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Time-dependent motion of a confined bubble in a tube: transition between two steady states

Published online by Cambridge University Press:  29 October 2018

Yingxian Estella Yu Open the ORCID record for Yingxian Estella Yu [Opens in a new window] ,
Lailai Zhu Open the ORCID record for Lailai Zhu [Opens in a new window] ,
Suin Shim ,
Jens Eggers Open the ORCID record for Jens Eggers [Opens in a new window]  and
Howard A. Stone
Yingxian Estella Yu
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Lailai Zhu
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Linné Flow Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, SE 10044 Stockholm, Sweden
Suin Shim
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Jens Eggers*
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email addresses for correspondence: jens.eggers@bris.ac.uk, hastone@princeton.edu
†Email addresses for correspondence: jens.eggers@bris.ac.uk, hastone@princeton.edu
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Abstract

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.

JFM classification

Drops and Bubbles: Bubble dynamics Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 857 , 25 December 2018 , R4
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Copyright
© 2018 Cambridge University Press 

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References

Atasi, O., Khodaparast, S., Scheid, B. & Stone, H. A. 2017 Effect of buoyancy on the motion of long bubbles in horizontal tubes. Phys. Rev. Fluids 2 (9), 094304.Google Scholar
Aussillous, P. & Quéré, D. 2000 Quick deposition of a fluid on the wall of a tube. Phys. Fluids 12 (10), 2367–2371.Google Scholar
Balestra, G., Zhu, L. & Gallaire, F. 2018 Viscous Taylor droplets in axisymmetric and planar tubes: from Bretherton’s theory to empirical models. Microfluid. Nanofluid. 22 (6), 67.Google Scholar
Bäumchen, O., Benzaquen, M., Salez, T., McGraw, J. D., Backholm, M., Fowler, P., Raphaël, E. & Dalnoki-Veress, K. 2013 Relaxation and intermediate asymptotics of a rectangular trench in a viscous film. Phys. Rev. E 88, 035001.Google Scholar
Blunt, M. J. 2001 Flow in porous media pore-network models and multiphase flow. Curr. Opin. Colloid Interface Sci. 6 (3), 197–207.Google Scholar
Boatto, S., Kadanoff, L. P. & Olla, P. 1993 Traveling-wave solutions to thin-film equations. Phys. Rev. E 48, 4423–4431.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10 (02), 166–188.Google Scholar
Eggers, J. & Fontelos, M. A. 2015 Singularities: Formation, Structure, and Propagation. Cambridge University Press.Google Scholar
Elveflow2009 Responsiveness of OB1 piezo–electric technologies. https://www.elveflow.com/microfluidic-tutorials/microfluidic-reviews-and-tutorials/responsiveness-of-flow-control-instruments .Google Scholar
Fairbrother, F. & Stubbs, A. E. 1935 119. Studies in electro-endosmosis. Part VI. The ‘bubble-tube’ method of measurement. J. Chem. Soc. 1, 527–529.Google Scholar
Gaver, D. P., Halpern, D., Jensen, O. E. & Grotberg, J. B. 1996 The steady motion of a semi-infinite bubble through a flexible-walled channel. J. Fluid Mech. 319, 25–65.Google Scholar
Hadikhani, P., Hashemi, S. M. H., Balestra, G., Zhu, L., Modestino, M. A, Gallaire, F. & Psaltis, D. 2018 Inertial manipulation of bubbles in rectangular microfluidic channels. Lab on a Chip 18 (7), 1035–1046.Google Scholar
Hazel, A. L. & Heil, M. 2003 Three-dimensional airway reopening: the steady propagation of a semi-infinite bubble into a buckled elastic tube. J. Fluid Mech. 478, 47–70.Google Scholar
Heil, M. 2001 Finite Reynolds number effects in the Bretherton problem. Phys. Fluids 13 (9), 2517–2521.Google Scholar
Khodaparast, S., Magnini, M., Borhani, N. & Thome, J. R. 2015 Dynamics of isolated confined air bubbles in liquid flows through circular microchannels: an experimental and numerical study. Microfluid. Nanofluid. 19 (1), 209–234.Google Scholar
Kotula, A. P. & Anna, S. L. 2012 Probing timescales for colloidal particle adsorption using slug bubbles in rectangular microchannels. Soft Matt. 8 (41), 10759–10772.Google Scholar
Lamstaes, C. & Eggers, J. 2017 Arrested bubble rise in a narrow tube. J. Stat. Phys. 167 (3–4), 656–682.Google Scholar
Leung, S. S. Y., Gupta, R., Fletcher, D. F. & Haynes, B. S. 2012 Gravitational effect on Taylor flow in horizontal microchannels. Chem. Engng Sci. 69 (1), 553–564.Google Scholar
Magnini, M., Ferrari, A., Thome, J. R. & Stone, H. A. 2017 Undulations on the surface of elongated bubbles in confined gas–liquid flows. Phys. Rev. Fluids 2 (8), 084001.Google Scholar
McGraw, J. D., Salez, T., Bäumchen, O., Raphaël, E. & Dalnoki-Veress, K. 2012 Self-similarity and energy dissipation in stepped polymer films. Phys. Rev. Lett. 109 (12), 128303.Google Scholar
Olgac, U. & Muradoglu, M. 2013 Effects of surfactant on liquid film thickness in the Bretherton problem. Intl J. Multiphase Flow 48, 58–70.Google Scholar
Park, C. W. 1992 Influence of soluble surfactants on the motion of a finite bubble in a capillary tube. Phys. Fluids A 4 (11), 2335–2347.Google Scholar
Quéré, D. 1999 Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31 (1), 347–384.Google Scholar
Ratulowski, J. & Chang, H. C. 1990 Marangoni effects of trace impurities on the motion of long gas bubbles in capillaries. J. Fluid Mech. 210, 303–328.Google Scholar
de Ryck, A. 2002 The effect of weak inertia on the emptying of a tube. Phys. Fluids 14 (7), 2102–2108.Google Scholar
Stebe, K. J. & Barthes-Biesel, D. 1995 Marangoni effects of adsorption–desorption controlled surfactants on the leading end of an infinitely long bubble in a capillary. J. Fluid Mech. 286, 25–48.Google Scholar
Stone, H. A. 2010 Interfaces: in fluid mechanics and across disciplines. J. Fluid Mech. 645, 1–25.Google Scholar
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10 (02), 161–165.Google Scholar
Yu, Y. E., Khodaparast, S. & Stone, H. A. 2017 Armoring confined bubbles in the flow of colloidal suspensions. Soft Matt. 13 (15), 2857–2865.Google Scholar
Yu, Y. E., Khodaparast, S. & Stone, H. A. 2018 Separation of particles by size from a suspension using the motion of a confined bubble. Appl. Phys. Lett. 112 (18), 181604.Google Scholar