Rising Bubbles in a Two-Dimensional Tube with Surface Tension

摘要

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) raised to the 1/2 power smaller than a critical value Fc approximately 0.36. In this paper accurate numerical solutions with T unequal to 0 are computed by series truncations. The results show that for each value of T unequal to 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 approximately = 0.23. Therefore, the degenerancy 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. (Author)