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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stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension \(T^\ast\).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.

76E05 Parallel shear flows in hydrodynamic stability
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
92C10 Biomechanics
Full Text: DOI
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