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Stability of plane Poiseuille flow to periodic disturbances of finite amplitude in the vicinity of the neutral curve

Published online by Cambridge University Press:  28 March 2006

Chaim L. Pekeris  and
Boris Shkoller
Chaim L. Pekeris
Affiliation:
Department of Applied Mathematics, The Weizmann Institute, Rehovoth, Israel
Boris Shkoller
Affiliation:
Department of Applied Mathematics, The Weizmann Institute, Rehovoth, Israel
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Abstract

Stuart (1960) has developed a theory of the stability of plane Poiseuille flow to periodic disturbances of finite amplitude which, in the neighbourhood of the neutral curve, leads to an equation of the Landau (1944) type for the amplitude A of the disturbance: \[ d|A|^2/dt = k_1|A|^2 - k_2|A|^4. \] If k2 is positive in the supercritical region (R > RC) where k1 is positive, then, according to Stuart, there is a possibility of the existence of periodic solutions of finite amplitude which asymptotically approach a constant value of (k1/k2)½. We have evaluated the coefficient k2 and found that there indeed exists a zone in the (α, R)-plane where it is positive. This is the zone inside the dashed curve shown in figure 1, with the region of instability predicted by the linear theory included inside the ‘neutral curve’. Stuart's theory and Eckhaus's generalization thereof could apply in the overlapping zone just above the lower branch of the neutral curve.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 29 , Issue 1 , 18 July 1967 , pp. 31 - 38
Copyright
© 1967 Cambridge University Press

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References

Eckhaus, W. 1965 Studies in Non-Linear Stability Theory, p. 98. New York: Springer.
Landau, L. 1944 C.R. Acad. Sci. U.R.S.S. 44, 311.
Meksyn, D. & Stuart, J. T. 1951 Proc. Roy. Soc. A, 208, 517.
Stuart, J. T. 1958 J. Fluid Mech. 4, 1.
Stuart, J. T. 1960 J. Fluid Mech. 9, 357.
Thomas, L. H. 1953 Phys. Rev. 91, 780.
Watson, J. 1960 J. Fluid Mech. 9, 371.