An island of stability in a sea of fingers: emergent large-scale features of the viscous flow instability

TL;DR

This paper investigates the large-scale features of viscous flow instabilities, revealing a stable inner region during fluid displacement that depends solely on viscosity ratio, independent of finger width.

Contribution

It introduces the concept of a stable inner region in viscous fingering, showing its size depends only on viscosity ratio, a novel insight into pattern formation.

Findings

Existence of a stable inner region during viscous displacement.

The ratio of finger length to stable region radius depends only on viscosity ratio.

This ratio is independent of the most unstable wavelength.

Abstract

The displacement of a more viscous fluid by a less viscous one in a quasi-two dimensional geometry leads to the formation of complex fingering patterns. This fingering has been characterized by a most unstable wavelength, $\lambda_c$, which depends on the viscosity difference between the two immiscible fluids and sets the characteristic width of the fingers. How the finger length grows after the instability occurs is an equally important, but previously overlooked, aspect that characterizes the global features of the patterns. As the lower viscosity fluid is injected, we show that there is a stable inner region where the outer fluid is completely displaced. The ratio of the finger length to the radius of this stable region depends only on the viscosity ratio of the fluids and is decoupled from $\lambda_c$.