Nonhomogeneous incompressible Bingham viscoplastic as a limit of nonlinear fluids

The equations for the nonhomogeneous incompressible Bingham fluid are considered and existence of a weak solution is proved for a two-dimensional boundary-value problem with periodic boundary conditions. The rheology of such a fluid is defined by an yield stress t* and a discontinuous stress-strain law. A fluid volume stiffens if its local stresses do not exceed t*, and a fluid behaves like a non-linear fluid otherwise. The flow equations are formulated in the stress-velocity-density-pressure setting. Our approach is different from that of Duvaut-Lions developed for the classical Bingham viscoplastic. We do not apply the variational inequality but make use an approximation of the generalized Bingham fluid by a non-Newtonian fluid with a continuous constitutive law. (c) 2006 Elsevier B.V. All rights reserved.