A computational study of rising plane Taylor bubbles

The problem of a plane bubble rising in a 2-D tube is revisited using Birkh off's formulation developed in 1957, The equations in this formulation have a one parameter (Froude number F) family of solutions which are divided in to three regimes characterized by distinct topologies at the apex. These eq uations are solved numerically using a conventional series representation m ethod and Newton's iterations. This numerical method fails for values of F in a range which contains the transition points. In this paper, it is demon strated through careful numerical computations how and why this method fail s. We also analyze the series and provide estimates of the transition point s. This strategy of estimating the transition points can be used for some p roblems where the conventional series representation method fails because i t does not adequately account for changes in the nature of the singularity that takes place as these transition points are approached in the parameter space. Furthermore, existence of two new critical Froude numbers is demons trated numerically. We further show that the previous results on this probl em have been incomplete by leaving out the characterization of the topology at the apex of the bubbles for values of F in the regime 0.234 < F < 0.357 8. We also resolve this issue in this paper. (C) 2000 Academic Press.