A nonlinear weighted least-squares finite element method for the Carreau-Yasuda non-Newtonian model

We study a nonlinear weighted least-squares finite element method for the Navier-Stokes equations governing non-Newtonian fluid flows by using the Carreau-Yasuda model. The Carreau-Yasuda model is used to describe the shear-thinning behavior of blood. We prove that the least-squares approximation converges to linearized solutions of the non-Newtonian model at the optimal rate. By using continuous piecewise linear finite element spaces for all variables and by appropriately adjusting the nonlinear weighting function, we obtain optimal L2-norm error convergence rates in all variables. Numerical results are given for a Carreau fluid in the 4-to-1 contraction problem, revealing the shear-thinning behavior. The physical parameter effects are also investigated.