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On the stability of viscous flow between rotating cylinders Part 2. Numerical analysis

Published online by Cambridge University Press:  28 March 2006

D. L. Harris  and
W. H. Reid
D. L. Harris
Affiliation:
The Electro Nuclear Systems Corporation, Bethesda, Maryland
W. H. Reid
Affiliation:
Brown University, Providence, Rhode Island Present address: Departments of Mathematics and the Geophysical Sciences, The University of Chicago.
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Abstract

A simple numerical method is presented for solving the eigenvalue problem which governs the stability of Couette flow. The method is particularly useful in obtaining the eigenfunctions associated with the various modes of instability. When the cylinders rotate in opposite directions, these eigenfunctions exhibit an exponentially damped oscillatory behaviour for sufficiently large values of − μ, where μ = Ω2/Ω1. In terms of the stream function which describes the motion in planes through the axis of the cylinders, this means that weak, viscously driven cells appear in the outer layes of the fluid which, according to Rayleigh's criterion, are dynamically stable. For μ = − 3, for example, four cells are present, the amplitudes of which are in the ratios 1·0:0·0172:0·013:0·00125.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 20 , Issue 1 , September 1964 , pp. 95 - 101
Copyright
© 1964 Cambridge University Press

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References

Chandrasekhar, S. 1954 Mathematika, 1, 5.
Davey, A. 1962 J. Fluid Mech. 14, 336.
Duty, R. L. & Reid, W. H. 1964 J. Fluid Mech. 20, 81.
Fox, L. 1957 The Numerical Solution of Two-Point Boundary Problems in Ordinary Differential Equations. Oxford University Press.
Hughes, T. H. & Reid, W. H. 1962 Tech. Rep. Div. of Appl. Math. Brown University, no. Nonr 562(07)/50.