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 τ∗τ∗ and a discontinuous stress–strain law. A fluid volume stiffens if its local stresses do not exceed τ∗τ∗, 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.