We implemented an adaptively refined least-squares finite element
approach for the Navier-Stokes equations that govern generalized
Newtonian fluid flows using the Carreau model. To capture the flow region, we developed an
adaptive mesh refinement approach based on the least-squares method.
The generated refined grids agree well with the physical attributes
of the flows. We also proved that the least-squares approximation
converges to the linearized versions solutions of the Carreau model
at the best possible rate. Model problems considered in the study are the flow past
a planar channel and 4-to-1 contraction problems.
We presented the numerical results of the model problems, revealing the
efficiency of the proposed scheme, and investigated the physical
parameter effects.