Fractal-to-nonfractal crossover for viscous fingers

We propose a position-space renormalization-group approach to the problem of viscous fingering in the absence of surface tension, with arbitrary viscosity ratio between the injected and displaced fluid. We find there are only two fixed points, the Eden and the diffusion-limited aggregation (DLA) points. The Eden point, which corresponds to a compact cluster with nonfractal surface, is stable in all directions, while the DLA fixed point is a saddle point. Hence if the viscosity of the injected fluid is not zero, the system must eventually cross over to a compact cluster. We also calculate the crossover exponent φ and crossover radius ${\mathit{R}}_{\times }$, and discuss possible experimental measurements.