Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index

We consider a system of nonlinear partial differential equations, modeling the motion of a viscous incompressible chemically reacting generalized Newtonian fluid in three space dimensions. The governing system consists of a steady convection-diffusion equation, for the concentration, and a generalized steady power-law-type fluid flow model, for the velocity and the pressure of the fluid, where the viscosity depends on both the shear-rate and the concentration through a concentration-dependent power-law index. The aim of the paper is to perform the mathematical analysis of a finite element approximation of this model. We consider a regularization of the model by introducing an additional term in the momentum equation and construct a finite element approximation of the regularized system. First, the convergence of the finite element method to a weak solution of the regularized model is shown, and we then prove that weak solutions of the regularized problem converge to a weak solution of the original problem.