Global stability analysis of flexible channel flow with a hyperelastic wall

We consider the stability of flux-driven flow through a long planar rigid channel, where a segment of one wall is replaced by a pre-tensioned hyperelastic (neo-Hookean) solid of finite thickness and subject to a uniform external pressure. We construct the steady configuration of the nonlinear system using Newton's method with spectral collocation and high-order finite differences. In agreement with previous studies, which use an asymptotically thin wall, we show that the thick-walled system always has at least one stable steady configuration, while for large Reynolds numbers the system exhibits three co-existing steady states for a range of external pressures. Two of these steady configurations are stable to non-oscillatory perturbations, one where the flexible wall is inflated (the upper branch) and one where the flexible wall is collapsed (the lower branch), connected by an unstable intermediate branch. We test the stability of these steady configurations to oscillatory perturbations using both a global eigensolver (constructed based on an analytical domain mapping technique) and also fully nonlinear simulations. We find that both the lower and upper branches of steady solutions can become unstable to self-excited oscillations, where the oscillating wall profile has two extrema. In the absence of wall inertia, increasing wall thickness partially stabilises the onset of oscillations, but the effect remains weak until the wall thickness becomes comparable to the width of the undeformed channel. However, with finite wall inertia and a relatively thick wall, higher-frequency modes of oscillation dominate the primary global instability for large Reynolds numbers.