The effects of surface tension and tube inclination on a two-dimensional rising bubble

The effects of surface tension σ and tube inclination β on the Froude number Fr of a large bubble rising in a two-dimensional duct is considered. It is found that there exists either one (for small σ and β > 0°) or a set (for any σ and β = 0°) of Fr -values for which the bubble has a continuous derivative at the nose. By selecting either this single Fr (or the maximum of the set), we obtain solutions in excellent agreement with both theoretical predictions and experimental results.