Selection Mechanism in Non-Newtonian Saffman–Taylor Fingers

Abstract

We present an analytical approach to the problem of predicting the finger width of a simple fluid driving a non-Newtonian (power-law) fluid. Our analysis is based on the Wentzel–Kramers–Brillouin approximation, by representing the deviation from the Newtonian viscosity as a singular perturbation in a parameter, leading to a solvability condition at the finger tip, which selects a unique finger width from the family of solutions. We find that the relation between the dimensionless finger width, , and the dimensionless group of parameters containing the viscosity and surface tension, , has the form for the shear thinning case and for the shear thickening case, in the limit of small . This theoretical estimate is compared with the existing experimental, finger width data as well as the one computed with the linearized model, and a good agreement is found near the power-law exponent, .