Rising bubbles in a two-dimensional tube with surface tension

The motion of a two-dimensional bubble rising at a constant velocity U in a tube of width h is considered. The acceleration of gravity g and the surface tension T are included in the dynamic boundary condition. For T=0, the results of Garabedian and Vanden-Broeck show that a solution exists for all values of the Froude number F=U/(gh)1/2 smaller than a critical value Fc∼0.36. In this paper accurate numerical solutions with T≠0 are computed by series truncations. The results show that for each value of T≠0, there exists a countably infinite number of solutions. Each of these solutions corresponds to a different value of F. As T tends to zero, all these solutions approach a unique limiting solution characterized by F=F*∼0.23. Therefore, the degeneracy associated with T=0 is removed by including the effect of surface tension. In addition, the profile corresponding to F=F* is found to be in good agreement with experimental data.