Rigorous Results in Existence and Selection of Saffman-Taylor Fingers by Kinetic Undercooling
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 ($ε^2 $). We rigorously conclude that for relative finger width $λ$ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling $ε^2 ~\rightarrow ~0$ if the Stokes multiplier for a relatively simple nonlinear differential equation is zero. This Stokes multiplier $S$ depends on the parameter $α\equiv \frac{2 λ-1}{(1-λ)}ε^{-\frac{4}{3}} $ and earlier calculations have shown this to be zero for a discrete set of values of $α$. 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.