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The development of Poiseuille flow

Published online by Cambridge University Press:  29 March 2006

S. Wilson
S. Wilson
Affiliation:
Department of Mathematics, University of Manchester
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Abstract

The problem considered is that of two-dimensional viscous flow in a straight channel. The steady Navier-Stokes equations are linearized on the assumption of small disturbance from the fully developed flow, leading to an eigenvalue equation resembling the Orr-Sommerfeld equation. This is solved in the limiting cases of small and large Reynolds number R, and an approximate method is proposed for moderate R. The main results are (i) the dominant mode of the disturbance velocity (i.e. that which persists longest) is antisymmetrical; (ii) for large R there are two sequences of eigenvalues. Both sequences are asymptotically real as R → ∞. The members of the first sequence are O(1) as R → ∞ and are complex for all finite R. The members of the second sequence are O(R−1) and the imaginary part is O(R−N) for all N. It is the eigenvalues of the second sequence which will dominate the flow at large R.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 38 , Issue 4 , 6 October 1969 , pp. 793 - 806
Copyright
© 1969 Cambridge University Press

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