  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.

Fluid Dynamics