Saffman-Taylor fingering instabilities in shear-thinning fluids displacing Newtonian fluids

We investigate the radial injection of a shear-thinning fluid, initially occupying a disk of a given radius, into the narrow gap between two parallel plates, displacing a viscous Newtonian fluid. In the Newtonian analogue, the Saffman-Taylor fingering instability occurs when a more viscous fluid is displaced by a less viscous one. Further complications arise when the displacing fluid is shear-thinning because its viscosity increases with radial distance from the source. A linear stability analysis, in which sinusoidal perturbations to the interface between the two fluids are considered, shows that each disturbance, characterised by a mode number, has a corresponding radius that governs the stability of that mode. These radii increase with mode number, approaching a critical radius. If the radius of the initial disk is greater than this critical radius, all modes are stable and the fingering instability is suppressed. This critical radius depends on the ratio of the viscosity of the displaced Newtonian fluid and another viscosity corresponding to the shear-thinning fluid, which we will refer to as the effective viscosity. The theoretical results are contrasted with observations from laboratory experiments. Significant discrepancies between theory and experiment suggest that the underlying physics governing stability is not yet fully understood. We consider various improvements to the model, and by once again comparing theory and experiment, we show that surface tension does not appear to play a pivotal role, but that wall-slip potentially leads to substantial changes in the theoretical predictions. • Instabilities in shear-thinning fluids displacing Newtonian fluids are investigated. • Sinusoidal interfacial disturbances are examined using a linear stability analysis. • Critical injection radii are derived for each mode. • The theoretical predictions are compared to laboratory experiments. • Further improvements to the mathematical model are explored.