Stability of pressure-driven collapsible-channel flow

We consider the stability of fluid flow along a planar finite-length flexible-walled channel driven by fixed upstream pressure and externally subject to a constant applied pressure. This pressure-driven system admits a steady state that is inflated for low external pressures, but which gradually collapses as the external pressure increases; collapse is accompanied by a reduction in the steady flow rate along the channel. However, for some parameters, this collapse is non-monotonic, and we compute two stable steady states: one with modest flow rate and another with very low flow rate. We show that both branches of steady states can become unstable to self-excited oscillations, where the corresponding neutral stability curve takes the form of a two-branch tongue. For steady solutions with modest flow rate, these oscillations are analogous to those observed in the flux-driven system from a mildly collapsed steady state, where the neutrally stable wall profile takes the form of a standing wave. Conversely, for low steady flow rates, the system instead exhibits violent “slamming” oscillations, where the flexible wall is transiently drawn toward the lower rigid wall for a short interval over every period. We show that both forms of oscillation involve a net increase in the upstream flow rate into the channel, which translates into an increase in the work done by the upstream driving pressure. However, the corresponding net energy flux extracted from the mean flow is significantly smaller and has a different sign between the two cases.