Two finite-difference schemes for calculation of Bingham fluid flows in a cavity

Abstract

Two finite-difference schemes are proposed in the paper for the calculation of a viscous incompressible Bingham fluid flow. The Duvaut–Lions variational inequality is considered as a mathematical model of the medium. One of the finite-difference schemes is a generalization of the well-known MAC scheme on staggered grids. The other scheme uses one grid for approximation of all velocity components and another grid for all components of the rate of deformation tensor and pressure. A special stabilizing term is introduced into this scheme, which provides stability and preserves the second order of convergence of the scheme. Additional consistency conditions for grid operators are introduced, which are necessary for the correctness of the difference method. The numerical solution of the problem of the Bingham fluid flow in a cavity is considered as a model example.