The interaction of multiple bubbles in a Hele-Shaw channel

TL;DR

This paper investigates the complex dynamics of two air bubbles in a Hele-Shaw channel, revealing unstable and stable steady states, bifurcation structures, and the influence of flow parameters through experiments and simulations.

Contribution

It introduces a novel edge-tracking method to compute weakly unstable two-bubble steady states and uncovers previously unknown stable two-bubble configurations.

Findings

Identification of unstable two-bubble steady states leading to coalescence or separation.

Discovery of stable two-bubble steady states with bubbles on opposite sides of the centreline.

Bifurcation structures similar to single-bubble cases, affecting multi-bubble dynamics.

Abstract

We study the dynamics of two air bubbles driven by the motion of a suspending viscous fluid in a Hele-Shaw channel with a small elevation along its centreline via physical experiment and numerical simulation of a depth-averaged model. For a single-bubble system we establish that, in general, bubble propagation speed monotonically increases with bubble volume so that two bubbles of different sizes, in the absence of any hydrodynamic interactions will either coalesce or separate in a finite time. However our experiments indicate that the bubbles interact and that an unstable two-bubble state is responsible for the eventual dynamical outcome: coalescence or separation. These results motivate us to develop an edge-tracking routine and calculate these weakly unstable two-bubble steady states from the governing equations. The steady states consist of pairs of `aligned' bubbles that appear on the same side of the centreline with the larger bubble leading. We also discover, through time-simulations and physical experiment, another class of two-bubble states which, surprisingly, consist of stable two-bubble steady states. In contrast to the `aligned' steady states, these bubbles appear either side of the centreline and are `offset' from each other. We calculate the bifurcation structures of both classes of steady states as the flow-rate and bubble volume ratio is varied. We find that they exhibit intriguing similarities to the single-bubble bifurcation structure, which has implications for the existence of $n$-bubble steady states.