Viscous Fingering in a Hele–Shaw Cell

  • Jian-Jun Xu
Part of the Springer Series in Synergetics book series (SSSYN)


In this chapter, we study another interfacial phenomenon: the formation of viscous fingers in a Hele–Shaw cell. This phenomenon occurs in an entirely different physical system from dendritic growth, but it raises similar issues and can thus be treated using the same approach established in the previous chapters. We are interested in the study of this phenomenon in this book, not only because the subject itself occupies an important position in the field of pattern formation, but also because the resolution for this problem provides a keystone for analytically studying further pattern formation problems in solidification, such as cellular growth and eutectic growth.


  1. 1.
    E. Ben-Jacob, R. Godbey, N.D. Goldenfield, J. Koplik, H. Levine, T. Mueller, L.M. Sander, Experimental demonstration of the role of anisotropy in interfacial pattern formation. Phys. Rev. Lett. 55, 1315–1318 (1985)ADSCrossRefGoogle Scholar
  2. 2.
    D. Bensimon, Stability of viscous fingering. Phys. Rev. A 33, 1302–1308 (1986)ADSCrossRefGoogle Scholar
  3. 3.
    D. Bensimon, P. Pelce, B.I. Shraiman, Dynamics of curved fronts and pattern selection. J. Phys. 48, 2081–2087 (1987)CrossRefGoogle Scholar
  4. 4.
    R.L. Chouke, P. Van Meurs, C. Van der Pol, The instability of slow immiscible viscous liquid-liquid displacements in permeable media. Trans. AIME 216, 188–194 (1959)Google Scholar
  5. 5.
    Y. Couder, N. Gerard, M. Rabaud, Narrow fingers in the Saffman–Taylor instability. Phys. Rev. A 34, 5175–5178 (1986)ADSCrossRefGoogle Scholar
  6. 6.
    A.J. DeGregoria, L.W. Schwartz, A boundary integral method for two-phase displacement in Hele–Shaw cell. J. Fluid Mech. 164, 383–400 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    S. Hill, Channeling in packed columns. Chem. Eng. Sci. 1, 247–253 (1952)MathSciNetCrossRefGoogle Scholar
  8. 8.
    G.M. Homsy, Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311 (1987)ADSCrossRefGoogle Scholar
  9. 9.
    D.C. Hong, J.S. Langer, Analytic theory of the selection mechanism in the Saffman–Taylor problem. Phys. Rev. Lett. 56, 2032–2035 (1986)ADSCrossRefGoogle Scholar
  10. 10.
    S.D. Howison, A.A. Lacey, J.R. Ockendon, Hele–Shaw free boundary problems with suction. Q. J. Mech. Appl. Math. 41, 183–193 (1988)MathSciNetCrossRefGoogle Scholar
  11. 11.
    D.A. Kessler, H. Levine, Stability of finger patterns in Hele–Shaw cells. Phys. Rev. A 32, 1930–1933 (1985)ADSCrossRefGoogle Scholar
  12. 12.
    D.A. Kessler, H. Levine, Theory of the Saffman–Taylor finger pattern. Phys. Rev. A 33, 2621–2633 (1986)ADSCrossRefGoogle Scholar
  13. 13.
    A.R. Kopf-Sill, G.M. Homsy, Narrow fingers in a Hele–Shaw cell. Phys. Fluids 30 (9), 2607–2609 (1987)ADSCrossRefGoogle Scholar
  14. 14.
    M. Matsushita, H. Yamada, Dendritic growth of single viscous finger under the influence of linear anisotropy. J. Cryst. Growth 99, 161–165 (1990)ADSCrossRefGoogle Scholar
  15. 15.
    J. Mclean, P.G. Saffman, The effect of surface tension on the shape of fingers in a Hele–Shaw cell. J. Fluid Mech. 102, 455–469 (1981)ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    E. Meiburg, G.M. Homsy, Nonlinear unstable viscous fingers in Hele–Shaw cell: numerical simulation. Phys. Fluids 31 (3), 429–439 (1988)ADSCrossRefzbMATHGoogle Scholar
  17. 17.
    L.A. Romero, PhD. Thesis, California Institute of Technology (1981)Google Scholar
  18. 18.
    P.G. Saffman, Exact solution for the growth of fingers from a flat interface between two fluids. Q. J. Mech. Appl. Math. 12, 146–150 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    P.G. Saffman, G.I. Taylor, The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245, 312–329 (1958)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    H. Segur, S. Tanveer, H. Levine (eds.), Asymptotics Beyond All Orders. NATO ASI Series, Series B: Physics, vol. 284 (Plenum, New York, 1991)Google Scholar
  21. 21.
    P. Tabling, G. Zocchi, A. Libchaber, An experimental study of the Saffman–Taylor instability. J. Fluid Mech. 177, 67–82 (1987)ADSCrossRefGoogle Scholar
  22. 22.
    S. Tanveer, Analytic theory for the selection of a symmetric Saffman–Taylor instability. Phys. Fluids 30 (8), 1589–1605 (1987)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    J.-M. Vanden-Broeck, Fingers in a Hele–Shaw cell with surface tension. Phys. Fluids 26 (8), 2033–2034 (1983)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    J.J. Xu, Global instability of viscous fingering in a Hele–Shaw cell: formation of oscillatory fingers. Eur. J. Appl. Math. 2, 105–132 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    J.J. Xu, Interfacial wave theory for oscillatory finger’s formation in a Hele–Shaw cell: a comparison with experiments. Eur. J. Appl. Math. 7, 169–199 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    J.J. Xu, Interfacial instabilities and fingering formation in Hele–Shaw flow. IMA J. Appl. Math. 57, 101–135 (1996)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    J. Yokoyama, Y. Kitagawa, H. Yamada M. Matsushita, Dendritic growth of viscous fingers under linear anisotropy. Phys. A 204, 789–799 (1994)CrossRefGoogle Scholar
  28. 28.
    P.A. Zhuravlev, On the motion of a fluid in channels. Zap. Leningr. Gorn. In-ta. 33 (3), 54–61 (1956)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Jian-Jun Xu
    • 1
  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

Personalised recommendations