Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2025-01-04T04:11:55.758Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of surface tension on the shape of fingers in a Hele Shaw cell

Published online by Cambridge University Press:  20 April 2006

J. W. McLean  and
P. G. Saffman
J. W. McLean
Affiliation:
Applied Mathematics, California Institute of Teclmology, Pasadena, California 91125
P. G. Saffman
Affiliation:
Applied Mathematics, California Institute of Teclmology, Pasadena, California 91125
Get access Rights & Permissions [Opens in a new window]

Abstract

The experimental results of Saffman & Taylor (1958) and Pitts (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the interface. Using free streamline techniques, the shape of the free surface is expressed as the solution of a nonlinear integro-differential equation. The equation is solved numerically and the solutions are compared with experimental results. The shapes of the profiles are very well predicted, but the dependence of finger width on surface tension is not quantitatively accurate, although the qualitative behaviour is correct. A conflict between the numerics and a formal singular perturbation analysis is noted but not resolved. The stability of the steady finger to small disturbances is also examined. Linearized stability analysis indicates that the two-dimensional fingers are not stabilized by the surface-tension effect, which disagrees with the experimental observations. A possible reason for the discrepancy between theory and experiment is suggested.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 102 , January 1981 , pp. 455 - 469
Copyright
© 1981 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

Carrier, G. F., Krook, M. & Pearson, C. E. 1966 Functions of a Complex Variable. McGraw-Hill.
Fairbrother, F. & Stubbs, A. E. 1935 The ‘bubble tube’ method of measurement. J. Chem. Soc. 1, 527–530.Google Scholar
Hele-Shaw, H. J. S. 1898 On the motion of a viscous fluid between two parallel plates. Nature 58, 34–36.Google Scholar
Jacquard, P. & Séguier, P. 1962 Mouvement de deux fluides en contact dans un milieu poreux. J. Méc. 1, 367–394.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.
Longuet-Higgins, M. S. 1978 The instabilities of gravity waves of finite amplitude in deep water. Proc. Roy. Soc. A 360, 471–488.Google Scholar
McLean, J. W. 1980 The fingering problem in flow through porous media. Ph.D. dissertation. California Institute of Technology.
Meng, J. C. S. & Thomson, J. A. L. 1978 Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods. J. Fluid Mech. 84, 433–453.Google Scholar
Pitts, E. 1980 Penetration of fluid into a Hele Shaw cell. J. Fluid Mech. 97, 53–64.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of fluid into a porous medium or Hele-Shaw cell. Proc. Roy. Soc. A 245, 312–329.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 fluids in narrow spaces. 2nd Symp. on Naval Hydrodynamics.
Wooding, R. A. & Morel-Seytoux, H. J. 1976 Multiphase fluid flow through porous media. Ann. Rev. Fluid Mech. 8, 233–274.Google Scholar