  Motivated by the reopening mechanics of strongly collapsed airways, we study
the steady propagation of an air finger through a collapsed oil-filled channel
with a single compliant wall. In a previous study using fully-compliant elastic
tubes, a `pointed' air finger was found to propagate at high speed and low
pressure, which may enable rapid reopening of highly collapsed airways with
minimal tissue damage (Heap & Juel 2008). In this paper, we identify the
selection mechanism of that pointed finger, which remained unexplained, by
conducting an experimental study in a rigid rectangular Hele-Shaw channel with
an elastic top boundary. The constitutive behaviour of this elasto-rigid
channel is broadly similar to that of an elastic tube, but unlike the tube the
channel's cross-section adopts self-similar shapes from the undeformed state to
the point of first near wall contact. The simplification of the vessel geometry
enables the systematic investigation of the reopening dynamics in terms of
initial collapse. We find that for low levels of initial collapse, a single
centred symmetric finger propagates in the channel and its shape is set by the
tip curvature. As the level of collapse increases, the channel cross-section
develops a central region of near opposite wall contact, and the finger shape
evolves smoothly towards a `flat-tipped' finger whose geometry is set by the
strong depth gradient near the channel walls. We show that the flat-tipped mode
of reopening is analogous to the pointed finger observed in tubes. Its
propagation is sustained by the vessel's extreme cross-sectional profile at
high collapse, while vessel compliance stabilises it. A simple scaling argument
based on the dissipated power reveals that this reopening mode is preferred at
higher propagation speeds when it becomes favourable to displace the elastic
channel wall rather than the viscous fluid.
