Selection Mechanism in Non-Newtonian Saffman–Taylor Fingers

.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, \(\Lambda\), and the dimensionless group of parameters containing the viscosity and surface tension, \(\nu\), has the form \(\Lambda \sim \frac{1}{2} - \mathcal{O}(\nu^{-1/2})\) for the shear thinning case and \(\Lambda \sim \frac{1}{2} + \mathcal{O}(\nu^{2/(4-n)})\) for the shear thickening case, in the limit of small \(\nu\). 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, \(n=1\).Keywordspattern formationfingering instabilityrheologyviscoelastic solutionMSC codes76A0576D4576E17