Pattern formation of the three-layer Saffman-Taylor problem in a radial Hele-Shaw cell

TL;DR

This study investigates the nonlinear dynamics of three-layer Saffman-Taylor fingering in a radial Hele-Shaw cell, revealing how initial layer thickness influences pattern morphology and stability, with results aligning with experiments and weakly nonlinear theory.

Contribution

It extends previous weakly nonlinear analysis by exploring full nonlinear interfacial dynamics using a spectral boundary integral method, highlighting new pattern formations and stability characteristics.

Findings

Less unstable structures with reduced finger tip-splitting as initial annulus thickness decreases

Development of fingers with low-amplitude bumps at tips due to high-frequency mode growth

Good agreement between nonlinear simulations, experiments, and weakly nonlinear analysis

Abstract

The three-layer Saffman-Taylor problem introduces two coupled moving interfaces separating the three fluids. A very recent weakly nonlinear analysis of this problem in a radial Hele-Shaw cell setup has shown that the morphologies of the emerging fingering patterns strongly depend on the initial thickness of the intermediate layer connecting the two interfaces. Here we go beyond the weakly nonlinear regime and explore full nonlinear interfacial dynamics using a spectrally accurate boundary integral method. We quantify the nonlinear instability of both interfaces as the relevant physical parameters (e.g., viscosities and surface tensions) are varied and show that our nonlinear computations are in good agreement with the experimental observations and the weakly nonlinear analysis. Nonlinear simulations reveal that due to the existence of a second interface, the classical highly branched morphologies are replaced by less unstable structures in which finger tip-splitting and finger competition phenomena are evidently restrained as the initial annulus' thickness is reduced. In addition, these novels patterns develop fingers with a series of low-amplitude bumps at their tips, associated with the enhanced growth of high-frequency modes promoted by the increasing coupling strength of interfaces.