Analytic solution for two bubbles in a hele-shaw cell

A four-parameter family of exact solutions is reported for two unequal bubbles moving steadily in a Hele-Shaw cell when surface tension is neglected. The solutions are given in closed form in terms of elliptic integrals and are found to be in good agreement with shapes observed in the experiments by Ikeda and Maxworthy [Phys. Rev. A 41, 4367 (1990)]. An analytic solution for a finger with a bubble is also obtained as a special case of the general two-bubble solution. The relevance of these exact solutions for a selection theory for the cases of a finger with a bubble and a pair of bubbles is also briefly discussed.