Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T13:25:22.987Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous fingering in Hele-Shaw cells

Published online by Cambridge University Press:  21 April 2006

P. G. Saffman
P. G. Saffman
Affiliation:
Applied Mathematics, 217-50, California Institute of Technology, Pasadena, CA 91125, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal, structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 173 , December 1986 , pp. 73 - 94
Copyright
© 1986 Cambridge University Press

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

Ben-Jacob, E., Godbey, R., Goldenfeld, N. D., Koplik, J., Levine, H., Mueller, T., & Sander L. M., 1985 Experimental demonstration of the role of anisotropy in interfacial pattern formation. Phys. Rev. Lett. 55, 1315–1318.Google Scholar
Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. I. & Tang, C. 1985 Viscous flows in two dimensions. Rev. Mod. Phys. (to appear).Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Chuoke, R. L., Meurs, P. van & Poel, C. van der 1959 The instability of slow immiscible viscous liquid—liquid displacements in permeable media. Trans. AIME 216, 188–194.Google Scholar
Combescot, R., Dombre, T., Hakim, V., Pomeau, Y. & Pumir, A. 1986 Shape selection of Saffman—Taylor fingers. Phys. Rev. Lett. (submitted).Google Scholar
Coyne, J. C. & Elrod, H. G. 1970 Conditions for the rupture of a lubricating film. Part 1. Theoretical model. Trans. ASME F: J. Lub. Tech. 92, 451–456Google Scholar
Degregoria, A. J. & Schwartz, L. W. 1985 Finger breakup in Hele-Shaw cells. Phys. Fluids 28, 2313–2314.Google Scholar
Degregoria, A. J. & Schwartz, L. W. 1986 A boundary integral method for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 164, 383–400.Google Scholar
Dombre, T., Hakim, V. & Pomeau, Y. 1986 Selection de la larguer des doigts dans l'instabilité de Saffman—Taylor. C. R. Acad Sci. Paris (to appear).Google Scholar
Fairbrother, F. & Stubbs, A. E. 1935 The bubble tube method of measurement. J. Chem. Soc. 1, 527–530.Google Scholar
Hele-Shaw, H. S. 1898 The flow of water. Nature 58, 34–36.Google Scholar
Hill, S. 1952 Channelling in packed columns. Chem. Engng. Sci. 1, 247–253.Google Scholar
Hong, D. C. & Langer, J. S. 1986 Analytic theory of the selection mechanism in the Saffman—Taylor problem. Phys. Rev. Lett. (submitted).Google Scholar
Howison, S. D. 1986 Fingering in Hele-Shaw cells. J. Fluid Mech. 167, 439–453.Google Scholar
Jacquard, P. & Seguier, P. 1962 Movement de deux fluides en contact dans un milieu poreux. J. Méc. 1, 367–394.Google Scholar
Kessler, D. A. & Levine, H. 1985 Stability of finger patterns in Hele-Shaw cells. Phys. Rev. A32, 1930–1933.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Mclean, J. W. & Saffman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 455–469.Google Scholar
Maher, J. V. 1985 The development of viscous fingering patterns. Phys. Rev. Lett. 54, 1498–1501.Google Scholar
Maxworthy, T. 1986 Bubble formation, motion and interaction in a Hele-Shaw cell. J. Fluid Mech. 173, 95–114.Google Scholar
Meng, J. C. S. & Thomson, J. A. L. 1978 Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods. J. Fluid Mech. 97, 53–64.Google Scholar
Nittman, J., Daccord, G. & Stanley, H. E. 1985 Fractal growth of viscous fingers: quantitative characterization of a fluid instability phenomenon. Nature 314, 141–144.Google Scholar
Park, C.-W. Gorell, S. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: experiments on viscously driven instabilities. J. Fluid Mech. 141, 275–287.Google Scholar
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Park, C. W. & Homsy, G. M. 1985 The instability of long fingers in Hele-Shaw flows. Phys. Fluids 28, 1583–1585.Google Scholar
Pitts, E. 1980 Penetration of fluid into a Hele-Shaw cell: the Saffman—Taylor experiment. J. Fluid Mech. 97, 53–64.Google Scholar
Pomeau, Y. 1980 The Saffman—Taylor fingers in the limit of very large surface tension. Stud Appl. Math. 73, 75–89.Google Scholar
Reinelt, D. A. 1986 Interface conditions for two phase flow displacement in Hele-Shaw cells. J. Fluid Mech. (submitted).Google Scholar
Reinelt, D. A. & Saffman, P. G. 1985 The penetration of a finger into a viscous fluid in a channel and tube. SIAM J. Sci. Stat. Comput. 6, 542–561.Google Scholar
Romero, L. A. 1981 The fingering problem in a Hele-Shaw cell. Ph.D. thesis. California Institute of Technology.
Saffman, P. G. 1959 Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium or Hele-Shaw cell. Q. J. Mech Appl. Math. 12, 146–150.Google Scholar
Saffman, P. G. 1982 Fingering in porous media. In Macroscopic Properties of Disordered media (ed. R. Burridge, S. Childress & G. Papanicolaou). Lecture Notes in Physics. vol. 154, pp. 208–215. Springer.
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into porous medium or Hele-Shaw cell containing a more viscouse liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Sarkar, S. K. 1985 Singularities in nonlocal dynamics. Phys. Rev. A 31, 3468–3472.Google Scholar
Shraiman, B. I. 1986 On velocity selection and the Saffman—Taylor problem. Phys. Rev. Lett. (submitted).Google Scholar
Shraiman, B. I. & Bensimon, D. 1984 Singularities in nonlocal interface dynamics. Phys. Rev. A 30, 2840–2843.Google Scholar
Tabeling, P. & Libchaber, A. 1986 Film draining and the Saffman—Taylor problem. Phys. Rev. A 33, 794–796.Google Scholar
Tabeling, P., Zocchi, G. & Libchaber, A. 1986 An experimental study of the Saffman—Taylor instability. J. Fluid Mech. (in press).Google Scholar
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161–165.Google Scholar
Taylor, G. I. & Saffman, P. G. 1958 Cavity flows of viscous liquids in narrow spaces. In Proc. 2nd Symp. on Naval Hydrodynamics, pp. 277–291.
Taylor, G. I. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Q. J. Mech. Appl. Math. 12, 265–279.Google Scholar
Tanveer, S. 1986 The effect of surface tension on the shape of a Hele-Shaw cell bubble. Phys. Fluids (to appear).Google Scholar
Vanden-Broeck, J.-M. 1983 Fingers in a Hele-Shaw cell with surface tension. Phys. Fluids 26, 2033–2034.Google Scholar