VOLUME 56s NUMBER PHYSICAL REVIEW LETTERS Shape Selection of Saffman-Taylor Fingers 12 MAY 1986 Roland Combescot, Thierry Dombre, Vincent Hakim, " and Yves PomeautbI Groupe de Physique des Solides, EcoIe NormaIe Superieure, F-7523'J Paris Cedex 05, France and Alain Pumir"' Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, New York I4853 {Received 10 March 1986) Among all problems of pattern selection the one posed by the Saffman-Taylor finger is yet un- solved, although numerics and experiments indicate the very simple property that the relative width of the finger tends to ~ in the low-surface-tension limit. %e explain this by performing an expan- sion beyond all orders leading to the formulation of a nonlinear eigenvalue problem with a discrete set of solutions. In particular we predict that the relative width of the finger tends to ~ as the sur- face tension to the po~er T. PACS numbers: 47, 10. +g, 03. 40.6c, 68. 10. — m Pattern selection in nonequilibrium phenomena, such as dendritic solidification or multiphase fluid flow, is a subject of much recent interest. The deter- mination of the width of Saffman-Taylor fingers is a primary example in that field. Saffman and Taylor' have studied the displacement of a viscous fluid by a less viscous one in a Hele-Shaw cell (a rectangular cav- ity between narrowly spaced glass plates) in an attempt to model displacement of oil by water in porous media. A single finger of the less-viscous fluid is eventually formed and propagates at constant velocity keeping a steady shape. In the absence of interfacial tension between the two fluids, Saffman and Taylor have com- puted explicitly the interface shape. They have ob- tained a continuous family of fingers parametrized by the relative width X between the finger and the chan- nel. What determined the actual shape of the finger was not understood. Then, McLean and Saffman2 derived the equations for the finger profile in the pres- ence of interfacial tension. Their numerical solutions3 give a discrete set of possible relative finger widths which surprisingly all decrease to —, ' when the dimen- sionless surface tension goes to zero. This was quite mysterious because attempts treating interfacial ten- sion as a conventional perturbation were unable to detect any sign of selection. Very recently, in the problem of velocity selection ln dendritic growth, some evidence has been produced numerically, " and analytically in simplifed models, that the selection is due to transcendentally small terms in the small parameter. It ~as proposed in a previous publication that analogous effects are present in the Saffman- Taylor problem and are responsible for the selection. This was done by a partly heuristic method similar to the one of Ref. 5, but important nonlinear effects that cannot be neglected are missed by this linear approach, as explained below. In this Letter, we give a fully non- linear treatment of the problem, free of ary arbitrary assumption, which is inspired by the work of Kruskal and Segur6 on the existence of needle crystals in geometric models of crystallization. Our approach to the nonlocal Saffman- Taylor model resolves the aforementioned difficulties of the linear treatment. It opens the way to a clear understanding of realistic selection phenomena in the broad area of diffusion- controlled interface motion. 8 We explain briefly our approach before proceeding to give more details on our analysis. We are interested in understanding how an arbitrarily small interfacial tension (k (( I) selects a discrete set of finger widths out of a continuum family. McLean and Saffman's equations [see Eqs. (Ia) and (lb) below] are equations for a half-finger profile. We find it convenient to in- troduce a new coordinate along the interface which makes apparent their symmetry with respect to reflec- tion along the center axis of the cell. Perturbation in powers of k gives an asymptotic expansion of the finger profile which respects this natural symmetry for any finger width. The crucial point is surprisingly that this does not give enough information to decide whether a given finger exists or not. Transcendentally small terms in k (i.e. , lying beyond all orders of the asymptotic expansion), whose existence is qualitatively explainable by the fact that the small parameter is in front of the highest derivative, may make the sym- metric continuation of the finger incompatible with a smooth behavior at its tip. We must therefore find a method to compute such terms. Generalizing the ap- proach of Ref. 6 to our problem, we define it in the complex plane. In the neighborhood of one of its singularities, the asymptotic expansion breaks down and small terms become large enough to be noticed and therefore are under control. Here we find a cou- ple of singularities and in their neighborhood we ob- 2036 1986 The American Physical Society