Communications in Mathematical Physics

, Volume 257, Issue 2, pp 303–317 | Cite as

Dynamic Scaling in Miscible Viscous Fingering

Article
First Online:
Received:
Accepted:

Abstract

We consider dynamic scaling in gravity driven miscible viscous fingering. We prove rigorous one-sided bounds on bulk transport and coarsening in regimes of physical interest. The analysis relies on comparison with solutions to one-dimensional conservation laws, and new scale-invariant estimates. Our bounds on the size of the mixing layer are of two kinds: a naive bound that is sharp in the absence of diffusion, and a more careful bound that accounts for diffusion as a selection criterion in the limit of vanishingly small diffusion. The naive bound is simple and robust, but does not yield the experimental speed of transport. In a reduced model derived by Wooding [20], we prove a sharp upper bound on the size of the mixing layer in accordance with his experiments. Wooding’s model also provides an example of a scalar conservation law where the entropy condition is not the physically appropriate selection criterion.

Keywords

Entropy Neural Network Statistical Physic Complex System Selection Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Batchelor, G.K., Moffatt, H.K., Worster, M.G., eds.: Perspectives in fluid dynamics. Cambridge: Cambridge University Press, 2000Google Scholar
  2. 2.
    Chouke, R., van Meurs P., van der Poel, C.: The instability of slow, immiscible, viscous liquid-liquid displacements. Trans. AIME 216, 188–194 (1958)Google Scholar
  3. 3.
    Constantin, P.: Some open problems and research directions in the mathematical study of fluid dynamics. In: Mathematics Unlimited–-2001 and beyond, Berlin: Springer, 2001, pp. 353–360Google Scholar
  4. 4.
    Constantin, P., Kiselev, A., Oberman, A., Ryzhik, L.: Bulk burning rate in passive-reactive diffusion. Arch. Ration. Mech. Anal. 154, 53–91 (2000)CrossRefGoogle Scholar
  5. 5.
    Dimonte, G.: Nonlinear evolution of the Rayleigh-Taylor and Richtmyer-Meshkov instabilities. Physics of Plasmas, 6, 2009–2015 (1999)Google Scholar
  6. 6.
    Doering, C.R., Constantin, P.: Bounds for heat transport in a porous layer. J. Fluid Mech. 376, 263–296 (1998)CrossRefGoogle Scholar
  7. 7.
    George, E., Glimm, J., Li, X.-L., Marchese, A., Xu, Z.-L.: A comparison of experimental, theoretical and numerical simulation of Rayleigh-Taylor mixing rates. PNAS 99, 2587–2592 (2002)CrossRefPubMedGoogle Scholar
  8. 8.
    Feller, W.: An introduction to probability theory and its applications. Vol. II. Second edition, New York: John Wiley & Sons Inc., 1971Google Scholar
  9. 9.
    Hill, S.: Channelling in packed columns. Chem. Eng. Sci. 1, 247–253 (1952)CrossRefGoogle Scholar
  10. 10.
    Homsy, G.M.: Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271–311 (1987)CrossRefGoogle Scholar
  11. 11.
    Howarth, L.N.: Bounds on flow quantities. Ann. Rev. Fluid. Mech. 4, 1972Google Scholar
  12. 12.
    Kohn, R.V., Yan, X.: Upper bound on the coarsening rate for an epitaxial growth model. Comm. Pure Appl. Math. 56, 1549–1564 (2003)CrossRefGoogle Scholar
  13. 13.
    Lax, P.D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Philadelphia, Pa.: Society for Industrial and Applied Mathematics, 1973Google Scholar
  14. 14.
    Lieb, E.H., Loss, M.: Analysis. Providence, RI: American Mathematical Society, 1997Google Scholar
  15. 15.
    Manickam, O., Homsy, G.M.: Fingering instabilities in vertical displacement flows in porous media. J. Fluid. Mech. 288, 75–102 (1995)Google Scholar
  16. 16.
    Otto, F.: Evolution of microstructure: an example. In: Ergodic theory, analysis, and efficient simulation of dynamical systems, Berlin: Springer, 2001, pp. 501–522Google Scholar
  17. 17.
    Otto, F.: Cross-over in scaling laws: a simple example from micromagnetics. In: Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Beijing: Higher Ed. Press, 2002, pp. 829–838Google Scholar
  18. 18.
    Saffman, P.G., Taylor, G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. Roy. Soc. London. Ser. A 245, 312–329 (1958) (2 plates)Google Scholar
  19. 19.
    Tanveer, S.: Surprises in viscous fingering. J. Fluid. Mech. 428, 511–545 (2000)Google Scholar
  20. 20.
    Wooding, R.A.: Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid. Mech. 39, 477–495 (1969)Google Scholar
  21. 21.
    Ziemer, W.P.: Weakly differentiable functions. Berlin-Heidelberg-NewYork: Springer-Verlag, 1989Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Institute for Applied MathematicsUniversity of BonnBonnGermany

Personalised recommendations