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Stability of two-dimensional collapsible-channel flow at high Reynolds number. (English) Zbl 1250.76067
Summary: 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stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension $$T^\ast$$. Far upstream the flow is parallel Poiseuille flow at Reynolds number $$Re$$; the width of the channel $$a$$ is and the length of the membrane is $$\lambda a$$ where $$1\ll Re^{1/7} \lesssim \lambda \lesssim Re$$. Steady flow was studied using interactive boundary-layer theory by J. C. Guneratne and T. J. Pedley [J. Fluid Mech. 569, 151–184 (2006; Zbl 1177.76083)] 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 $$T^\ast$$. 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.

MSC:
 76E05 Parallel shear flows in hydrodynamic stability 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 92C10 Biomechanics
Software:
Algorithm 838; Algorithm 822
Full Text:
References:
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