Journal of Fluid Mechanics

Two-phase displacement in Hele Shaw cells: theory

C.-W.  Park a1 and G. M.  Homsy a1
a1 Department of Chemical Engineering, Stanford University, Stanford, California 94305

Article author query
park cw   [Google Scholar] 
homsy gm   [Google Scholar] 


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.

(Published Online April 20 2006)
(Received June 21 1983)
(Revised October 4 1983)