  We address the problem of steady laminar flow of a shear-thinning fluid in
rectangular ducts, which is encountered in many systems, in particular, in
microfluidic and biomedical devices. However, an exact solution for the flow of
non-Newtonian fluids that considers a realistic shear-thinning rheological
behavior is not available in the literature. In this study, an accurate
solution for the case of Carreau fluid is obtained and investigated
numerically. The numerical solution allows us to analyze the effects of the
fluid rheology and the aspect ratio of the rectangular duct on the velocity
field and pressure gradient that drives the flow. The relationship between the
pressure gradient and the Carreau number is found to follow the rheological
curve of the shear-thinning fluid. The analysis shows that the fluid rheology
and the aspect ratio have independent contributions to the integral flow
characteristics. Moreover, separate consideration of these contributions allows
us to arrive at universal scaling and general formulae for the pressure
gradient and friction factor for various rheological parameters of the fluid
and aspect ratios of rectangular ducts.
