Non-unique bubble dynamics in a vertical capillary with an external flow

We study bubble motion in a vertical capillary tube under an external flow. Bretherton (J. Fluid Mech., vol. 10, issue 2, 1961, pp. 166-188) has shown that, without external flow, a bubble can spontaneously rise when the Bond number (Bo = rho gR(2)/gamma) is above the critical value Bo(cr) = 0.842, where rho is the liquid density, g the gravitational acceleration, R the tube radius and gamma the surface tension. It was then shown by Magnini et al. (Phys. Rev. Fluids, vol. 4, issue 2, 2019, 023601) that the presence of an imposed liquid flow, in the same (upward) direction as buoyancy, accelerates the bubble and thickens the liquid film around it. In this work we carry out a systematic study of the bubble motion under a wide range of upward and downward external flows, focusing on the inertialess regime with Bond numbers above the critical value. We show that a rich variety of bubble dynamics occurs when an external downward flow is applied, opposing the buoyancy-driven rise of the bubble. We reveal the existence of a critical capillary number of the external downward flow (Ca-l = mu U-l/gamma, where mu is the fluid viscosity and U-l is the mean liquid speed) at which the bubble arrests and changes its translational direction. Depending on the relative direction of gravity and the external flow, the thickness of the film separating the bubble surface and the tube inner wall follows two distinct solution branches. The results from theory, experiments and numerical simulations confirm the existence of the two solution branches and reveal that the two branches overlap over a finite range of Ca-l, thus suggesting non-unique, history-dependent solutions for the steady-state film thickness under the same external flow conditions. Furthermore, inertialess symmetry-breaking shape profiles at steady state are found as the bubble transits near the tipping points of the solution branches, which are shown in both experiments and three-dimensional numerical simulations.

Automated summary

The research systematically studies the motion of bubbles in a vertical capillary tube under external flows, revealing diverse dynamic phenomena when an external downward flow is applied to the bubble. The study discovers the existence of two distinct solution branches for the film thickness separating the bubble surface and the tube inner wall, as well as non-unique, history-dependent solutions for the steady-state film thickness under the same external flow conditions. Furthermore, inertialess symmetry-breaking shape profiles at steady state are found as the bubble transits near the tipping points of the solution branches, which are verified in experiments and numerical simulations.