Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T07:33:59.406Z Has data issue: false hasContentIssue false
  • English
  • Français

Interfacial wave theory for oscillatory finger formation in a Hele–Shaw cell: a comparison with experiments

Published online by Cambridge University Press:  26 September 2008

Jian-Jun Xu
Jian-Jun Xu
Affiliation:
Department of Mathematics and Statistics, McGill University, Montreal, Quebec, Canada H3A 2K6
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper is devoted to an analysis of the formation of oscillatory viscous fingers in a Hele-Shaw cell on the basis of the interfacial wave theory, previously established for the pattern formation dynamics in dendrite growth, as well as in the classic Saffman–Taylor flow. In particular, we study the problem of selection and persistence of oscillatory fingers with a tiny bubble at the finger tip. We obtain uniformly valid asymptotic solutions for this problem, and derive the linear, global wave instability mechanism for this more complicated system. The global, neutrally stable modes are computed in a large region of parameters, which select the form of oscillatory fingers in the later stage of evolution. We have compared the theoretical predictions with the experimental data by Couder et al. (1986) and by Kopf-Sill & Homsy (1987), and found excellent quantitative agreement.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 7 , Issue 2 , April 1996 , pp. 169 - 199
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Zhuravlev, P. A. 1956 Zap Leningr. Com. Inst. 33, 54–61.Google Scholar
[2] Couder, Y., Gerard, N. & Rabaud, M. 1986 Narrow fingers in the Saffman–Taylor instability. Phys. Rev. A 34, 5175.CrossRefGoogle ScholarPubMed
[3] Kopf-Sill, A. R. & Homsy, G. M. 1987 Narrow fingers in a Hele–Shaw cell. Phys. Fluids 30 (9), 2607.CrossRefGoogle Scholar
[4] Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid. Proc. R. Soc. London Ser. A 245, 312.Google Scholar
[5] Bensimon, D., Pelce, P. & Shraiman, B. I. 1987 Dynamics of curved fronts and pattern selection. J. Physique 48, 2081.CrossRefGoogle Scholar
[6] Romero, L. A. 1981 PhD Thesis, California Institute Technology.Google Scholar
[7] Vanden-Broeck, J.-M. 1983 Fingers in a Hele–Shaw cell with surface tension. Phvs. Fluids 26 (8), 2033.CrossRefGoogle Scholar
[8] Tanveer, S. 1987 Analytic theory for the selection of a symmetric Saffman–Taylor instability. Phys. Fluid 30 (8), 1589.CrossRefGoogle Scholar
[9] Hong, D. C. & Langer, J. S. 1986 Analytic theory of the selection mechanism in the Saffman–Taylor problem. Phys. Rev. Lett. 56, 2032.CrossRefGoogle ScholarPubMed
[10] Kessler, D. A. & Levine, H. 1985 Stability of finger patterns in Hele–Shaw cells. Phys. Rev. A 32, 1930. (Also: 1986 Theory of the Saffman–Taylor ‘finger’ pattern. Phys. Rev. A 323, 2632.)CrossRefGoogle ScholarPubMed
[11] McLean, J. & Safpman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele–Shaw cell. J. Fluid Mech. 102, 455.CrossRefGoogle Scholar
[12] DeGregoria, A. J. & Schwartz, L. W. 1986 A boundary integral method for two-phase displacement in Hele–Shaw cell. J. Fluid Mech. 164, 383.CrossRefGoogle Scholar
[13] Meiburg, E. & Homsy, G. M. 1988 Phys. Fluids 31 (3), 429.CrossRefGoogle Scholar
[14] Tabling, P., Zocch, G. & Libchaber, A. 1987 An experimental study of the Saffman–Taylor instability. J. Fluid Mech. 177, 67.CrossRefGoogle Scholar
[15] Homsy, G. M. 1987 Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271.CrossRefGoogle Scholar
[16] Howison, S. D., Lacey, A. A. & Ockendon, J. R. 1988 Hele–Shaw free boundary problems with suction. Q. J. Mech. Appl. Math. 41, 183.CrossRefGoogle Scholar
[17] Xu, J. J. 1989 Interfacial wave theory for dendritic structure of a growing needle crystal: (I) local instability mechanism; (Il) wave emission mechanism at turning point. Phys. Rev. A 40 (3), 1599, 1609.CrossRefGoogle Scholar
[18] Xu, J. J. 1990 Global neutral stability of dendrite growth and selection condition of tip velocity. J. Crystal Growth 100, 481.CrossRefGoogle Scholar
[19] Xu, J. J. 1990 Asymptotic theory of steady axi-symmetrical needlike crystal growth. Studies in Appl. Math. 83, 71.Google Scholar
[20] Xu, J.J. 1991 Interfacial wave theory of solidification: dendritic pattern formation and selection of growth velocity. Physical Review A 15 43 (2), 930–947.CrossRefGoogle ScholarPubMed
[21] Xu, J. J. & Pan, Z. X. 1993 Interfacial wave theory of dendritic growth from a binary mixture: a comparison with experiment. J. Crystal Growth 129, 666.CrossRefGoogle Scholar
[22] Xu, J.J. 1991 Global instability of viscous fingering in a Hele–Shaw cell: formation of oscillatory fingers. Euro. J. Appl. Math. 2, 105–132.CrossRefGoogle Scholar
[23] Xu, J. J., Pan, Z. X. & Cole, J. D. 1993 Numerical investigation of global instability mechanism of dendrite growth. Euro. J. Appl. Math, (submitted).Google Scholar
[24] Segur, H., Tanveer, S. and Levine, H. (eds.) 1991 Asymptotics Beyond All Orders NATO ASI Series, Series B: Physics, Vol. 284. Plenum Press, New York.CrossRefGoogle Scholar
[25] Xu, J. J. 1996 Interfacial instabilities and fingering formation in Hele–Shaw flow. IMAJ. Appl. Math, (to appear).CrossRefGoogle Scholar