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Inertial dynamics of air bubbles crossing a horizontal fluid–fluid interface

Published online by Cambridge University Press:  13 July 2012

Romain Bonhomme ,
Jacques Magnaudet ,
Fabien Duval  and
Bruno Piar
Romain Bonhomme
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); Allée Camille Soula, F-31400 Toulouse, France CNRS; IMFT; F-31400 Toulouse, France Institut de Radioprotection et de Sûreté Nucléaire, BP 3, 13115 St Paul lez Durance CEDEX, France
Jacques Magnaudet*
Affiliation:
Université de Toulouse; INPT, UPS; IMFT (Institut de Mécanique des Fluides de Toulouse); Allée Camille Soula, F-31400 Toulouse, France CNRS; IMFT; F-31400 Toulouse, France
Fabien Duval
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire, BP 3, 13115 St Paul lez Durance CEDEX, France
Bruno Piar
Affiliation:
Institut de Radioprotection et de Sûreté Nucléaire, BP 3, 13115 St Paul lez Durance CEDEX, France
*
†Email address for correspondence: magnau@imft.fr
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Abstract

The dynamics of isolated air bubbles crossing the horizontal interface separating two Newtonian immiscible liquids initially at rest are studied both experimentally and computationally. High-speed video imaging is used to obtain a detailed evolution of the various interfaces involved in the system. The size of the bubbles and the viscosity contrast between the two liquids are varied by more than one and four orders of magnitude, respectively, making it possible to obtain bubble shapes ranging from spherical to toroidal. A variety of flow regimes is observed, including that of small bubbles remaining trapped at the fluid–fluid interface in a film-drainage configuration. In most cases, the bubble succeeds in crossing the interface without being stopped near its undisturbed position and, during a certain period of time, tows a significant column of lower fluid which sometimes exhibits a complex dynamics as it lengthens in the upper fluid. Direct numerical simulations of several selected experimental situations are performed with a code employing a volume-of-fluid type formulation of the incompressible Navier–Stokes equations. Comparisons between experimental and numerical results confirm the reliability of the computational approach in most situations but also points out the need for improvements to capture some subtle but important physical processes, most notably those related to film drainage. Influence of the physical parameters highlighted by experiments and computations, especially that of the density and viscosity contrasts between the two fluids and of the various interfacial tensions, is discussed and analysed in the light of simple models and available theories.

JFM classification

Interfacial Flows (free surface): Capillary flows Drops and Bubbles: Drops and Bubbles
Type
Papers
Information
Journal of Fluid Mechanics , Volume 707 , 25 September 2012 , pp. 405 - 443
Copyright
Copyright © Cambridge University Press 2012

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References

1. Allan, R. S., Charles, G. E. & Mason, S. G. 1961 The approach of gas bubbles to a gas/liquid interface. J. Colloid Sci. 16, 150–165.Google Scholar
2. Bart, E. 1968 The slow unsteady settling of a fluid sphere toward a flat fluid interface. Chem. Engng Sci. 23, 193–210.Google Scholar
3. Bonometti, T. & Magnaudet, J. 2006 Transition from spherical caps to toroidal bubbles. Phys. Fluids 18, 052102.Google Scholar
4. Bonometti, T. & Magnaudet, J. 2007 An interface-capturing method for incompressible two-phase flows: validation and application to bubble dynamics. Intl J. Multiphase Flow 33, 109–133.CrossRefGoogle Scholar
5. Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modelling surface tension. J. Comput. Phys. 100, 335–354.Google Scholar
6. Bush, J. W. M. & Eames, I. 1998 Fluid displacement by high Reynolds number bubble motion in a thin gap. Intl J. Multiphase Flow 24, 411–430.CrossRefGoogle Scholar
7. Camassa, R., Falcon, C., Lin, J., McLaughlin, R. M. & Parker, R. 2009 Prolonged residence times for particles settling through stratified miscible fluids in the Stokes regime. Phys. Fluids 21, 031702.Google Scholar
8. Charles, G. E. & Mason, S. G. 1960 The coalescence of liquid drops with flat liquid/liquid interfaces. J. Colloid Sci. 15, 236–267.Google Scholar
9. Chi, B. K. & Leal, L. G. 1989 A theoretical study of the motion of a viscous drop toward a planar wall at low Reynolds number. J. Fluid Mech. 201, 123–146.Google Scholar
10. Cranga, J., Gardin, P., Huin, D. & Magnaudet, J. 2001 Influence of surface pressure and slag layer on bubble bursting in degasser systems. In Computational Modelling of Materials, Minerals, and Metals Processing (ed. Cross, M., Evans, J. W. & Bailey, C. ), pp. 105–114. The Minerals, Metals & Materials Society.Google Scholar
11. Davies, R. M. & Taylor, G. I. 1950 The mechanics of large bubbles rising through liquids in tubes. Proc. R. Soc. Lond. A 200, 375–390.Google Scholar
12. Debrégeas, G., de Gennes, P.-G. & Brochard-Wyart, F. 1998 The life and death of ‘bare’ viscous bubbles. Science 279, 1704–1707.Google Scholar
13. Dietrich, N., Poncin, S., Pheulpin, S. & Li, H. Z. 2008 Passage of a bubble through a liquid–liquid interface. AIChE J. 54, 594–600.CrossRefGoogle Scholar
14. Eames, I. & Duursma, G. 1997 Displacement of horizontal layers by bubbles injected into fluidized beds. Chem. Engng Sci. 52, 2697–2705.Google Scholar
15. Geller, A. S., Lee, S. H. & Leal, L. G. 1986 The creeping motion of a spherical particle normal to a deformable interface. J. Fluid Mech. 169, 27–69.CrossRefGoogle Scholar
16. de Gennes, P.-G., Brochard-Wyart, F. & Quéré, D. 2004 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls and Waves. Springer.Google Scholar
17. Greene, G. A., Chen, J. C. & Conlin, M. T. 1988 Onset of entrainment between immiscible liquid layers due to rising gas bubbles. Intl J. Heat Mass Transfer 31, 1309–1317.Google Scholar
18. Greene, G. A., Chen, J. C. & Conlin, M. T. 1991 Bubble induced entrainment between stratified liquid layers. Intl J. Heat Mass Transfer 34, 149–157.CrossRefGoogle Scholar
19. Hartland, S. 1968 The approach of a rigid sphere to a deformable liquid/liquid interface. J. Colloid Interface Sci. 26, 383–394.CrossRefGoogle Scholar
20. Hartland, S. 1969 The profile of the draining film between a rigid sphere and a deformable fluid–liquid interface. Chem. Engng Sci. 24, 987–995.Google Scholar
21. Jones, A. F. & Wilson, S. D. R. 1978 The film drainage problem in droplet coalescence. J. Fluid Mech. 287, 279–298.Google Scholar
22. Joseph, D. D. 2003 Rise velocity of a spherical cap bubble. J. Fluid Mech. 488, 213–223.CrossRefGoogle Scholar
23. Kemiha, M., Olmos, E., Fei, W., Poncin, S. & Li, H. Z. 2007 Passage of a gas bubble through a liquid–liquid interface. Ind. Engng Chem. Res. 46, 6099–6104.CrossRefGoogle Scholar
24. Kim, J. 2007 Phase field computations for ternary fluid flows. Comput. Meth. Appl. Mech. Engng 196, 4779–4788.CrossRefGoogle Scholar
25. Kim, J. 2009 A generalized continuous surface tension force formulation for phase-field models for multi-component immiscible fluid flows. Comput. Meth. Appl. Mech. Engng 198, 3105–3112.CrossRefGoogle Scholar
26. Kim, J. & Lowengrub, J. 2005 Phase field modelling and simulation of three-phase flows. Interface Free Bound. 7, 435–466.CrossRefGoogle Scholar
27. Kobayashi, S. 1993 Iron droplet formation due to bubbles passing through molten iron/slag interface. ISIJ Int. 33, 577–582.CrossRefGoogle Scholar
28. Leal, L. G. & Lee, S. H. 1982 Particle motion near a deformable fluid interface. Adv. Colloid Interface Sci. 17, 61–81.Google Scholar
29. Lee, D. G. & Kim, H. Y. 2011 Sinking of small sphere at low Reynolds number through interface. Phys. Fluids 23, 072104.Google Scholar
30. Lundgren, T. S. & Mansour, N. N. 1991 Vortex ring bubbles. J. Fluid Mech. 87, 263–288.Google Scholar
31. Magnaudet, J. & Mougin, G. 2007 Wake instability of a fixed spheroidal bubble. J. Fluid Mech. 572, 311–337.Google Scholar
32. Manga, M. & Stone, H. A. 1995 Low Reynolds number motion of bubbles, drops and rigid spheres through fluid–fluid interfaces. J. Fluid Mech. 287, 279–298.CrossRefGoogle Scholar
33. Manga, M., Stone, H. A. & O’Connell, R. L. 1993 The interaction of plume heads with compositional discontinuities in the Earth’s mantle. J. Geophys. Res. 98, 19979–19990.Google Scholar
34. Maru, H. C., Wasan, D. T. & Kintner, R. C. 1971 Behavior of a rigid sphere at a liquid–liquid interface. Chem. Engng Sci. 26, 1615–1628.CrossRefGoogle Scholar
35. Meiron, D. I. 1989 On the stability of gas bubbles rising in an inviscid fluid. J. Fluid Mech. 198, 101–114.CrossRefGoogle Scholar
36. Miksis, M., Vanden-Broeck, J. M. & Keller, J. B. 1981 Axisymmetric bubble or drop in a uniform flow. J. Fluid Mech. 108, 89–100.Google Scholar
37. Mohamed-Kassim, Z. & Longmire, E. K. 2004 Drop coalescence through a liquid–liquid interface. Phys. Fluids 16, 2170–2181.Google Scholar
38. Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.Google Scholar
39. Moore, D. W. 1965 The velocity of rise of distorted gas bubbles in a liquid of small viscosity. J. Fluid Mech. 23, 749–766.CrossRefGoogle Scholar
40. Natarajan, R. & Acrivos, A. 1993 The instability of the steady flow past spheres and disks. J. Fluid Mech. 254, 323–344.Google Scholar
41. Pedley, T. J. 1968 The toroidal bubble. J. Fluid Mech. 32, 97–112.CrossRefGoogle Scholar
42. Pigeonneau, F. & Sellier, A. 2011 Low-Reynolds-number gravity-driven migration and deformation of bubbles near a free surface. Phys. Fluids 23, 092102.Google Scholar
43. Poggi, D., Minto, M. & Davenport, W. G. 1969 Mechanisms of metal entrapment in slags. J. Met. 21, 40–45.Google Scholar
44. Popinet, S. 2003 Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries. J. Comput. Phys. 190, 572–600.Google Scholar
45. Popinet, S. 2009 An accurate adaptive solver for surface-tension-driven interfacial flows. J. Comput. Phys. 228, 5838–5866.CrossRefGoogle Scholar
46. Princen, H. M. 1963 Shape of a fluid drop at a liquid–liquid interface. J. Colloid Sci. 18, 178–195.CrossRefGoogle Scholar
47. Princen, H. M. & Mason, S. G. 1965a Shape of a fluid drop at a fluid–liquid interface. Part 1. Extension and test of two-phase theory. J. Colloid Sci. 20, 156–172.CrossRefGoogle Scholar
48. Princen, H. M. & Mason, S. G. 1965b Shape of a fluid drop at a fluid–liquid interface. Part 2. Theory for three-phase systems. J. Colloid Sci. 20, 246–266.Google Scholar
49. Reiter, G. & Schwerdtfeger, K. 1992a Observation of physical phenomena occurring during passage of bubbles through liquid/liquid interfaces. ISIJ Int. 32, 50–56.CrossRefGoogle Scholar
50. Reiter, G. & Schwerdtfeger, K. 1992b Characteristics of entrainment at liquid/liquid interfaces due to rising bubbles. ISIJ Int. 32, 57–65.Google Scholar
51. Shah, S. T., Wasan, D. T. & Kintner, R. C. 1972 Passage of a liquid drop through a liquid–liquid interface. Chem. Engng Sci. 27, 881–893.CrossRefGoogle Scholar
52. Shopov, P. J. & Minev, P. D. 1992 The unsteady motion of a bubble or drop towards a liquid–liquid interface. J. Fluid Mech. 235, 123–141.Google Scholar
53. Smith, P. G. & Van de Ven, T. G. M. 1984 The effect of gravity on the drainage of a thin liquid film between a solid sphere and a liquid/fluid interface. J. Colloid Interface Sci. 100, 456–464.Google Scholar
54. Srdić-Mitrović, A. N., Mohamed, N. A. & Fernando, H. J. S. 1999 Gravitational settling of particles through density interfaces. J. Fluid Mech. 381, 175–198.Google Scholar
55. Taylor, T. D. & Acrivos, A. 1964 On the deformation and drag of a falling viscous drop at low Reynolds number. J. Fluid Mech. 18, 466–476.CrossRefGoogle Scholar
56. Thomas, S., Esmaeeli, A. & Tryggvason, G. 2010 Multiscale computations of thin films in multiphase flows. Intl J. Multiphase Flow 36, 71–77.Google Scholar
57. Tsai, S. S. H., Wexler, J. S., Wan, J. & Stone, H. A. 2011 Conformal coating of particles in microchannels by magnetic forcing. Appl. Phys. Lett. 99, 153509.Google Scholar
58. Walters, J. K. & Davidson, J. F. 1963 The initial motion of a gas bubble formed in a inviscid liquid. Part 2. The three-dimensional bubble and the toroidal bubble. J. Fluid Mech. 17, 321–336.Google Scholar
59. Zalesak, S. T. 1979 Fully multidimensional Flux-Corrected Transport algorithms for fluids. J. Comput. Phys. 31, 335–362.Google Scholar