A general solution of the Hele-Shaw problem for flows in a channel

The partial derivatives of the function which maps the auxiliary plane into the physical plane are rational functions for all known exact solutions of the problem of fingering in a Hele-Shaw cell. Using methods of complex analysis a general form of the solution is constructed which possesses this property and, unlike existing solutions, is not necessarily symmetrical about the central axis of the cell. The generalized problem of the dynamics of the system is written for the free parameters of the solution and it is shown to be completely integrable.