When a confined bubble translates steadily in a cylindrical capillary tube, without the consideration of gravity effects, a uniform thin film of liquid separates the bubble surface and the tube wall. In this work, we investigate how this steady state is established by considering the transitional motion of the bubble as it adjusts its film thickness profile between two steady states, characterized by two different bubble speeds. During the transition, two uniform film regions coexist, separated by a step-like transitional region. The transitional motion also requires modification of the film solution near the rear of the bubble, which depends on the ratio of the two capillary numbers. These theoretical results are verified by experiments and numerical simulations.
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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 ( ${Bo} \equiv \rho g R^2 / \gamma$ ) 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 ( ${Ca}_l \equiv \mu U_l/\gamma$ , 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.
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- Award ID(s):
- 1804863
- NSF-PAR ID:
- 10268472
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 911
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/jfm.2020.1027
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