Mean-field diffusion-limited aggregation and the Saffman-Taylor problem in three dimensions

Motivated by the connection to diffusion-limited aggregation (DLA), we study the Saffman-Taylor problem in an axisymmetric tube geometry. We find that the surface tension selects an allowed set of finger widths from the continuous set present at zero surface tension, as is expected by analogy with the channel geometry. However, the solution branches merge and disappear in pairs as surface tension is lowered. This behavior was first seen in self-similar solutions for sector-shaped cells in two dimensions, but this is the first occurrence of this effect for steady-state patterns. The implications of our results for DLA and for flow in porous media are discussed.