Scaling properties of viscous fingering

We present a study of viscous fingering using the Volume Of Fluid method and a central injection geometry, assuming a Laplacian field and a simple surface tension law. As in experiments we see branched structures resulting from the Saffman-Taylor instability. We find that the area $A$ of a viscous-fingering cluster varies as a simple power law $A\sim L^{\alpha }$ of its interface length $L$. Our results are compared to previously published simulations in which the viscosity of the invading fluid is vanishing. We find differences in exponent $\alpha$ and in the appearance of detached droplets and bubbles.