Video: Life and fate of a bubble in a constricted Hele-Shaw channel

Abstract

Motivated by the desire to understand complex transient behaviour in fluid flows, we study the dynamics of an air bubble driven by the steady motion of a suspending viscous fluid within a Hele-Shaw channel with a centred depth perturbation. Using both experiments and numerical simulations of a depth-averaged model, we investigate the evolution of an initially centred bubble of prescribed volume as a function of flow rate and initial shape. The experiments exhibit a rich variety of organised transient dynamics, involving bubble break up as well as aggregation and coalescence of interacting neighbouring bubbles. The long-term outcome is either a single bubble or multiple separating bubbles, positioned along the channel in order of increasing velocity. Up to moderate flow rates, the life and fate of the bubble are reproducible and can be categorised by a small number of characteristic behaviours that occur in simply-connected regions of the parameter plane. Increasing the flow rate leads to less reproducible time evolutions with increasing sensitivity to initial conditions and perturbations in the channel. Timedependent numerical simulations that allow for break up and coalescence are found to reproduce most of the dynamical behaviour observed experimentally including enhanced sensitivity at high flow rate. An unusual feature of this system is that the set of steady and periodic solutions can change during temporal evolution because both the number of bubbles and their size distribution evolve due to break up and coalescence events. Calculation of stable and unstable solutions in the single-and two-bubble cases reveals that the transient dynamics are orchestrated by weakly-unstable solutions of the system termed edge states that can appear and disappear as the number of bubbles changes. † Email address for correspondence: anne.juel@manchester.ac.uk arXiv:2005.13959v1 [physics.flu-dyn] 28 May 2020

The interaction of multiple bubbles in a Hele-Shaw channel

“…As a model 'playground' to test these ideas we consider the steady state structure and transient dynamics of two finite air bubbles propagating in a Hele-Shaw channel with a prescribed depth perturbation when the surrounding fluid is extracted at a constant flow rate (see figure 1). In a previous work [9], we showed that a single bubble may break up into two (or more) bubbles depending on its initial spatial configuration and on the flow rate and that, post breakup, the bubbles may either merge back into a single or compound bubble or separate indefinitely (see figure 2). A key result of this study was that the post breakup dynamics were strongly influenced by the existence of weakly unstable steady states that are specific to the two-bubble system.…”

“…Transient behaviour and steady propagation modes have been investigated extensively in the case of a single-bubble using a mixture of analytical and numerical techniques [see, for example 10,17,20,[32][33][34] and experiments [see, for example 18,21,22,31,38,39]. If a depth-perturbation is added to the bottom of the channel as shown in figure 1, the range of existence and stability of steady propagation modes changes dramatically, as mapped out by Franco-Gómez et al [7,8], Gaillard et al [9], Keeler et al [14]. The solution branches interact in a highly nontrivial manner, resulting in a number of different bifurcations and regions of bi-stability in the system; features absent when there is no geometric perturbation in the channel.…”

“…Recently it has been shown that the transient behaviour of a single-bubble in a perturbed Hele-Shaw channel is heavily influenced by so-called 'edge-states' of the system whose stable and unstable manifolds separate different dynamical outcomes [9,14].…”

Stability and Bifurcation of Dynamic Contact Lines in Two Dimensions

“…If so, the unstable branch is not just an insignificant consequence of the fold bifurcation but provides unique insight into the system dynamics. The influence and importance of unstable states in fluid dynamics systems has been investigated in many different contexts, including shear flow [11], droplets [17], finite air bubbles [16,26] and a slide-coating flow [8]. Indeed, as shown in figure 2(b), where the phase-plane is sketched for a generic system with an stable ('attractor') and weakly unstable ('saddle-node') state, the unstable state can act as a separator of dynamical outcomes; the stable manifold is a dividing 'line' and the unstable manifold connects to the stable state.…”

Bifurcations of drops and bubbles propagating in variable-depth Hele-Shaw channels

“…Recent 3D studies for strongly confined drops [12,13] confirm that the films which confine the drops are not uniform in thickness. These thin films affect the drag on the drop [14][15][16][17] and have an important role in systems with surfactants as they dominate the surface area of the drop, and hence, any surface transport processes [18,19].…”