Advertisement

Journal of Statistical Physics

, Volume 114, Issue 5–6, pp 1501–1536 | Cite as

The Effect of Finiteness in the Saffman–Taylor Viscous Fingering Problem

  • Darren Crowdy
  • Saleh Tanveer
Article

Abstract

We derive a family of exact time-evolving solutions for the the evolution of a finite blob of fluid confined to a channel in a Hele–Shaw cell. We show rigorously that, for large fluid volume, there are solutions for which one of the interfaces approaches the steady Saffman–Taylor finger solution of arbitrary width λ∈(0, 1). On the basis of this, we argue that the far-field effects of a displaced second interface do not provide a selection mechanism for the formation of a width-\( \frac{1}{2} \) finger when surface tension, or any other regularization, is ignored.

Saffman–Taylor fingers selection pattern formation Hele–Shaw flow distant interface effects 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

REFERENCES

  1. 1.
    P. G. Saffman and G. I. Taylor, The penetration of a fluid in a porous medium of Hele-Shaw cell containing a more viscous fluid, Proc. Roy. Soc. A 245:312–329 (1958).Google Scholar
  2. 2.
    P. G. Saffman, Viscous fingering in a Hele-Shaw cell, J. Fluid Mech. 173:73(1986).Google Scholar
  3. 3.
    D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman, and C. Tang, Viscous flow in two-dimensions, Rev. Modern Phys. 58:977(1986).Google Scholar
  4. 4.
    G. M. Homsy, Viscous fingering in porous media, Ann. Rev. Fluid Mech. 19:271(1987).Google Scholar
  5. 5.
    D. Kessler, J. Koplik, and H. Levine, Patterned selection in fingered growth phenomena, Adv. Phys. 37:255(1988).Google Scholar
  6. 6.
    P. Pelce, Dynamics of Curved Fronts (Academic Press, New York, 1988).Google Scholar
  7. 7.
    J. W. McLean and P. G. Saffman, The effect of surface tension on the shape of fingers in a Hele-Shaw cell, J. Fluid Mech. 102:455(1981).Google Scholar
  8. 8.
    J. M. Vanden-Broeck, Fingers in a Hele-Shaw cell with surface tension, Phys. Fluids 26:2033(1983).Google Scholar
  9. 9.
    D. Kessler and H. Levine, The theory of Saffman-Taylor finger, Phys. Rev. A 32:1930(1985).Google Scholar
  10. 10.
    D. Kessler and H. Levine, Stability of finger patterns in Hele-Shaw cells, Phys. Rev. A 33:2632(1986).Google Scholar
  11. 11.
    R. Combescot, T. Dombre, V. Hakim, Y. Pomeau, and A. Pumir, Shape selection for Saffman-Taylor fingers, Phys. Rev. Lett. 56:2036(1986).Google Scholar
  12. 12.
    D. C. Hong and J. S. Langer, Analytic theory for the selection of Saffman-Taylor finger, Phys. Rev. Lett. 56:2032(1986).Google Scholar
  13. 13.
    B. I. Shraiman, On velocity selection and the Saffman-Taylor problem, Phys. Rev. Lett. 56:2028(1986).Google Scholar
  14. 14.
    A. T. Dorsey and O. Martin, Saffman-Taylor fingers with anisotropic surface tension, Phys. Rev. A 35:3989(1987).Google Scholar
  15. 15.
    S. Tanveer, Analytic theory for the selection of symmetric Saffman-Taylor finger, Phys. Fluids 30:1589(1987b).Google Scholar
  16. 16.
    S. Tanveer, Analytic theory for the linear stability of Saffman-Taylor finger, Phys. Fluids 30:2318(1987c).Google Scholar
  17. 17.
    S. J. Chapman, On the role of Stokes line in the selection Saffman-Taylor fingers with small surface tension, Eur. J. Appl. Math. 10:513(1999).Google Scholar
  18. 18.
    X. Xie and S. Tanveer, Rigorous results in steady state selection, Arch. Rat. Mech. (2003).Google Scholar
  19. 19.
    S. Tanveer and X. Xie, Analyticity and nonexistence of steady Hele-Shaw fingers, Comm. Pure Appl. Math. (2003).Google Scholar
  20. 20.
    S. Tanveer, Surprises in viscous fingering, J. Fluid Mech. 409:273–308 (2000).Google Scholar
  21. 21.
    M. Feigenbaum, Pattern selection: Determined by symmetry and modifiable by long-range effects, J. Stat. Phys., to appear.Google Scholar
  22. 22.
    M. Feigenbaum, I. Procaccia, and B. Davidovich, Dynamics of finger formation in Laplacian growth without surface tension, J. Stat. Phys. 103(2001).Google Scholar
  23. 23.
    P. Zhuravlev, Zap Leningrad Com. Inst. 133:54(1956).Google Scholar
  24. 24.
    P. G. Saffman, Exact solution for the growth of fingers from a flat interface between two fluids in a porous medium or Hele-Shaw cell, Quart. J. Mech. Appl. Math. 12:146–150 (1959).Google Scholar
  25. 25.
    D. Crowdy, A theory of exact solutions for the evolution of a fluid annulus in a rotating Hele-Shaw cell, Quart. Appl. Math 60:11–36 (2002).Google Scholar
  26. 26.
    S. Richardson, Hele-Shaw flows with time-dependent free boundaries in infinite and semi-infinite strips, Q. J. Mech. Appl. Math. 35:531–548 (1982).Google Scholar
  27. 27.
    S. Richardson, Hele-Shaw flows with free boundaries driven along infinite strips by a pressure difference, Eur. J. Appl. Math. 7:345–366 (1996).Google Scholar
  28. 28.
    S. Tanveer, Evolution of Hele-Shaw interface for small surface tension, Proc. Roy. Soc. London Ser. A 343:155–204 (1993).Google Scholar
  29. 29.
    G. Valiron, fr Théorie des fonctions (Masson, Paris, 1948).Google Scholar
  30. 30.
    S. Howison, Fingering in Hele-Shaw cells, J. Fluid Mech. 167:439–453 (1986).Google Scholar
  31. 31.
    L. A. Caffarelli, A remark on the Hausdorff measure of a free boundary, and convergence of coincidence sets, Boll. U.M.I (5) 18-A:109–113 (1981).Google Scholar
  32. 32.
    V. Isakov, Inverse source problems, AMS Math Surveys and Monographs, Vol. 34 (AMS, Providence, Rhode Island, 1990).Google Scholar
  33. 33.
    A. N. Varchenko and P. I. Etingof, Why the boundary of a round drop becomes a curve of order four, AMS University Lecture Series, Vol. 3 (AMS, Providence, Rhode Island, 1992).Google Scholar
  34. 34.
    B. Gustafsson, An ill-posed MBP for doubly-connected domains, Ark. Mat. 25:231–253 (1978).Google Scholar
  35. 35.
    S. Tanveer and P. G. Saffman, Stability of bubbles in a Hele-Shaw cell, Phys. Fluids 30:2624–2635 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • Darren Crowdy
    • 1
  • Saleh Tanveer
    • 2
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridge
  2. 2.Department of MathematicsOhio State UniversityColumbus

Personalised recommendations