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Two-phase displacement in Hele Shaw cells: theory

Published online by Cambridge University Press:  20 April 2006

C.-W. Park  and
G. M. Homsy
C.-W. Park
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, California 94305
G. M. Homsy
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, California 94305
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Abstract

A theory describing two-phase displacement in the gap between closely spaced planes is developed. The main assumptions of the theory are that the displaced fluid wets the walls, and that the capillary number Ca and the ratio of gap width to transverse characteristic length ε are both small. Relatively mild restrictions apply to the ratio M of viscosities of displacing to displaced fluids; in particular the theory holds for M = o(Ca−1/3). We formulate the theory as a double asymptotic expansion in the small parameters ε and Ca1/3. The expansion in ε is uniform while that in Ca1/3 is not, necessitating the use of matched asymptotic expansions. The previous work of Bretherton (1961) is clarified and extended, and both the form and the constants in the effective boundary condition of Chouke, van Meurs & van der Poel (1959) and of Saffman & Taylor (1958) are determined.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 139 , February 1984 , pp. 291 - 308
Copyright
© 1984 Cambridge University Press

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References

Atherton, R. W. & Homsy, G. M. 1976 On the derivation of evolution equations for interfacial waves Chem. Engng Commun. 2, 57.Google Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes J. Fluid Mech. 10, 166.Google Scholar
Chouke, R. L., VAN MEURS, P. & VAN DER POEL, C. 1959 The instability of slow, immiscible, viscous liquid—liquid displacements in permeable media Trans. AIME 216, 188.Google Scholar
Hele Shaw, H. S. 1898 Investigation of the nature of surface resistance of water and of stream-line motion under certain experimental conditions Trans. Instn Nav. Archit., Lond. 40, 21.Google Scholar
Landau, L. D. & Levich, V. G. 1942 Dragging of a liquid by a moving plate Acta Physicochim. URSS 17, 42.Google Scholar
Mclean, J. W. & Saffman, P. G. 1981 The effect of the surface tension on the shape of fingers in a Hele Shaw cell J. Fluid Mech. 102, 455.Google Scholar
Moore, A. D. 1949 Fields from fluid flow mappers J. Appl. Phys. 20, 790.Google Scholar
Renk, F., Wayner, P. C. & Homsy, G. M. 1978 On the transition between a wetting film and a capillary meniscus J. Coll. Interface Sci. 67, 408.Google Scholar
Ruschak, K. J. 1974 The fluid mechanics of coating flows. Ph.D. dissertation, University of Minnesota.
Ruschak, K. J. & Scriven, L. E. 1977 Developing flow on a vertical wall J. Fluid Mech. 81, 305.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, p. 208. Springer.
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.Google Scholar
Van Dyke, M. 1964 Perturbation Methods in Fluid Mechanics. Academic.
Van Dyke, M. 1982 An Album of Fluid Motion. Parabolic.
Wilson, S. D. R. 1982 The drag-out problem in film coating theory J. Engng Maths 16, 209.Google Scholar