Saffman-Taylor Instabilities in Shear-Thinning Fluids

In this talk, we consider a simple mathematical model of the radial injection of a shear-thinning fluid, initially occupying a circular region of some initial injection radius, which displaces a viscous Newtonian fluid within a Hele-Shaw cell. For Newtonian fluids, Saffman-Taylor instabilities are known to develop when the displacing fluid is less viscous than the displaced fluid. However, when the displacing fluid is shear-thinning, its viscosity depends on radial distance from the source. Using a linear stability analysis, we show that there exists a critical radius where for injection radii larger than this critical value, the flow remains stable. In addition, we derive a series of critical radii for each mode corresponding to a sinusoidal perturbation of the fluid-fluid interface. The theoretical results are compared to a set of Hele-Shaw laboratory experiments in which a shear-thinning fluid called Xanthan gum is injected radially into a small gap between two plates, displacing a viscous Newtonian silicone oil. This comparison shows significant discrepancies between theory and experiment, and we delve into this problem further to locate the missing physics.