Rigorous Results in Existence and Selection of Saffman-Taylor Fingers by Kinetic Undercooling

Abstract

The selection of Saffman-Taylor fingers by surface tension has been extensively investigated. In this paper we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele-Shaw cell by small but non-zero kinetic undercooling (${\epsilon }^2$). We rigorously conclude that for relative finger width $\lambda$ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling ${\epsilon }^2\to 0$ if the Stokes multiplier for a relatively simple nonlinear differential equation is zero. This Stokes multiplier $S$ depends on the parameter $\alpha \equiv \frac{2\lambda -1}{(1-\lambda )}{\epsilon }^{-\frac{4}{3}}$ and earlier calculations have shown this to be zero for a discrete set of values of $\alpha$. While this result is similar to that obtained previously for Saffman-Taylor fingers by surface tension, the analysis for the problem with kinetic undercooling exhibits a number of subtleties as pointed out by Chapman and King (2003) [The selection of Saffman-Taylor fingers by kinetic undercooling, Journal of Engineering Mathematics 46, 1-32]. The main subtlety is the behavior of the Stokes lines at the finger tip, where the analysis is complicated by non-analyticity of coefficients in the governing equation.