A spherical-cap bubble moving at terminal velocity in a viscous liquid

Abstract

The nonlinear Navier–Stokes equations governing steady, laminar, axisymmetric flow past a deformable bubble are solved by the Galerkin finite-element method simultaneously with a set of elliptic partial differential equations governing boundary-fitted mesh. For Reynolds number 20 ≤ Re ≤ 500, numerical solutions of spherical-cap bubbles are obtained at capillary number Ca = 1. Increasing Ca to 2 leads to a highly curved, cusp-like bubble rim that seems to correspond to skirt formation. The computed steady, axisymmetric spherical-cap bubbles with closed, laminar wakes compare reasonably with the available experimental results, especially for Re ≤ 100. By exploring the parameter space (for Re ≤ 200), a sufficient condition for steady axisymmetric solutions of bubbles with the spherical-cap shape is found to be roughly Ca > 0.4. The basic characteristics of spherical-cap bubbles of Ca ≥ 0.5, for a given Re ≥ 50, are found to be almost independent of the value of Ca (or Weber number We ≡ Re Ca). At a fixed Re ≥ 50, continuation by increasing Ca (or We) from a spherical bubble solution cannot lead to solutions of spherical-cap bubbles, but rather to a turning point at We slightly greater than 10 where the solution branch folds back to reduced values of Ca (or We). Yet continuation by reducing Ca (or We) from a spherical-cap bubble solution cannot arrive at a spherical bubble solution for Re ≥ 50, but rather at solutions with bubbles having more complicated shapes such as a sombrero, etc. Without thorough examinations of the solution stability, multiple steady axisymmetric solutions are shown to exist in the parameter space for a given set of parameters.