a1 Department of Mathematics, Bangalore University, Bangalore-560 001, India
a2 Department of Mathematics, Karnatak University, Dharwad-580 003, India
a3 Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
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stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension . Far upstream the flow is parallel Poiseuille flow at Reynolds number ; the width of the channel is and the length of the membrane is , where . 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 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 . An unexpected finding is that the flow is always unstable, with a growth rate that increases with . In other words, the stability problem is ill-posed. However, when the pressure difference is held fixed () 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.
(Received February 17 2011)
(Reviewed January 11 2012)
(Accepted January 16 2012)
(Online publication February 16 2012)