  We consider flow along a finite-length collapsible channel driven by a fixed
upstream flux, where a section of one wall of a planar rigid channel is
replaced by a plane-strain elastic beam subject to uniform external pressure. A
modified constitutive law is used to ensure that the elastic beam is
energetically conservative. We apply the finite element method to solve the
fully nonlinear steady and unsteady systems. In line with previous studies, we
show that the system always has at least one static solution and that there is
a narrow region of the parameter space where the system simultaneously exhibits
two stable static configurations: an (inflated) upper branch and a (collapsed)
lower branch, connected by a pair of limit point bifurcations to an unstable
intermediate branch. Both upper and lower static configurations can each become
unstable to self-excited oscillations, initiating either side of the region
with multiple static states. As the Reynolds number increases along the upper
branch the oscillatory limit cycle persists into the region with multiple
steady states, where interaction with the intermediate static branch suggests a
nearby homoclinic orbit. These oscillations approach zero amplitude at the
upper branch limit point, resulting in a stable tongue between the upper and
lower branch oscillations. Furthermore, this new formulation allows us to
calculate a detailed energy budget over a period of oscillation, where we show
that both upper and lower branch instabilities require an increase in the work
done by the upstream pressure to overcome the increased dissipation.
