Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Fractal growth viscous fingers: quantitative characterization of a fluid instability phenomenon

Abstract

What happens when one attempts to push water through a fluid of higher viscosity? Under appropriate experimental conditions, the water breaks through in the form of highly branched patterns called viscous fingers. Water was used to push a more viscous but miscible, non-newtonian fluid in a Hele-Shaw cell. The resulting viscous finger instability was found to be a fractal growth phenomenon. Reproducible values of the fractal dimension df were found and were interpreted using a modification of the diffusion limited aggregation model.

This is a preview of subscription content

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

References

  1. 1

    Saffman, P. G. & Taylor, G. I. Proc. R. Soc. A245, 312–329 (1958).

    ADS  CAS  Google Scholar 

  2. 2

    Chuoke, R. L., Van Meurs, P. & Van der Poel, C. J. petrol. Tech. 11, 64–70 (1959).

    Google Scholar 

  3. 3

    Stalkup, F. I. Miscible Displacements (Soc. of Petroleum Eng., AIME, N.Y., 1983).

    Google Scholar 

  4. 4

    Hele Shaw, J. S. S. Nature 58, 34–36 (1898).

    ADS  Article  Google Scholar 

  5. 5

    Mandelbrot, B. B. The Fractal Geometry of Nature (Freeman, San Francisco, 1982).

    MATH  Google Scholar 

  6. 6

    Brady, R. M. & Ball, R. C. Nature 309, 225–229 (1984).

    ADS  CAS  Article  Google Scholar 

  7. 7

    Avnir, D., Farin, D. & Pfeiffer, P. Nature 308, 261–263 (1984).

    ADS  CAS  Article  Google Scholar 

  8. 8

    Matsushita, M., Sano, M., Hayakawa, Y., Honjo, H. & Sawada, Y. Phys. Rev. Lett. 53, 286–289 (1984).

    ADS  CAS  Article  Google Scholar 

  9. 9

    Powles, J. G. & Quirke, N. Phys. Rev. Lett. 52, 1571–1574 (1984).

    ADS  CAS  Article  Google Scholar 

  10. 10

    Forrest, S. L. & Witten, T. A. J. Phys. A12, L109–112 (1979).

    ADS  CAS  Google Scholar 

  11. 11

    Lamb, H. Hydrodynamics (Cambridge University Press, London, 1932).

    MATH  Google Scholar 

  12. 12

    Chandrasekhar, S. Rev. mod. Phys. 15, 1–89 (1943).

    ADS  Article  Google Scholar 

  13. 13

    Witten, T. A. & Sander, L. M. Phys. Rev. Lett. 47, 1499–1501 (1981).

    Article  Google Scholar 

  14. 14

    Witten, T. A. & Sander, L. M. Phys. Rev. B27, 5685–5697 (1983).

    Article  Google Scholar 

  15. 15

    Herrmann, H. J. Phys. Rep. (in the press).

  16. 16

    Stanley, H. E. Fractals in Statistical Physics (Oxford University Press, New York, 1985).

    Google Scholar 

  17. 17

    Meakin, P. Phys. Rev. A27, 1495–1507 (1983).

    ADS  CAS  Article  Google Scholar 

  18. 18

    Peters, E. J. & Flock, D. L. J. Soc. petrol. Engng 21, 249–258 (1981).

    CAS  Article  Google Scholar 

  19. 19

    Paterson, L. J. Fluid Mech. 113, 513–529 (1981).

    ADS  Article  Google Scholar 

  20. 20

    Todd, M. R. & Longstaff, W. J. J. petrol. Tech. 253, 874–882 (1972).

    Article  Google Scholar 

  21. 21

    Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, 1971).

    Google Scholar 

  22. 22

    Niemeyer, L., Pietronero, L. & Wiesmann, H. J. Phys. Rev. Lett. 52, 1033–1036 (1984).

    ADS  MathSciNet  Article  Google Scholar 

  23. 23

    Paterson, L. Phys. Rev. Lett. 52, 1621–1624 (1984).

    ADS  CAS  Article  Google Scholar 

  24. 24

    Maher, J. V. Phys. Rev. Lett. (submitted).

  25. 25

    Tang, C. Phys. Rev. A (submitted).

  26. 26

    Kadanoff, L. P. J. stat. Phys. (submitted).

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nittmann, J., Daccord, G. & Stanley, H. Fractal growth viscous fingers: quantitative characterization of a fluid instability phenomenon. Nature 314, 141–144 (1985). https://doi.org/10.1038/314141a0

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing