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We study the singular effects of vanishingly small surface tension on thedynamics of finger competition in the Saffman-Taylor problem, using theasymptotic techniques described in [S. Tanveer, Phil. Trans. R. Soc. Lond. A343, 155 (1993)]and [M. Siegel, and S. Tanveer, Phys. Rev. Lett. 76, 419(1996)] as well as direct numerical computation, following the numerical schemeof [T. Hou, J. Lowengrub, and M. Shelley,J. Comp. Phys. 114, 312 (1994)]. Wedemonstrate the dramatic effects of small surface tension on the late timeevolution of two-finger configurations with respect to exact (non-singular)zero surface tension solutions. The effect is present even when the relevantzero surface tension solution has asymptotic behavior consistent with selectiontheory.Such singular effects therefore cannot be traced back to steady stateselection theory, and imply a drastic global change in the structure ofphase-space flow. They can be interpreted in the framework of a recentlyintroduced dynamical solvability scenario according to which surface tensionunfolds the structually unstable flow, restoring the hyperbolicity ofmultifinger fixed points.Comment: 16 pages, 15 figures, submitted to Phys. Rev

Effects of small surface tension in Hele-Shaw multifinger dynamics: An analytical and numerical study