RESUMEN
A dynamical systems approach to competition of Saffman-Taylor fingers in a Hele-Shaw channel is developed. This is based on global analysis of the phase space flow of the low-dimensional ordinary-differential-equation sets associated with the classes of exact solutions of the problem without surface tension. Some simple examples are studied in detail. A general proof of the existence of finite-time singularities for broad classes of solutions is given. Solutions leading to finite-time interface pinchoff are also identified. The existence of a continuum of multifinger fixed points and its dynamical implications are discussed. We conclude that exact zero-surface tension solutions taken in a global sense as families of trajectories in phase space are unphysical because the multifinger fixed points are nonhyperbolic, and an unfolding does not exist within the same class of solutions. Hyperbolicity (saddle-point structure) of the multifinger fixed points is argued to be essential to the physically correct qualitative description of finger competition. The restoring of hyperbolicity by surface tension is proposed as the key point to formulate a generic dynamical solvability scenario for interfacial pattern selection.
RESUMEN
We find that solvability theory selects a set of stationary solutions of the Saffman-Taylor problem with coexistence of two unequal fingers advancing with the same velocity but with different relative widths lambda1 and lambda2 and different tip positions. For vanishingly small dimensionless surface tension d0, an infinite discrete set of values of the total filling fraction lambda=lambda1+lambda2 and of the relative individual finger width p=lambda1/lambda are selected out of a two-parameter continuous degeneracy. They scale as lambda-1/2 approximately d0(2/3) and p-1/2 approximately d0(1/3). The selected values of lambda differ from those of the single finger case. Explicit approximate expressions for both spectra are given.