Analytic theory for the selection of a symmetric Saffman–Taylor finger in a Hele–Shaw cell

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Abstract
An analytic theory is presented for the selection mechanism of a symmetric finger with a discrete set of possible width from a continuum of nonsymmetric Saffman–Taylor finger solutions of arbitrary width. Both linear and nonlinear analyses have been carried out and in almost all cases, it is shown analytically that nonsymmetric solutions do not exist for nonzero surface tension if the finger boundary is assumed to be smooth. In the other cases, numerical calculations appear to support the same conclusions. In the asymptotic range, the predicted set of countably infinite possible finger widths according to nonlinear analysis agree with the previous analytical results of Combescot et al. [Phys. Rev. Lett. 5 6, 2036 (1986)] for a finger assumed to be symmetric about the channel centerline. The predicted coefficient of the power law dependence of finger width on surface tension for the first few of these solutions, i.e., on the branches first calculated by Mclean–Saffman [J. Fluid Mech. 1 0 2, 455 (1980)], Romero (Ph.D. thesis, California Institute of Technology, 1982) and Vanden‐Broeck [Phys. Fluids 2 6, 2033 (1983)], are in agreement with Tanveer’s [Phys. Fluids 3 0, 651 (1987)] estimate of this coefficient based on numerical calculations of bubbles. It is also shown that the nonlinear analysis has the same qualitative features as the linear analysis and, in the asymptotic range, the agreement is close.

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