An iterative method for computing the flows of a viscoplastic Bingham medium

A finite-difference scheme for computing the planar flows of a viscoplastic Bingham medium is considered. The Duvaut–Lions variational inequality is used as a mathematical model of the medium. The velocity components are approximated on the main mesh, whereas the pressure and the components of the strain-rate and stress tensors are approximated on the nodes of the semi-staggered grid. It is shown that the Uzawa-like iterative method used to solve the variational inequality requires a special adaptation in the case of the discrete problem. As a model example, the numerical solution of the lid-driven cavity problem for a viscoplastic medium is discussed. The obtained results are compared with the known ones. The paper was prepared by E.A. Muravleva when visiting Max Planck Institute for Mathematics in the Sciences, 04103, Leipzig, Germany.