  We study the dynamics of two air bubbles driven by the motion of a suspending
viscous fluid in a Hele-Shaw channel with a small elevation along its
centreline via physical experiment and numerical simulation of a depth-averaged
model. For a single-bubble system we establish that, in general, bubble
propagation speed monotonically increases with bubble volume so that two
bubbles of different sizes, in the absence of any hydrodynamic interactions
will either coalesce or separate in a finite time. However our experiments
indicate that the bubbles interact and that an unstable two-bubble state is
responsible for the eventual dynamical outcome: coalescence or separation.
These results motivate us to develop an edge-tracking routine and calculate
these weakly unstable two-bubble steady states from the governing equations.
The steady states consist of pairs of `aligned' bubbles that appear on the same
side of the centreline with the larger bubble leading. We also discover,
through time-simulations and physical experiment, another class of two-bubble
states which, surprisingly, consist of stable two-bubble steady states. In
contrast to the `aligned' steady states, these bubbles appear either side of
the centreline and are `offset' from each other. We calculate the bifurcation
structures of both classes of steady states as the flow-rate and bubble volume
ratio is varied. We find that they exhibit intriguing similarities to the
single-bubble bifurcation structure, which has implications for the existence
of $n$-bubble steady states.
