Bubble propagation in Hele-Shaw channels with centred constrictions

TL;DR

This study investigates how finite bubbles behave in a Hele-Shaw channel with a centered constriction, revealing the stability conditions and bifurcation structures that determine whether bubbles stay centered or shift off-center during propagation.

Contribution

The paper introduces a combined experimental and numerical analysis of bubble dynamics in a Hele-Shaw channel with a rail, highlighting the stability regimes and bifurcation phenomena for different bubble sizes and flow conditions.

Findings

Steady symmetric bubble propagation is stable for large bubbles spanning the channel.

For smaller bubbles, static solutions are unstable and bubbles tend to shift off-center.

The depth-averaged model accurately predicts steady modes and bifurcation structures.

Abstract

We study the propagation of finite bubbles in a Hele-Shaw channel, where a centred occlusion (termed a rail) is introduced to provide a small axially-uniform depth constriction. For bubbles wide enough to span the channel, the system's behaviour is similar to that of semi-infinite fingers and a symmetric static solution is stable. Here, we focus on smaller bubbles, in which case the symmetric static solution is unstable and the static bubble is displaced towards one of the deeper regions of the channel on either side of the rail. Using a combination of experiments and numerical simulations of a depth-averaged model, we show that a bubble propagating axially due to a small imposed flow rate can be stabilised in a steady symmetric mode centred on the rail through a subtle interaction between stabilising viscous forces and destabilising surface tension forces. However, for sufficiently large capillary numbers Ca, the ratio of viscous to surface tension forces, viscous forces in turn become destabilising thus returning the bubble to an off-centred propagation regime. With decreasing bubble size, the range of Ca for which steady centred propagation is stable decreases, and eventually vanishes through the coalescence of two supercritical pitchfork bifurcations. The depth-averaged model is found to accurately predict all the steady modes of propagation observed experimentally, and provides a comprehensive picture of the underlying steady bifurcation structure. However, for sufficiently large imposed flow rates, we find that initially centred bubbles do not converge onto a steady mode of propagation. Instead they transiently explore weakly unstable steady modes, an evolution which results in their break-up and eventual settling into a steady propagating state of changed topology.