  Deformable bubbles propagated by the flow of a viscous liquid in a planar
Hele-Shaw channel of uniform depth tend to travel steadily along the channel's
streamwise axis and pairs of neighbouring bubbles will either separate or
coalesce because an individual bubble's propagation speed increases
monotonically with its size. Thus, any group of bubbles will eventually
rearrange itself in order of decreasing size and all of the bubbles will
separate. We show that, by introducing a small geometric perturbation to the
channel in the form of an axially-uniform depth reduction along its centreline,
the system supports a multitude of stable bubble formations and this can
disrupt the usual reordering by bubble size. The constituent bubbles of a
stable formation lie in alternation on opposite sides of the
depth-perturbation, retain fixed shapes and propagate steadily at the same
speed. A stable formation is always led by the smallest of its constituent
bubbles, which would be the slowest bubble in isolation. The leading bubble
propagates as if it were isolated and the trailing bubbles reduce their speeds
by adjusting their shapes and overlaps of the depth-perturbation in the
perturbation fields of their preceding nearest-neighbours in order to match
that of the leading bubble. The trailing bubbles can be arranged in any order
and, hence, the number of stable formations increases factorially as the number
of bubbles is increased.
