A stabilized mixed finite element method for the incompressible shear-rate dependent non-Newtonian fluids: Variational Multiscale framework and consistent linearization

This paper presents a stabilized mixed finite element method for shear-rate dependent incompressible fluids. The viscosity of the fluid is considered a function of the second invariant of the rate-of-deformation tensor, thus making the shear-stress shear-strain relation nonlinear. The weak form of the generalized Navier-Stokes equations is cast in the Variational Multiscale (VMS) framework that leads to a two-level description of the problem. Consistent linearization of the fine-scale problem with respect to the fine-scale velocity field and the use of bubble functions to expand the fine-scale trial and test functions lead to an analytical expression for the fine-scale velocity along with a definition of the stabilization tensor τ. The ensuing nonlinear stabilized form is presented and the consistent tangent tensor is derived. Numerical convergence of the proposed method on structured and unstructured meshes that are composed of linear triangles and bilinear quadrilaterals are presented. Shear-thinning and shear-thickening effects are investigated via the backward facing step problem and the effects of geometric parameters on the flow characteristics are highlighted. Time dependent features are investigated via the transient vortex-shedding problem and the accuracy and stability properties of the new method are shown.