Numerical analysis of solution branchings of the Hele-Shaw problem for steadily moving bubble

It is well known that a free boundary problem for the bubble steady motion in a Hele-Shaw cell is nonregular in the limit of zero surface tension. Through this nonregularity a degeneracy of the solution appears: for a given area of the bubble P. G. Saffman and G. I. Taylor found a family of exact solutions. S. Tanveer showed that the solution degeneracy is removed by taking into account the effect of surface tension, but he gave no clear mathematical explanation for such removal. In addition, S. Tanveer found several branches of the bubble solution. To find all solution branches, we have defined a modified Hele-Shaw problem by analogy with J.-M. Vanden-Broeck's approach to the problem of steady fingers. Numerical solution of this modified problem has been found. A unique solution has been obtained for a given area of the bubble. This solution coincides with the main branch of S. Tanveer's solution. No other solution branches have been found. An explanation for this disagreement is that S. Tanveer's solutions may include the nonunivalent physical plane, while we have found only univalent solutions. In this paper we give a clear explanation for the reasons of the degeneracy removal when surface tension is introduced. In the physical plane the flow domain has two characteristic points: at infinity on the left and at infinity on the right, at which the domain width is assigned. Both these values have to be defined by single integration of the main boundary equation. With only one integration constant the two conditions cannot be satisfied because the bubble contour shape has no fore and aft symmetry. Thus a solvability condition appears.