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The selection of Saffman-Taylor fingers by surface tension has beenextensively investigated. In this paper we are concerned with the existence andselection of steadily translating symmetric finger solutions in a Hele-Shawcell by small but non-zero kinetic undercooling ($\epsilon^2 $). We rigorouslyconclude that for relative finger width $\lambda$ near one half, symmetricfinger solutions exist in the asymptotic limit of undercooling $\epsilon^2~\rightarrow ~0$ if the Stokes multiplier for a relatively simple nonlineardifferential equation is zero. This Stokes multiplier $S$ depends on theparameter $\alpha \equiv \frac{2 \lambda-1}{(1-\lambda)}\epsilon^{-\frac{4}{3}} $ and earlier calculations have shownthis to be zero for a discrete set of values of $\alpha$. While this result issimilar to that obtained previously for Saffman-Taylor fingers by surfacetension, the analysis for the problem with kinetic undercooling exhibits anumber of subtleties as pointed out by Chapman and King (2003) [The selectionof Saffman-Taylor fingers by kinetic undercooling, Journal of EngineeringMathematics 46, 1-32]. The main subtlety is the behavior of the Stokes lines atthe finger tip, where the analysis is complicated by non-analyticity ofcoefficients in the governing equation.

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