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We consider flow along a finite-length collapsible channel driven by a fixedupstream flux, where a section of one wall of a planar rigid channel isreplaced by a plane-strain elastic beam subject to uniform external pressure. Amodified constitutive law is used to ensure that the elastic beam isenergetically conservative. We apply the finite element method to solve thefully nonlinear steady and unsteady systems. In line with previous studies, weshow that the system always has at least one static solution and that there isa narrow region of the parameter space where the system simultaneously exhibitstwo stable static configurations: an (inflated) upper branch and a (collapsed)lower branch, connected by a pair of limit point bifurcations to an unstableintermediate branch. Both upper and lower static configurations can each becomeunstable to self-excited oscillations, initiating either side of the regionwith multiple static states. As the Reynolds number increases along the upperbranch the oscillatory limit cycle persists into the region with multiplesteady states, where interaction with the intermediate static branch suggests anearby homoclinic orbit. These oscillations approach zero amplitude at theupper branch limit point, resulting in a stable tongue between the upper andlower branch oscillations. Furthermore, this new formulation allows us tocalculate a detailed energy budget over a period of oscillation, where we showthat both upper and lower branch instabilities require an increase in the workdone by the upstream pressure to overcome the increased dissipation.

Energetics of collapsible channel flow with a nonlinear fluid-beam model