Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis

TL;DR

This paper presents a weakly nonlinear analysis of the Saffman-Taylor viscous fingering instability in a Hele-Shaw cell, explaining complex pattern formation phenomena like finger competition, spreading, and splitting.

Contribution

It introduces a weakly nonlinear theoretical framework that predicts finger interactions and pattern dynamics beyond linear stability analysis.

Findings

Enhanced growth of sub-harmonic perturbations causes finger competition.

Harmonic modes lead to spreading and splitting of fingers.

Nonlinear mode-coupling with phase relations amplifies perturbations.

Abstract

The Saffman-Taylor viscous fingering instability occurs when a less viscous fluid displaces a more viscous one between narrowly spaced parallel plates in a Hele-Shaw cell. Experiments in radial flow geometry form fan-like patterns, in which fingers of different lengths compete, spread and split. Our weakly nonlinear analysis of the instability predicts these phenomena, which are beyond the scope of linear stability theory. Finger competition arises through enhanced growth of sub-harmonic perturbations, while spreading and splitting occur through the growth of harmonic modes. Nonlinear mode-coupling enhances the growth of these perturbations with appropriate relative phases, as we demonstrate through a symmetry analysis of the mode coupling equations. We contrast mode coupling in radial flow with rectangular flow geometry.