Fingering in a Hele-Shaw Cell as an obstacle-like problem

By casting the Saffman-Taylor problem for the Hele -Shaw fingering, into a dam - like problem, gives new insight on the ? = 1/2 selection mechanism. In particular the thrust of the fluid across the x-section of the channel at the nose of a symmetric finger is maximized for ? = 1/2. This and other results are derived from a precise asymptotic expansion of the solution for x -> ?. The latter is made possible by a novel variational - like setting of the problem, formally similar to an obstacle problem. We establish that such variational - like solutions exist even for quite irregular fingers. These solutions are shown to be unique up to a translation of the x-axis.