Prolegomena to variational inequalities and numerical schemes for compressible viscoplastic fluids

Firstly, a summary of the development of the constitutive equation for an incompressible Bingham fluid, the variational inequality and an operator-splitting numerical method for the solution of isothermal flow problems is presented. Next, a variational inequality is derived for compressible viscoplastic fluids in which the viscosity and yield stress depend on the pressure, temperature and the three invariants of the first Rivlin–Ericksen tensor, and inertial effects are present. Based on a known version for compressible Newtonian fluids, an extension of the operator-splitting scheme to the flows of compressible viscoplastic fluids, when inertia and thermal effects are manifest, is proposed. Finally, this scheme is employed to examine the isothermal flow of a Bingham material in a square cavity when the fluid is slightly compressible, with its density depending linearly or exponentially on the pressure. The results are compared with those for an incompressible fluid for small Bingham numbers.