Viscous fingering in a shear-thinning fluid

We study the Saffman-Taylor instability in a rectangular Hele-Shaw cell. The driven fluid is a dilute (or semidilute) polymer solution, with a viscosity that exhibits shear thinning. Other non-Newtonian properties such as elastic effects are negligible under the present experimental conditions; the system thus allows for separate investigation of the influence of shear thinning on the instability. The experiments show that, for weak shear-thinning, the results for the width of the fingers as a function of the capillary number collapse onto the universal curve for Newtonian fluids, provided the shear-thinning viscosity is used to calculate the capillary number. For stronger shear thinning, narrower fingers are found. The experiment allows also for a study of the applicability of Darcy's law to shear thinning fluids. For Newtonian fluids, this law gives the finger velocity as a function of the pressure gradient. For weakly shear-thinning fluids, we find that an effective Darcy's law, in which the constant viscosity is replaced by the shear-thinning viscosity, gives good agreement with the experiments. For stronger shear thinning, the predictions from the effective Darcy's law deteriorate. Satisfactory agreement with experimental data can be obtained when using a "shear-thinning" Darcy's law, which can be derived using a power law model for the shear rate dependence of the viscosity. (C) 2000 American Institute of Physics. [S1070-6631(00)01702-5].