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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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References

Chandrasekhar, S. 1958 The stability of viscous flow between rotating cylinders. Proc. Roy. Soc A 246, 301–311.Google Scholar
Coles, D. 1965 Transition in circular Couette flow J. Fluid. Mech. 21, 385–425.Google Scholar
Davey, A. 1962 The growth of Taylor vortices in flow between rotating cylinders J. Fluid Mech. 14, 336–368.Google Scholar
Donnelly, R. J., 1958 Experiments on the stability of viscous flow between rotating cylinders. I. Torque measurements. Proc. Roy. Soc A 246, 312–325.Google Scholar
Howard, L. N. 1963 Heat transport by turbulent convection J. Fluid Mech. 17, 405–432.Google Scholar
Joseph, D. D. 1966 Nonlinear stability of the Boussinesq equations Arch. Rational Mech. Anal. 22, 163–184.Google Scholar
Malkus, W. V. R. & Veronis, G. 1958 Finite amplitude cellular convection J. Fluid Mech. 4, 225–260.Google Scholar
Orr, W. F. M. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Proc. Roy. Irish Acad A 17, 9–138.Google Scholar
Serrin, J. 1959 On the stability of viscous fluid motions Arch. Rational Mech. Anal. 3, 1–13.Google Scholar
Sparrow, E. M., Munro, W. D. & Jonsson, V. K. 1964 Instability of the flow between rotating cylinders: the wide-gap problem. J. Fluid Mech. 20, 35–46.Google Scholar
Stuart, J. T. 1958 On the non-linear mechanics of hydrodynamic stability J. Fluid Mech. 4, 1–21.Google Scholar
Taylor, G. I. 1936 Fluid friction between rotating cylinders. Part 1. Torque measurements. Proc. Roy. Soc A 157, 546–595.Google Scholar
Wendt, F. 1933 Turbulents Stromungen zwischen zwei rotierenden conaxialen Zylindern Ingen. Arch. 4, 577–595.Google Scholar