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 (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)-> 0 if the Stokes multiplier for a relatively simple non-linear differential equation is zero. This Stokes multiplier S depends on the parameter alpha 2 lambda - 1/(1 - lambda)epsilon(-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 behaviour of the Stokes lines at the finger tip, where the analysis is complicated by non-analyticity of coefficients in the governing equation.