Analytic theory of the Saffman-Taylor fingers

A complete analytic solution is presented for the selection of symmetric Saffman-Taylor fingers, in the small--surface-tension limit. In a previous paper [Combescot et al., Phys. Rev. Lett. 56, 2036 (1986)] we showed that the selection can be understood in terms of a nonlinear eigenvalue problem that describes the region where ordinary perturbation theory fails to converge. This ``inner'' problem is completely solved by systematically studying the asymptotic series of the (formal) solution of the problem. Two methods are proposed to extract the selection mechanism from this asymptotic series. Numerical results agree very well with other existing numerical results, and, somewhat surprisingly, with the results that we obtained by a WKB solution of the inner problem. Last, we complete our WKB treatment by determining explicitly the additive constant appearing in the WKB formula.