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

Abstract

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 $|m|=1$ .

Stable production of fluid jets with vanishing diameters via tip streaming

“…In both cases, the solution reaches a saddle-node bifurcation, which corresponds to a turning point of the curve D(C). The solution for the unstable branch beyond the turning point is calculated by imposing the droplet interface length and calculating the corresponding capillary number (Herrada et al 2023). The pinning condition suppresses the central pinching mode, considerably increasing the critical capillary number.…”

“…We use fourth-order finite differences with equally spaced points to discretize the mapped axial direction for Q > 0. Details of the discretization used with intrinsic coordinates for Q = 0 can be found elsewhere (Herrada et al 2023). The MATLAB eigs function is applied to find the eigenfrequencies around a reference value ω 0 .…”

“…This coordinate system is used to analyse the stability of the microjetting mode. The equations in intrinsic coordinates for the closed droplet can be found elsewhere (Herrada, Yu & Stone 2023).…”

“…Details of the discretization used with intrinsic coordinates for can be found elsewhere (Herrada et al. 2023). The MATLAB eigs function is applied to find the eigenfrequencies around a reference value .…”