Selection rules for the tip-splitting instability.

The local destabilization of a Saffman-Taylor viscous finger occurs by a splitting of its tip and results in the formation of two branches separated by a fjord. The accumulation of such instabilities leads to complex patterns. In this paper we present a detailed analysis of a dynamical model that accounts for the selection of both the width and the orientation of the fjords growing in a wedge of angle ${\ensuremath{\theta}}_{0}.$ It is shown that the selection rules have a dynamical origin and are related to the existence of attracting sets that disappear in the absence of surface tension. We also infer the existence of a critical angle ${\ensuremath{\theta}}_{c}=60\ifmmode^\circ\else\textdegree\fi{}$ such that if ${\ensuremath{\theta}}_{0}l{\ensuremath{\theta}}_{c},$ the symmetric tip-splitting becomes unstable.