  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
shear-dependent fluids. The model is applied to perform simulations of viscous
fingering in shear- thinning fluids, and it is found to be capable of
describing the complete crossover from the Newtonian regime at low shear rate
to the strongly shear-thinning regime at high shear rate. The width selection
of a single steady-state finger is studied in detail for a 2-plateaux
shear-thinning law (Carreau law) in both its weakly and strongly shear-thinning
limits, and the results are related to previous analyses. In the strongly
shear-thinning regime a rescaling is found for power-law (Ostwald-de-Waehle)
fluids that allows for a direct comparison between simulations and experiments
without any adjustable parameters, and good agreement is obtained.
