Self-similarity and structure of DLA and viscous fingering clusters

Abstract

Diffusion-limited aggregation clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree-like fractals. The different branches may be assigned a branch order in a way that exhibits scaling, and permits a self-similar characterisation in terms of the bifurcation ratio r, and the length ratio r, of branches of different orders. The fractal dimension is given by D = ln(rN)/ln(l/rL). Good agreement between experiments and simulations is found. A crossover function characterises the branch orders.