Fingering versus stability in the limit of zero interfacial tension

TL;DR

This study explores the nonlinear growth of fluid invasion patterns in a quasi-two-dimensional setup without interfacial tension, revealing how viscosity ratios influence the transition from fractal fingering to stable structures.

Contribution

It uncovers the global nonlinear behaviors and pattern transitions driven by viscosity ratios, beyond traditional wavelength-focused analyses.

Findings

High viscosity ratios lead to stable, blunt structures.

Low viscosity ratios produce highly-branched fractal fingers.

Complete displacement occurs around viscosity ratio of 0.3.

Abstract

The invasion of one fluid into another of higher viscosity in a quasi-two dimensional geometry typically produces complex fingering patterns. Because interfacial tension suppresses short-wavelength fluctuations, its elimination by using pairs of miscible fluids would suggest an instability producing highly ramified singular structures. Previous studies focused on wavelength selection at the instability onset and overlooked the striking features appearing more globally. Here we investigate the non-linear growth that occurs after the instability has been fully established. We find a rich variety of patterns that are characterized by the viscosity ratio between the inner and the outer fluid, $\eta_{in}$/$\eta_{out}$, as distinct from the most-unstable wavelength, which determines the onset of the instability. As $\eta_{in}$/$\eta_{out}$ increases, a regime dominated by long highly-branched fractal fingers gives way to one dominated by blunt stable structures characteristic of proportionate growth. Simultaneously, a central region of complete outer-fluid displacement grows until it encompasses the entire pattern at $\eta_{in}$/$\eta_{out}$$\approx$0.3.