Nonhomogeneous Incompressible Herschel–Bulkley Fluid Flows Between Two Eccentric Cylinders

Abstract

The equations for the nonhomogeneous incompressible Herschel–Bulkley fluid are considered and existence of a weak solution is proved for a boundary-value problem which describes three-dimensional flows between two eccentric cylinders when in each two-dimensional cross-section annulus the flow characteristics are the same. The rheology of such a fluid is defined by a yield stress τ* and a discontinuous stress-strain law. A fluid volume stiffens if its local stresses do not exceed τ*, and a fluid behaves like a nonlinear 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 fluids. We do not apply the variational inequality but make use of an approximation of the generalized Bingham fluid by a non-Newtonian fluid with a continuous constitutive law.