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stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension T*. Far upstream the flow is parallel Poiseuille flow at Reynolds number Re; the width of the channel is a and the length of the membrane is lambda a, where 1 << Re-1/7 less than or similar to lambda << 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*. 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.