Saffman-Taylor instability of shear thinning fluids

A theoretical study of the Saffman-Taylor instability criterion in a Hele-Shaw cell when an inviscid fluid pushes a shear thinning fluid, is presented. The shear thinning fluid model consists of three independent fluids, two of which are Newtonian while the viscosity of the third is linearly dependent of the shear rate. The corresponding Darcy’s law is computed, and then the dispersion relation is obtained in a linear perturbative analysis. We show how the dispersion relation characteristic of the Saffman-Taylor instability is controlled by a “shear thinning characteristic number” that takes into account the rheological behavior of the shear thinning fluid in the Hele-Shaw cell. The model is discussed compared with the experimental pattern formation.