Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-07T08:06:11.923Z Has data issue: false hasContentIssue false
  • English
  • Français

Fingering in Hele-Shaw cells

Published online by Cambridge University Press:  21 April 2006

S. D. Howison
S. D. Howison
Affiliation:
Mathematical Institute, Oxford University, 24–29 St. Giles, Oxford OXl 3LB, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

A large class of explicit solutions for Hele-Shaw flow with a free surface is presented. The results are valid when surface-tension effects in the plane of the cell are negligible. Most of the solutions given produce fingers, both in channel flow and on a growing air bubble. Possible behaviour of these fingers is described, and a qualitative comparison with published experimental results is made.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 167 , June 1986 , pp. 439 - 453
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

Bensimon, D., Kadanoff, L. P., Liang, S., Shraiman, B. I. & Tang, C. 1985 Viscous flows in two dimensions, University of Chicago preprint.
Davis, P. J. 1974 The Schwarz function and its applications. Carus mathematical monographs, 17. Mathematical Association of America.
Dibenedetto, E. & Friedman, A. 1985 Bubble growth in porous media, preprint.
Galin, L. A. 1945 Unsteady filtration with a free surface. Dok. Akad. Nauk. SSSR 47, 246–249.Google Scholar
Howison, S. D. 1986a Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Math. 46, 20–26.Google Scholar
Howison, S. D. 1986b Bubble growth in porous media and Hele-Shaw cells. Proc. R. Soc. Edin. A (to appear).Google Scholar
Howison, S. D., Lacey, A. A. & Ockendon, J. R. 1985 Singularity development in moving boundary problems. Q. J. Mech. Appl. Maths 38, 343–360.Google Scholar
Jacquard, P. & Séguier, P. 1962 Mouvement de deux fluides en contact dans un milieu poreux, J. Méc. 1, 367–394.
Kessler, D. A., Koplik, J. & Levine, H. 1985 Geometric models of interface evolution 3: dendritic growth Phys. Rev. A 31, 1712–1717.Google Scholar
Lacey, A. A. 1982 Moving boundary problems in the flow of liquid through porous media. J. Austral. Math. Soc. B 24, 171–193.Google Scholar
Langer, J. S. 1980 Instabilities and pattern formation in crystal growth. Rev. Mod. Phys. 52,1–28.Google Scholar
Maclean, 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
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell. J. Fluid Mech. 113, 513–529.Google Scholar
Richardson, S. 1972 Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618.Google Scholar
Saffman, P. G. 1983, 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. Maths. 12, 146–150.
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. Lond. A 245, 312–329.Google Scholar
Shraiman, B. & Bensimon, D. 1984 Singularities in non-local interface dynamics. Phys. Rev. A 30, 2840–2842.Google Scholar
Taylor, G. I. & Saffman, P. G. 1958 Cavity flows of viscous liquids in narrow spaces. In Proc. 2nd Symp. on Naval Hydrodynamics.
Taylor, G. I. & Saffman, P. G. 1983, A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Q. J. Mech. Appl. Maths 12, 265–279.
Tryggvason, G. & Aref, H. 1985 Finger-interaction mechanisms in stratified Hele-Shaw flow. J. Fluid Mech. 154, 287–301.Google Scholar
Vandenbroeck, J. M. 1983, Fingers in a Hele-Shaw cell with surface tension. Phys. Fluids 26, 2033–2034.