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 ${D}_{\mathrm{VF}}\ensuremath{\simeq}{D}_{\mathrm{DLA}}\ensuremath{\simeq}1.3$, but the mass amplitudes obey $\frac{{A}_{\mathrm{VF}}}{{A}_{\mathrm{DLA}}}\ensuremath{\simeq}1.3$. The effective ${D}_{\mathrm{VF}}^{\mathrm{eff}}$ seems to increase for wider pores, while ${D}_{\mathrm{DLA}}^{\mathrm{eff}}$ remains 1.3. For $p>{p}_{c}$, the crossover to homogeneous behavior is very slow, and $1.3<{D}_{\mathrm{DLA}}^{\mathrm{eff}}<{D}_{\mathrm{VF}}^{\mathrm{eff}}<1.7$.