The interaction of multiple bubbles in a Hele-Shaw channel

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, the 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 to 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-dependent simulations and physical experiment, another class of two-bubble states that, surprisingly, are stable. In contrast to the ‘aligned’ steady states, these bubbles appear on 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 are 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.

On the selection of Saffman–Taylor viscous fingers for divergent flow in a wedge

“…Finally, there are many recent experimental and numerical results for different Hele-Shaw problems, some of which appear to have similar selected branches of permitted solutions (Gaillard et al 2021;Keeler et al 2022). We expect that the techniques used here can also be applied to these problems to derive selection conditions, governed by exponentially small components of the solutions.…”

“…2021; Keeler et al. 2022). We expect that the techniques used here can also be applied to these problems to derive selection conditions, governed by exponentially small components of the solutions.…”

The unpredictable nature of bubble evolution

“…Based on these results, we infer that a continuum of size fractions between the resulting post-breakup bubbles is possible. Although we do not pursue the resulting post-breakup dynamics in this paper, based on a previous study of two-bubble dynamics by Keeler et al 32 , it is expected that the simulations would capture the added multiplicity of long-term behaviours that were observed experimentally. In Fig.…”

“…The validity of this model in describing the dynamics of single air bubbles has been verified in Franco-Gomez et al 9,36 and Keeler et al 10 . Multiple-bubble interactions have also recently been studied in both Gaillard et al 11 and Keeler et al 32 . Here, we recall the important details.…”

“…The dynamics of multiple interacting bubbles are non-trivial and are not pursued in this paper. Instead, a comprehensive study can be found in Keeler et al 32 .…”