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.

Section (1) is devoted to a discussion of the model-fitting problem, which finds its explicit solution in equation (1.13). In section (2) the maximum likelihood, (ML), estimates of the model parameters are investigated, and for the class of series considered shown to possess the same optimum properties as in the case of independent series. Next, the covariance matrix of the parameter estimates is expressed in terms of the spectral function of the generating process (eq. 3.7). The last section is concerned with certain working approximations to the ML statistics.

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We prove curvature estimates for general curvature functions. As an application we show the existence of closed, strictly convex hypersurfaces with prescribed curvature $F$, where the defining cone of $F$ is $\C_+$. $F$ is only assumed to be monotone, symmetric, homogeneous of degree $1$, concave and of class $C^{m,\al}$, $m\ge4$.