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A rigorous numerical solution for steady laminar flows of shear-thinning fluids in rectangular ducts is presented for the first time. We derive universal scaling laws for the effective viscosity that depends on the dimensionless rheological parameters and the effective channel size that is a function of the aspect ratio of the duct. These allow us to generalize the classical formula for the friction factor of Newtonian flows (12/Re) to Carreau fluids flowing in rectangular ducts of any aspect ratio, where the Reynolds number is based on the effective channel size and the effective viscosity.
Barmak et al. (Tue,) studied this question.