A bi-projection scheme for unsteady visco-plastic Bingham medium flows

A new numerical scheme to compute isothermal and unsteady flow of an incompressible viscoplastic Bingham medium will be presented. The main difficulty, for both theoretical and numerical approaches, is due to the non-differentiability of the plastic part of the stress tensor in regions where the rate-of-strain tensor vanishes. This is handled by reformulating the definition of the plastic stress tensor in terms of a projection. A new time scheme, based on the classical incremental projection method for the Newtonian Navier-Stokes equations, is proposed. The plastic tensor is treated implicitly in the first sub-step of the projection scheme and is computed by using a fixed point procedure. A pseudo-time relaxation is added into the Bingham projection whose effect is to ensure a geometric convergence of the fixed point algorithm. Stability and error analyses of the numerical scheme will be shown. Numerical results, obtained on the well-known two-dimensional lid-driven cavity test case, will be detailed for Reynolds number up to 10 000 and Bingham number up to 100.