Stability of finite-length collapsible channel flow to spanwise perturbations

We consider flow through a finite-length flexible-walled channel formed by removing a portion from one wall of a wide rigid channel and replacing by a pre-tensioned hyperelastic sheet of finite thickness. The flow is driven by a prescribed upstream flow rate which is uniform along the channel inlet, and so with periodic boundary conditions in the transverse direction the system admits a steady state with a wall profile which is deformed in the streamwise direction but spatially uniform in the spanwise direction. Identical to the planar case, this system exhibits two stable static configurations: an upper branch where the flexible wall is mostly inflated and a lower branch where the wall is highly collapsed. We consider the stability of these steady states to spanwise perturbations, showing that both can become unstable to distinct families of self-excited oscillations. In particular, for large spanwise wavenumbers, these steady states are unstable to oscillatory normal modes where the perturbation wall profile has a single oscillating hump in the streamwise direction, not previously seen in collapsible channel flows driven by fixed upstream flux. Furthermore, for sufficiently large wavenumbers and no wall pre-tension in the spanwise direction, this system is always unstable to a new spanwise non-uniform static configuration, which arises due to the merging of a pair of unstable oscillatory normal modes. However, with non-zero spanwise wall pre-tension, there is always a region of parameter space where the spanwise uniform steady state is stable for all wavenumbers.