On steady solutions of Hele-Shaw problem

The problem of mathematical modeling of two phase flow in a Hele-Shaw cell is observed. Our study focuses on steadily bubbles and fingers moving in a Hele-Shaw channel in the absence of surface tension. The formalization of hodograph transformation method is suggested. The formalization allows to restore the known exact solutions, to construct new solutions, and to estimate the degeneracy of solution without need of constructing full solution. In particular the degree of degeneracy of the single bubble solution is derived and the single bubble solution without fore and aft symmetry is proved to not exist.