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

Bubble propagation in Hele-Shaw channels with centred constrictions