Viscous Fingers and Diffusion-Limited Aggregates near Percolation

Abstract

Viscous fingers (VF) and diffusion-limited aggregates (DLA) are shown to exhibit different features when they grow on dilute percolating networks. On the infinite incipient cluster at the site-percolation threshold $p_c$ on the square lattice, both fractal dimensionalities are $D_{\mathrm{VF}}\simeq D_{\mathrm{DLA}}\simeq 1.3$, but the mass amplitudes obey $\frac{A_{\mathrm{VF}}}{A_{\mathrm{DLA}}}\simeq 1.3$. The effective $D_{\mathrm{VF}}^{\mathrm{eff}}$ seems to increase for wider pores, while $D_{\mathrm{DLA}}^{\mathrm{eff}}$ remains 1.3. For $p>\mathrm{}{p}_c$, the crossover to homogeneous behavior is very slow, and $1.3<\mathrm{}{D}_{\mathrm{DLA}}^{\mathrm{eff}}<D_{\mathrm{VF}}^{\mathrm{eff}}<1.7\mathrm{}$.