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

Published online by Cambridge University Press:  01 February 2021

Yingxian Estella Yu Open the ORCID record for Yingxian Estella Yu [Opens in a new window] ,
Mirco Magnini Open the ORCID record for Mirco Magnini [Opens in a new window] ,
Lailai Zhu Open the ORCID record for Lailai Zhu [Opens in a new window] ,
Suin Shim Open the ORCID record for Suin Shim [Opens in a new window]  and
Howard A. Stone Open the ORCID record for Howard A. Stone [Opens in a new window]
Yingxian Estella Yu
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA
Mirco Magnini
Affiliation:
Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, NottinghamNG7 2RD, UK
Lailai Zhu
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA Department of Mechanical Engineering, National University of Singapore, 117575, Republic of Singapore
Suin Shim
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA
*
†Email address for correspondence: hastone@princeton.edu
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Abstract

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.

JFM classification

Drops and Bubbles: Bubble dynamics Low-Reynolds-number flows: Lubrication theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 911 , 25 March 2021 , A34
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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