Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-26T13:09:35.055Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of confinement on the motion of a single clean bubble

Published online by Cambridge University Press:  10 December 2008

B. FIGUEROA-ESPINOZA ,
R. ZENIT  and
D. LEGENDRE
B. FIGUEROA-ESPINOZA
Affiliation:
Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Circuito Exterior s/n, Ciudad Universitaria, 04510 México, D.F., México
R. ZENIT*
Affiliation:
Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México, Circuito Exterior s/n, Ciudad Universitaria, 04510 México, D.F., México
D. LEGENDRE
Affiliation:
Institut de Mécanique des Fluides de Toulouse, Alee du Prof. Soula, 31400 Toulouse, France
*
†Email address for correspondence: zenit@servidor.unam.mx
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of confining a gas bubble between two parallel walls was investigated for the inertia-dominated regime characterized by high Reynolds and low Weber numbers. Single bubble experiments were performed with non-polar liquids such that the bubble surface could be considered clean; hence, shear free. The drag coefficient was found to be the result of two main effects: the Reynolds number and the confinement. The total drag could be written as the product of the corresponding unconfined drag, which depended mainly on the Reynolds number, and a function F(s)=1 + κs3. The confinement parameter s was defined as the ratio of the bubble radius to the gap width. The value of the constant κ depended on the way in which the bubbles moved within the gap, which was found to be either in a rectilinear (κ≈8) or oscillatory trajectory (κ≈80). For Re < 70, and a range of values of the confinement parameter, the bubbles followed a rectilinear path. For this regime, numerical simulations were performed to obtain the drag force on the bubble directly; a reasonable agreement was found with experiments. Moreover, a comparison of these results with a potential-flow-based model indicated that the vorticity produced at the walls induced a significant part of the drag. For Re > 70, oscillations were observed in the bubble trajectory. In all cases, the oscillation occurred in a zigzag manner. Near the transition the bubbles oscillated but did not reach the walls; for larger Reynolds numbers, the bubbles collided repeatedly with the walls as they ascended. The instability, which is different from the well-known unconfined path instability, resulted from the reversal of sign of the wall-induced lift force: for low Reynolds number, the walls have a stabilizing effect because of the repulsive nature of the lift force between the walls and the bubble, while for high Reynolds number the lift is attractive and trajectories become unstable. Considering a model for the lift force of a bubble moving near a wall, the conditions for the transition were identified. A reasonable agreement between the model and experiments was found.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 616 , 10 December 2008 , pp. 419 - 443
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Arfken, G. B. & Weber, H. J. 1995 Mathematical Methods for Physicists. Academic Press.Google Scholar
Bairstow, L., Cave, B. M. & Lang, E. D. 1922 The two-dimensional slow motion of viscous fluids. Proc. R. Soc. Lond. A 705, 394.Google Scholar
Biesheuvel, A. & van Wijngaarden, L. 1982 The motion of pairs of gas bubble in a perfect liquid. J. Engng Math. 16, 349–365.CrossRefGoogle Scholar
Brücker, C. 1999 Structure and dynamics of the wake of bubbles and its relevance for bubble interaction. Phys. Fluids 11 (7), 1781–1796.CrossRefGoogle Scholar
Clift, R., Grace, R. J. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic Press.Google Scholar
De Vries, A. W. G. 2001 Path and wake of a rising bubble. PhD thesis, University of Twente, the Netherlands.Google Scholar
Duineveld, P. C. 1995 The rise velocity and shape of bubbles in pure water at high Reynolds number. J. Fluid Mech. 292, 325–332.CrossRefGoogle Scholar
Faxen, H. 1922 Der widerstand gegen die bewegung einer starren kugel in einer zahen flussigkeit, die zwischen zwei parallen ebenen wanden eingeschlossen ist. Ann. Phys. 68, 89.CrossRefGoogle Scholar
Hadamard, J. 1911 Mouvement permanent lent dune sphere liquide et visqueuse dans un liquide visqueux C. R. Acad. Sci. Paris 152, 1735.Google Scholar
Happel, J. & Brenner, H. 1991 Low Reynolds Number Hydrodynamics. Kluwer.Google Scholar
Hobson, E. W. 1931 The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press.Google Scholar
Kok, J. B. W. 1993 Dynamics of a pair of gas bubbles moving through liquid. Part I. Theory. Eur. J. Mech. B Fluids 12, 515–540.Google Scholar
Kumaran, V. & Koch, D. L. 1993 The effect of hydrodynamic interactions on the average properties of a bidisperse suspension of high-Reynolds-number, low Weber number bubbles. Phys. Fluids 5, 1123–1134.CrossRefGoogle Scholar
Legendre, D. & Magnaudet, J. 1998 The lift force on a spherical bubble in a viscous linear shear flow. J. Fluid Mech. 368, 81–126.CrossRefGoogle Scholar
Legendre, D., Magnaudet, J. & Mougin, G. 2003 Hydrodynamic interactions between two spherical bubbles rising side by side in a viscous liquid. J. Fluid Mech. 497, 133–66.CrossRefGoogle Scholar
Legendre, D. 2007 On the relation between the drag and the vorticity produced on clean bubble. Phys. Fluids 19, 018102.CrossRefGoogle Scholar
Levich, V. G. 1948 The motion of bubbles at high Reynolds numbers. Zh. Eksptl. Teor. Fiz. 19, 18.Google Scholar
Levich, V. G. 1962 Physico–Chemical Hydrodynamics. Prentice Hall.Google Scholar
Magnaudet, J. & Mougin, G. 2007 Wake instability of a fixed spheroidal bubble. J. Fluid Mech. 572, 311–337.CrossRefGoogle Scholar
Moctezuma, M. F., Lima-Ochoterena, R. & Zenit, R. 2005 Velocity fluctuations resulting from the interaction of a bubble with a vertical wall. Phys. Fluids 17, 098106.CrossRefGoogle Scholar
Moore, D. W. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.CrossRefGoogle Scholar
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
Mougin, G. & Magnaudet, J. 2002 Path instability of a rising bubble. Phys. Rev. Lett. 88, 014502.CrossRefGoogle ScholarPubMed
Rybczynski, W. 1911 Uber die Fortschreitende Bwegung einer-ussigen Kugel in einem zahen Medium. Bull. Acad. Sci. Cracovie Ser. A 1 40, 1.Google Scholar
Sato, A., Shirota, M., Sanada, T., Watanabe, M. & Ruzicka, M. 2007 Path and wake of a pair of bubbles rising side by side. In Sixth International Conference on Multiphase Flow, S1_Wed_D_45, Leipzig, Germany.CrossRefGoogle Scholar
Shew, W., Poncet, S. & Pinton, J.-F. 2006 Force measurements on rising bubbles. J. Fluid Mech. 569, 51–60.CrossRefGoogle Scholar
Takemura, F. & Magnaudet, J. 2003 The transverse force on clean and contaminated bubbles rising near a vertical wall at moderate Reynolds number. J. Fluid Mech. 495, 235–253.CrossRefGoogle Scholar
Veldhuis, C. 2007 Leonardo's paradox: Path and shape instabilities of particles and bubbles. PhD thesis, University of Twente, the Netherlands.Google Scholar
van Wijngaarden, L. 1976 Bubble interactions between bubbles in liquid. J. Fluid Mech. 77, 27–44.CrossRefGoogle Scholar
Yang, B. & Prospereti, A. 2007 Linear stability of the flow past a spheroidal bubble. J. Fluid Mech. 582, 53–78.CrossRefGoogle Scholar
Zenit, R. & Magnaudet, J. 2008 Path instability of rising spheroidal air bubbles: a shape-controlled process. Phys. Fluids 20, 061702.CrossRefGoogle Scholar