On the oscillatory dynamics of a Saffman--Taylor finger with a bubble at its tip

TL;DR

This study investigates how a small bubble at the tip of a propagating air finger in a Hele-Shaw channel causes oscillatory and disordered interfacial dynamics, revealing conditions that lead to different behaviors.

Contribution

It provides a systematic experimental analysis of the influence of bubble size and flow rate on the oscillatory and steady states of the finger-bubble system in Hele-Shaw flows.

Findings

Oscillations occur when bubble size matches finger tip curvature.

Disordered dynamics emerge at high flow rates with irregular finger meandering.

Oscillations cease when the bubble becomes large enough to form a compound finger.

Abstract

The complex behaviour of air-liquid interfaces driven into Hele-Shaw channels at high speeds could arise from oscillatory dynamics; yet, both the physical and dynamical mechanisms that lead to interfacial oscillations remain unclear. We extend the experiments by Couder \textit{et. al.} (\textit{Phys. Rev. A}, vol. 34, 1986, p. 5175) to present a systematic investigation of the dynamics that result when a small air bubble is placed at the tip of a steadily propagating air finger in a Hele-Shaw channel. The system can exhibit steady and oscillatory behaviour, and we show that these different behaviours each occur in well-defined regions of the phase space defined by flow rate and bubble size. For sufficiently large flow rates, periodic finger oscillations give way to disordered dynamics characterised by an irregular meandering of the finger's tip. We demonstrate that at a fixed flow rate, the oscillations commence when the bubble size is increased sufficiently so that the decreased in-plane curvature of the bubble tip matches the in-plane curvature of the finger tip. The equality between the two in-plane curvatures causes the axial pressure gradient across the bubble, which drives the finger, to vanish, thus rendering the finger susceptible to lateral perturbations. Differing timescales for finger and bubble restoral under perturbation allow sustained oscillations to develop in the finger-bubble system. The oscillations cease when the bubble is sufficiently large that it can act as the tip of a compound finger. The disordered dynamics at high flow rates are consistent with the transient exploration of unstable periodic states, which suggests that similar dynamics may underlie the observed disordered dynamics in viscous fingering.