Stability of two-dimensional collapsible-channel flow to inviscid perturbations

We consider the linear stability of two-dimensional inviscid but vortical flow in a rigid, parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension $T$. Far upstream the flow is parallel Poiseuille flow; the width of the channel is $a$ and the length of the membrane is $\lambda a$, where $\lambda \gg 1$. Steady flow at high Reynolds number $Re$ was studied using interactive boundary-layer theory by Guneratne \& Pedley (J. Fluid Mech. \textbf{569}, 151-184, 2006) for various values of the pressure difference $P_e$ across the membrane at its upstream end. Here we study small-amplitude, unsteady perturbations to the trivial steady solution for $P_e = 0$. An unexpected finding is that the flow is always unstable, with a growth rate that increases with $T$. In other words, the stability problem is ill-posed. However, when the pressure difference is held fixed (= 0) at the downstream end of the membrane, the problem is well-posed and all solutions are stable. The physical mechanisms underlying these findings are explored; the crucial factor in the fluid dynamics is the vorticity gradient across the incoming Poiseuille flow. Similar results are found for the viscous problem at high $Re$.