Pattern selection and tip perturbations in the Saffman-Taylor problem

We present an analytic approach to the Saffman-Taylor problem of predicting the width of a viscous finger in a Hele-Shaw cell. Our first purpose is to provide a systematic description of the way in which the singular perturbation introduced by capillary forces leads to a solvability mechanism for pattern selection. We then show how recent experimental observations by Couder e al. may be interpreted in terms suggested by this mechanism. In particular, we show that the remarkable changes in pattern selection induced by bubbles trapped at the tips of fingers can be explained by assuming that the perturbation caused by a bubble is effectively equivalent to a positive opening angle at the tip.