Among all problems of pattern selection the one posed by the Saffman-Taylor finger is yet unsolved, although numerics and experiments indicate the very simple property that the relative width of the finger tends to {} in the low-surface-tension limit. We explain this by performing an expansion beyond all orders leading to the formulation of a nonlinear eigenvalue problem with a discrete set of solutions. In particular we predict that the relative width of the finger tends to {} as the surface tension to the power ⅔.
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Combescot et al. (1986) studied this question.