# The influence of invariant solutions on the transient behaviour of an air bubble in a Hele-Shaw channel

## TL;DR

This study explores how invariant solutions influence the transient dynamics of an air bubble in a Hele-Shaw channel, revealing complex shape transitions and bifurcations through a mathematical model aligned with experimental observations.

## Contribution

It demonstrates the application of dynamical systems concepts, including invariant solutions and bifurcations, to model and analyze bubble shape transitions in fluid mechanics.

## Key findings

- Existence of stable asymmetric bubble solutions across flow rates.
- Development of a second stable solution branch above a critical flow rate.
- Identification of unstable periodic orbits as edge states influencing transient behaviour.

## Abstract

We hypothesize that dynamical systems concepts used to study the transition to turbulence in shear flows are applicable to other transition phenomena in fluid mechanics. In this paper, we consider a finite air bubble that propagates within a Hele-Shaw channel containing a depth-perturbation. Recent experiments revealed that the bubble shape becomes more complex, quantified by an increasing number of transient bubble tips, with increasing flow rate. Eventually the bubble changes topology, breaking into two or more distinct entities with non-trivial dynamics. We demonstrate that qualitatively similar behaviour to the experiments is exhibited by a previously established, depth-averaged mathematical model; a consequence of the model's intricate solution structure. For the bubble volumes studied, a stable asymmetric bubble exists for all flow rates of interest, whilst a second stable solution branch develops above a critical flow rate and transitions between symmetric and asymmetric shapes. The region of bistability is bounded by two Hopf bifurcations on the second branch. By developing a method for a numerical weakly nonlinear stability analysis we show that unstable periodic orbits emanate from the Hopf bifurcation at the lower flow rate and, moreover, that these orbits are edge states that influence the transient behaviour of the system.