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No steady water waves of small amplitude are supported by a shear flow with a still free surface

Published online by Cambridge University Press:  01 February 2013

Vladimir Kozlov  and
Nikolay Kuznetsov
Vladimir Kozlov
Affiliation:
Department of Mathematics, Linköping University, S-581 83 Linköping, Sweden
Nikolay Kuznetsov*
Affiliation:
Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy pr. 61, St. Petersburg 199178, Russian Federation
*
†Email address for correspondence: [email protected]
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Abstract

The two-dimensional free-boundary problem describing steady gravity waves with vorticity on water of finite depth is considered. It is proved that no small-amplitude waves are supported by a horizontal shear flow whose free surface is still, that is, it is stagnant in a coordinate frame such that the flow is time-independent in it. The class of vorticity distributions for which such flows exist includes all positive constant distributions, as well as linear and quadratic ones with arbitrary positive coefficients.

JFM classification

Waves/Free-surface Flows: Channel flow Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 717 , 25 February 2013 , pp. 523 - 534
Copyright
©2013 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Agmon, S., Douglis, A. & Nirenberg, L. 1959 Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Maths 12, 623–727.Google Scholar
Benjamin, T. B. 1995 Verification of the Benjamin–Lighthill conjecture about steady water waves. J. Fluid Mech. 295, 337–356.Google Scholar
Constantin, A. & Strauss, W. 2004 Exact steady periodic water waves with vorticity. Commun. Pure Appl. Maths 57, 481–527.Google Scholar
Ehrnström, M., Escher, J. & Wahlén, E. 2011 Steady water waves with multiple critical layers. SIAM J. Math. Anal. 43, 1436–1456.Google Scholar
Hur, V. M. 2008 Exact solitary water waves with vorticity. Arch. Rat. Mech. Math. Anal. 188, 213–244.Google Scholar
Kamke, E. 1959 Differentialgleichungen, I. Gewönliche Differentialgleichungen. Teubner.Google Scholar
Keady, G. & Norbury, J. 1978 Waves and conjugate streams with vorticity. Mathematika 25, 129–150.Google Scholar
Kozlov, V. & Kuznetsov, N. 2008 On behaviour of free-surface profiles for bounded steady water waves. J. Math. Pures Appl. 90, 1–14.CrossRefGoogle Scholar
Kozlov, V. & Kuznetsov, N. 2010 The Benjamin–Lighthill conjecture for near-critical values of Bernoulli’s constant. Arch. Rat. Mech. Math. Anal. 197, 433–488.Google Scholar
Kozlov, V. & Kuznetsov, N. 2011a The Benjamin–Lighthill conjecture for steady water waves (revisited). Arch. Rat. Mech. Anal. 201, 631–645.Google Scholar
Kozlov, V. & Kuznetsov, N. 2011b Steady free-surface vortical flows parallel to the horizontal bottom. Q. J. Mech. Appl. Maths 64, 371–399.Google Scholar
Kozlov, V. & Kuznetsov, N. 2012 Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents. Arch. Rat. Mech. Math. Anal. (submitted); also available online at http://arXiv.org/abs/1207.5181.Google Scholar
Lavrentiev, M. & Shabat, B. 1980 Effets Hydrodynamiques et Modèles Mathématiques. Mir.Google Scholar
Michlin, S. G. 1978 Partielle Differentialgleichungen in der mathematischen Physik. Harri Deutsch.Google Scholar
Strauss, W. 2010 Steady water waves. Bull. Amer. Math. Soc. 47, 671–694.Google Scholar
Swan, C., Cummins, I. & James, R. 2001 An experimental study of two-dimensional surface water waves propagating in depth-varying currents. J. Fluid Mech. 428, 273–304.Google Scholar
Thomas, G. P. 1990 Wave-current interactions: an experimental and numerical study. J. Fluid Mech. 216, 505–536.Google Scholar
Wahlén, E. 2009 Steady water waves with a critical layer. J. Differ. Equ. 246, 2468–2483.Google Scholar