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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
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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

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