Journal of Statistical Physics

, Volume 167, Issue 3–4, pp 656–682 | Cite as

Arrested Bubble Rise in a Narrow Tube

  • Catherine Lamstaes
  • Jens Eggers


If a long air bubble is placed inside a vertical tube closed at the top it can rise by displacing the fluid above it. However, Bretherton found that if the tube radius, R, is smaller than a critical value \(R_{c}=0.918 \; \ell _c\), where \(\ell _c=\sqrt{\gamma /\rho g}\) is the capillary length, there is no solution corresponding to steady rise. Experimentally, the bubble rise appears to have stopped altogether. Here we explain this observation by studying the unsteady bubble motion for \(R<R_{c}\). We find that the minimum spacing between the bubble and the tube goes to zero in limit of large t like \(t^{-4/5}\), leading to a rapid slow-down of the bubble’s mean speed \(U \propto t^{-2}\). As a result, the total bubble rise in infinite time remains very small, giving the appearance of arrested motion.


Singularities Thin film flow Lubrication theory Surface tension 



We are grateful to Howard Stone for pointing out to us the paradox of the stuck bubble, and for enlightening discussions. Discussions with John Kolinski and Hyoungsoo Kim on the possibility of experiments are also gratefully acknowledged.


  1. 1.
    Bretherton, F.P.: The motion of long bubbles in tubes. J. Fluid Mech. 10, 166 (1961)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Dupont, T.F., Goldstein, R.E., Kadanoff, L.P., Zhou, S.-M.: Finite-time singularity formation in Hele-Shaw systems. Phys. Rev. E 47, 4182 (1993)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Constantin, P., Dupont, T.F., Goldstein, R.E., Kadanoff, L.P., Shelley, M.J., Zhou, S.-M.: Droplet breakup in a model of the Hele-Shaw cell. Phys. Rev. E 47, 4169 (1993)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Bertozzi, A.L., Brenner, M.P., Dupont, T.F., Kadanoff, L.P.: Singularities and similarities in interface flows. In: Sirovich, L. (ed.) Applied Mathematics Series, vol. 100, p. 155. Springer, New York (1994)Google Scholar
  5. 5.
    Boatto, S., Kadanoff, L.P., Olla, P.: Traveling-wave solutions to thin-film equations. Phys. Rev. E 48, 4423 (1993)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Kadanoff, L.P.: Singularities and blowups. Phys. Today 50(9), 11–12 (1997)CrossRefGoogle Scholar
  7. 7.
    Jones, A.F., Wilson, S.D.R.: The film drainage problem in droplet coalescence. J. Fluid Mech. 87, 263 (1978)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    Eggers, J., Fontelos, M.A.: Singularities: Formation, Structure, and Propagation. Cambridge University Press, Cambridge (2015)CrossRefzbMATHGoogle Scholar
  9. 9.
    Hinch, E.J.: Perturbation Methods. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar
  10. 10.
    Wilson, S.D.R., Jones, A.F.: The entry of a falling film into a pool and the air-entrainment problem. J. Fluid Mech. 128, 219 (1983)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Oron, A., Davis, S.H., Bankoff, S.G.: Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980 (1997)ADSCrossRefGoogle Scholar
  12. 12.
    Wilson, S.D.R.: The drag-out problem in film coating theory. J. Engg. Math. 16, 209 (1982)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Eggers, J., Stone, H.A.: Characteristic lengths at moving contact lines for a perfectly wetting fluid: the influence of speed on the dynamic contact angle. J. Fluid Mech. 505, 309–321 (2004)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Eggers, J., Dupont, T.F.: Drop formation in a one-dimensional approximation of the Navier-Stokes equation. J. Fluid Mech. 262, 205 (1994)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Duchemin, L., Lister, J.R., Lange, U.: Static shapes of levitated viscous drops. J. Fluid Mech. 533, 161–170 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Bender, C.M., Orszag, S.A.: Advanced Mathematical Methods for Scientists and Engineers. Mc Graw-Hill, New York (1978)zbMATHGoogle Scholar
  17. 17.
    Yiantsios, S.G., Davis, R.H.: Close approach and deformation of two viscous drops due to gravity and van der waals forces. J. Colloid Interf. Sci. 144, 412–433 (1991)CrossRefGoogle Scholar
  18. 18.
    Almgren, R., Bertozzi, A.L., Brenner, M.P.: Stable and unstable singularities in the unforced Hele-Shaw cell. Phys. Fluids 8, 1356 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Bonn, D., Eggers, J., Indekeu, J., Meunier, J., Rolley, E.: Wetting and spreading. Rev. Mod. Phys. 81, 739 (2009)ADSCrossRefGoogle Scholar
  20. 20.
    Hammoud, N., Trinh, P.H., Howell, P.D., Stone, H.A.: The influence of van der Waals interactions on a bubble moving in a tube (2016). arXiv:1601.00726
  21. 21.
    Eggers, J., Stone, H.A.: unpublished manuscript (2015)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of MathematicsUniversity of Bristol, University WalkBristolUK

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