Finite element approximation of steady flows of generalized Newtonian fluids with concentration-dependent power-law index
Provides a rigorous numerical analysis framework for a complex fluid model, but the results are theoretical and incremental for the numerical analysis community.
The paper develops and analyzes a finite element approximation for a system of PDEs describing chemically reacting generalized Newtonian fluids with concentration-dependent viscosity. The method converges to weak solutions of the original problem.
We consider a system of nonlinear partial differential equations describing the motion of an 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, 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 a mathematical analysis of a finite element approximation of this model. We formulate a regularization of the model by introducing an additional term in the conservation-of-momentum equation and construct a finite element approximation of the regularized system. We show the convergence of the finite element method to a weak solution of the regularized model and prove that weak solutions of the regularized problem converge to a weak solution of the original problem.