Global stability analysis of bubbles rising in a vertical capillary with an external flow

We study the linear stability of bubbles in a capillary tube under external flow. Yu et al. (J. Fluid Mech., vol. 911, 2021, pp. 1-19) showed that a rich variety of bubble dynamics occurs when a downward external flow is applied, opposing the buoyancy-driven ascent of the bubble. They found experimentally and numerically the existence of two branches of solutions that overlap over a finite range of the capillary number of the downward external flow in cases where the Reynolds number is small and the Bond number is larger than the critical value for which the bubble can rise spontaneously (Bretherton, J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166-188). Furthermore, inertialess, symmetry-breaking steady-state shapes were found as the bubble transits near the tipping points of the solution branches. In this work, using steady axisymmetric simulations, we show that the reported multiplicity of solutions can be described using bifurcation diagrams with three branches of steady axisymmetric solutions and two limit points. The linear global stability analysis of the different branches of the stationary axisymmetric solutions demonstrates that the symmetry breaking is due to the development of three-dimensional instabilities with azimuthal wavenumber.