Pushing a non-Newtonian fluid in a Hele-Shaw cell: From fingers to needles

We make a theoretical study of the finger behavior of a simple fluid displacing a non-Newtonian fluid confined in a Hele–Shaw cell. We study the Saffman–Taylor instability when the viscosity of the displaced fluid changes with shear. Our results predict a decrease of the finger width that goes to zero for large values of the velocity. An analytical treatment allows the predictions of the dynamics in radial growth.