Finite element approximation of flow of fluids with shear-rate-and pressure-dependent viscosity - Hirn, Adrian; Lanzendörfer, Martin; Stebel, Jan (IMA journal of numerical analysis, 2012)
In this paper we consider a class of incompressible viscous fluids whose viscosity depends on the shear rate and pressure. We deal with isothermal steady flow and analyse the Galerkin discretization of the corresponding equations. We discuss the existence and uniqueness of discrete solutions and their convergence to the solution of the original problem. In particular, we derive a priori error estimates, which provide optimal rates of convergence with respect to the expected regularity of the solution. Finally, we demonstrate the achieved results by numerical experiments. The fluid models under consideration appear in many practical problems, for instance, in elastohydrodynamic lubrication where very high pressures occur. Here we consider shear-thinning fluid models similar to the power-law/Carreau model. A restricted sublinear dependence of the viscosity on the pressure is allowed. The mathematical theory concerned with the self-consistency of the governing equations has emerged only recently. We adopt the established theory in the context of discrete approximations. To our knowledge, this is the first analysis of the finite element method for fluids with pressure-dependent viscosity. The derived estimates coincide with the optimal error estimates established recently for Carreau-type models, which are covered as a special case.