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Stability of the unsteady viscous flow in a curved pipe

Published online by Cambridge University Press:  21 April 2006

Demetrius Papageorgiou
Demetrius Papageorgiou
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA
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Abstract

The linear stability of the flow of an incompressible viscous fluid through a curved pipe of circular cross-section is considered. There is a sinusoidal pressure gradient, with zero mean, acting down the pipe. The flow is shown to be unstable to a Taylor-Görtler mode of instability, with vortices aligned with the basic flow first appearing at the outer bend of the pipe when a critical value of the Taylor number is exceeded. A WKBJ perturbation solution is constructed and the form of the vortex amplitude is determined. This solution is found to break down in the vicinity of the pipe's outer bend, and an inner solution is presented to overcome this. The solution is determined by identifying a saddle point in the complex plane of the cross-sectional angle coordinate. This leads to an eigenvalue problem for the Taylor number, for fixed wavenumber and cross-sectional angle coordinate, which in turn leads to the determination of the critical Taylor number above which instability sets in.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 182 , September 1987 , pp. 209 - 233
Copyright
© 1987 Cambridge University Press

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