×

zbMATH — the first resource for mathematics

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stability of two-dimensional high-Reynolds-number flow in a rigid parallel-sided channel, of which part of one wall has been replaced by a flexible membrane under longitudinal tension \(T^\ast\).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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1016/j.euromechflu.2009.03.002 · Zbl 1167.76329
[2] Abramowitz, Handbook of Mathematical Functions (1970)
[3] DOI: 10.1093/qjmam/29.3.365 · Zbl 0359.76027
[4] DOI: 10.1093/qjmam/29.3.343 · Zbl 0359.76026
[5] DOI: 10.1115/1.3426281
[6] DOI: 10.1017/S002211200300394X · Zbl 1049.76015
[7] DOI: 10.1017/S0022112085003512
[8] DOI: 10.1017/S0022112090003408 · Zbl 0708.76056
[9] Pedley, Perspectives in Fluid Dynamics pp 105– (2000)
[10] DOI: 10.1017/S0022112006002655 · Zbl 1177.76083
[11] DOI: 10.1017/CBO9780511896996
[12] DOI: 10.1145/592843.592847 · Zbl 1070.65511
[13] DOI: 10.1017/S0022112098001062 · Zbl 0924.76023
[14] DOI: 10.1017/S0022112096000286 · Zbl 0875.76264
[15] DOI: 10.1145/1039813.1039818 · Zbl 1074.33017
[16] DOI: 10.1017/S0022112008000293 · Zbl 1151.76455
[17] DOI: 10.1109/TBME.1969.4502660
[18] DOI: 10.1016/S0006-3495(69)86451-9
[19] DOI: 10.1093/qjmam/36.2.271 · Zbl 0542.76053
[20] DOI: 10.1016/0889-9746(91)90421-K
[21] DOI: 10.1017/S0022112010003277 · Zbl 1205.76077
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.