Bingham fluid with viscosity and yield stress depending on the density

In this note we present a mathematical model for the channel flow of a Bingham fluid in which the rheological parameters are not constant. We consider two different situations: (i) the yield stress and Bingham viscosity depend on the density which is not constant; (ii) the yield stress and Bingham viscosity depend on a parameter that is not spatially uniform but the density of the fluid is constant. Model (i) can be used to describe the mud flow in coastal and estuarine waters. Model (ii) can be used to describe blood flow in vessels. After formulating the mathematical model, which turns out to be a moving boundary problem, we solve the problem numerically by means of an implicit finite-difference scheme, determining the evolution of the yield surface (free boundary). We perform numerical simulations for cases (i) and (ii) using experimental data that are available in the literature.