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Nonlinear wave interactions in shear flows. Part 1. A variational formulation

Published online by Cambridge University Press:  29 March 2006

J. R. Usher  and
A. D. D. Craik
J. R. Usher
Affiliation:
Department of Mathematics, Teesside Polytechnic, Middlesbrough, England Present address: Department of Mathematics, Glasgow College of Technology, Glasgow, Scotland.
A. D. D. Craik
Affiliation:
Department of Applied Mathematics, University of St Andrews, Fife, Scotland
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Abstract

A modified version of Bateman's variational formulation of the incompressible Navier-Stokes equations and boundary conditions (see Dryden, Murnaghan & Bateman 1956) is introduced. This is employed to examine a particular nonlinear problem of hydrodynamic stability which was treated previously, using a ‘direct’ approach, by Craik (1971). This problem concerns the resonant interaction at second order of a triad of wave modes in a parallel shear flow.

The present method is conceptually attractive; it also has the major advantage over the ‘direct’ method of a substantial reduction in algebraic complexity, which allows results to be derived far more readily. Also, some further improvements are made upon Craik's previous analysis. Such a variational approach may often be simpler than present conventional methods of tackling nonlinear viscous-flow problems. The present paper shows how other problems of nonlinear stability and wave interactions may be tackled in this way.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 66 , Issue 2 , 6 November 1974 , pp. 209 - 221
Copyright
© 1974 Cambridge University Press

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References

Benney, D. J. & Newell, A. C. 1967 J. Math. & Phys. 46, 133.
Boyd, T. J. M. & Turner, J. G. 1972 J. Phys. A, 5, 881.
Boyd, T. J. M. & Turner, J. G. 1973 J. Phys. A, 6, 272.
Craik, A. D. D. 1968 J. Fluid Mech. 34, 531.
Craik, A. D. D. 1971 J. Fluid Mech. 50, 393.
Davey, A. 1972 J. Fluid Mech. 53, 769.
Diprima, R. C., Eckhaus, W. & Segel, L. A. 1971 J. Fluid Mech. 49, 705.
Dryden, H. L., Murnaghan, F. P. & Bateman, H. 1956 Hydrodynamics. Dover.
Finlayson, B. A. 1972a Phys. Fluids, 15, 963.
Finlayson, B. A. 1972b The Method of Weighted Residuals and Variational Principles. Academic.
Glansdorff, P. & Prigogine, I. 1964 Physica, 30, 351.
Hocking, L. M. & Stewartson, K. 1971 Mathematika, 18, 219.
Hocking, L. M. & Stewartson, K. 1972 Proc. Roy. Soc. A, 326, 289.
Hocking, L. M., Stewartson, K. & Stuart, J. T. 1972 J. Fluid Mech. 51, 705.
Krylov, N. & Bogoliubov, N. N. 1947 Introduction to Non-Linear Mechanics. Princeton University Press.
Mcgoldrick, L. F. 1965 J. Fluid Mech. 21, 305.
Mahony, J. J. 1972 Proc. Roy. Soc. A, 330, 349.
Millikan, C. B. 1929 Phil. Mag. 7, 641.
Phillips, O. M. 1960 J. Fluid Mech. 9, 193.
Schechter, R. S. 1966 In Non-equilibrium Thermodynamics, Variational Techniques and Stability (ed. R. J. Donnelly, R. Herman & I. Prigogine), pp. 55–69. Chicago University Press.
Simmons, W. F. 1969 Proc. Roy. Soc. A, 309, 551.
Stewartson, K. & Stuart, J. T. 1971 J. Fluid Mech. 48, 529.
Usher, J. R. 1974 Ph.D. dissertation, University of St. Andrews.
Whitham, G. B. 1965 J. Fluid Mech. 22, 273.
Whitham, G. B. 1967 Proc. Roy. Soc. A, 299, 6.