Extremum principles for selection in the Saffman-Taylor finger and Taylor-Saffman bubble problems

We consider the Saffman-Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele-Shaw configuration, and the Taylor-Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of traveling wave solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the channel through which it propagates. To select the "correct" member of the family of solutions we propose two related extremum principles. Employing these principles for a symmetric family with zero surface tension (σ=0) selects the solution (finger or bubble, viscous or inviscid) which happens to be the same as that obtained by taking the limit σ →0 of the nonzero isotropic surface tension problem. For other problems, e.g., perturbation by anisotropic surface tension, the fingers selected by the two approaches are not necessarily the same. We claim that the finger selected by our criteria describes what will be observed on an intermediate time scale in which the effect of perturbations is neglected, whereas the finger selected by taking the limit of vanishingly small perturbations describes what will be observed on the asymptotic time scale O(1/σ) for the problem with the perturbation included. The intermediate time scale is much shorter than the asymptotic time scale. For infinitesimal σ the asymptotic time scale may be well beyond any reasonable observational time scale.