Weakly nonlinear investigation of the Saffman-Taylor problem in a rectangular Hele-Shaw cell

TL;DR

This paper investigates the early nonlinear dynamics of viscous fingering in a rectangular Hele-Shaw cell, revealing how symmetry breaking and mode interactions influence finger growth and tip-splitting.

Contribution

It provides a weakly nonlinear analysis linking interface asymmetry, viscosity contrast, and mode coupling, explaining finger behavior not captured by linear theory.

Findings

Symmetry breaking occurs via sub-harmonic mode growth.

Absence of finger tip-splitting in early stages is explained.

Growth rates saturate compared to linear stability predictions.

Abstract

We analyze the Saffman-Taylor viscous fingering problem in rectangular geometry. We investigate the onset of nonlinear effects and the basic symmetries of the mode coupling equations, highlighting the link between interface asymmetry and viscosity contrast. Symmetry breaking occurs through enhanced growth of sub-harmonic perturbations. Our results explain the absence of finger tip-splitting in the early flow stages, and saturation of growth rates compared with the predictions of linear stability.