Stability of two-dimensional collapsible-channel flow at high Reynolds number

We study the linear stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension ${T}^{\ensuremath{\ast} } $ . Far upstream the flow is parallel Poiseuille flow at Reynolds number $\mathit{Re}$ ; the width of the channel is $a$ and the length of the membrane is $\lambda a$ , where $1\ll {\mathit{Re}}^{1/ 7} \lesssim \lambda \ll \mathit{Re}$ . Steady flow was studied using interactive boundary-layer theory by Guneratne & Pedley (J. Fluid Mech., vol. 569, 2006, pp. 151–184) for various values of the pressure difference ${P}_{e} $ across the membrane at its upstream end. Here unsteady interactive boundary-layer theory is used to investigate the stability of 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}^{\ensuremath{\ast} } $ . 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, or a little further downstream, the problem is well-posed and all solutions are stable. The physical mechanisms underlying these findings are explored using a simple inviscid model; the crucial factor in the fluid dynamics is the vorticity gradient across the incoming Poiseuille flow.