FINGER COMPETITION AND VISCOSITY CONTRAST IN VISCOUS FINGERING - A TOPOLOGICAL APPROACH

Finger competition in rectangular Hele-Shaw cells is discussed with particular focus on topological features of the velocity field. A reduced description of the interface dynamics is provided in terms of the motion of topological defects which obey simple conservation rules. Based on the existence of a topological invariant, a time-dependent quantity S(t), directly related to the number of topological defects, is proposed as a global characterization of the finger competition. This quantity is not simply related to the interface shape but may be used as a measure of the 'complexity' of evolving patterns and to elucidate the conditions for eventual approach to a single finger stationary solution. Some representative numerical simulations for two extreme limits of the viscosity contrast parameter are presented and analyzed in terms of the dynamics of topological defects, Two specific sequences or histories for the creation/annihilation of defects are then identified, corresponding to radically different behavior of competing fingers. With the help of a local vorticity analysis, we conclude that, in the limit of low Viscosity contrast, the Saffman-Taylor finger has a drastically reduced but non-vanishing basin of attraction.