Finite element simulation of nonlinear fluids

This thesis is concerned with new numerical and algorithmic tools for flows with pressure and shear dependent viscosity together with the necessary background of the generalized Navier-Stokes equations. In general, the viscosity can depend on the density, for instance in the case of compressible flow, and if the relation between the pressure and the density is reversible, the viscosity may depend on the pressure. This, however, is not the case for incompressible powder flow where the viscous stresses may change with varying pressure whereas density changes remain negligible. Meanwhile, the viscosity of the flow may change as the shear rate is varied, for instance in Bingham flow. Both dependencies can occur together, something that holds for the so-called nonflowing or slow-flowing materials, namely slow-flowing smaller-sized bulk powder. The Navier-Stokes equations in primitive variables (velocity-pressure) are regarded as the privilege answer to incorporate these phenomena. The modification of the viscous stresses leads to generalized Navier-Stokes equations extending the range of their validity to such flow. The resulting equations are mathematically more complex than the Navier- Stokes equations. From the numerical point of view several problems arise. Firstly, the difficulty of approximating incompressible velocity fields. Secondly, poor conditioning and possible lack of differentiability of the involved nonlinear functions due to the material laws and finally, the eventual domination of convective effects.