A family of exponents from a fractal model of viscous fingering and DLA

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Abstract
The authors present a simple fractal model of the interface between oil and water in viscous fingering or the cluster boundary in diffusion-limited aggregation (DLA). The continuum equations for immiscible displacement of Newtonian fluids with no surface tension in a porous medium are solved analytically to find the flow velocity on the boundary. The moments of the velocity distribution scale with the size of the system yielding a family of exponents. The scaling depends on the geometry of the interface and the oil/water viscosity ratio. For an infinite viscosity ratio, which corresponds to DLA, the form of the scaling is in qualitative agreement with the numerical results of Amitrano et al. (1986). The method can be easily extended to study other systems with a fractal boundary condition.