Phase-field simulations of viscous fingering in shear-thinning fluids

A phase-field model for the Hele-Shaw flow of non-Newtonian fluids is developed. It extends a previous model for Newtonian fluids to a wide range of fluids with a shear-dependent viscosity. The model is applied to simulate viscous fingering in shear-thinning fluids and found to capture the complete crossover from the Newtonian regime at low-shear rate to the strongly shear-thinning regime. The width selection of a single steady-state finger is studied in detail for a two-plateau shear-thinning law (Carreau's law) in both its weakly and strongly shear-thinning limits, and the results are related to the previous analyses. For power-law (Ostwald-de Waele) fluids in the strongly shear-thinning regime, good agreement with experimental data from the literature is obtained.