Scaling properties of viscous fingering

TL;DR

This study investigates the scaling behavior of viscous fingering patterns using numerical simulations, revealing power-law relationships and differences from previous models with vanishing viscosity.

Contribution

It introduces a detailed numerical analysis of viscous fingering with finite viscosity, highlighting new scaling laws and morphological differences from prior idealized models.

Findings

Area scales as a power law with interface length

Differences observed compared to vanishing viscosity models

Detached droplets and bubbles appear in simulations

Abstract

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.