Flow of a shear-thinning fluid in a rectangular duct

We address the problem of steady laminar flow of a shear-thinning fluid in a rectangular duct, which is encountered in many systems, particularly 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 paper, 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 formulas for the pressure gradient and friction factor for various rheological parameters of the fluid and aspect ratios of rectangular ducts.