces to the prototypical problem of stagnating flow against an infinite planar boundary, varying only with the stagnation angle with which the flow approaches the boundary. We compute numerical solutions of this idealised problem, using the augmented-Lagrangian algorithm, and determine the geometry of the stagnation-point plug as a function of this stagnation angle. As the angle decreases, the plug becomes larger, is elongated in the flow direction, and becomes increasingly asymmetric. However, for all angles, the plug features a right-angle at its vertex, a result that we demonstrate numerically and prove direct from the model equations. We also show how local stagnation plugs are embedded in global flows, illustrating the results from the specific case studies of recirculating flow in a sharp corner and uniform flow around an elliptic cylinder.