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On bounds and non-existence in the problem of steady waves with vorticity

Published online by Cambridge University Press:  15 January 2015

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

For the problem describing steady gravity waves with vorticity on a two-dimensional unidirectional flow of finite depth the following results are obtained. (i) Bounds are found for the free-surface profile and for Bernoulli’s constant. (ii) If only one parallel shear flow exists for a given value of Bernoulli’s constant, then there are no wave solutions provided the vorticity distribution is subject to a certain condition.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 765 , 25 February 2015 , R1
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Copyright
© 2015 Cambridge University Press 

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References

Amick, C. J. & Toland, J. F. 1981 On solitary waves of finite amplitude. Arch. Rat. Mech. Anal. 76, 9–95.Google Scholar
Benjamin, T. B. 1971 A unified theory of conjugate flows. Phil. Trans. R. Soc. Lond. A 269, 587–643.Google Scholar
Constantin, A. & Strauss, W. 2004 Exact steady periodic water waves with vorticity. Commun. Pure Appl. Maths 57, 481–527.Google Scholar
Constantin, A. & Strauss, W. 2011 Periodic travelling gravity water waves with discontinuous vorticity. Arch. Rat. Mech. Anal. 202, 133–175.Google Scholar
Gilbarg, D. & Trudinger, N. S. 2001 Elliptic Partial Differential Equations of Second Order. Springer.Google Scholar
Groves, M. D. & Wahlén, E. 2008 Small-amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity. Physica D 237, 1530–1538.Google Scholar
Keady, G. & Norbury, J. 1975 Water waves and conjugate streams. J. Fluid Mech. 70, 663–671.Google Scholar
Keady, G. & Norbury, J. 1978 Waves and conjugate streams with vorticity. Mathematika 25, 129–150.Google Scholar
Keady, G. & Norbury, J. 1982 Domain comparison theorems for flows with vorticity. Q. J. Mech. Appl. Maths 35, 17–32.CrossRefGoogle Scholar
Kozlov, V. & Kuznetsov, N. 2009 Fundamental bounds for steady water waves. Math. Ann. 345, 643–655.Google Scholar
Kozlov, V. & Kuznetsov, N. 2010 The Benjamin–Lighthill conjecture for near-critical values of Bernoulli’s constant. Arch. Rat. Mech. Anal. 197, 433–488.CrossRefGoogle Scholar
Kozlov, V. & Kuznetsov, N. 2011 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 Bounds for steady water waves with vorticity. J. Differ. Equ. 252, 663–691.Google Scholar
Kozlov, V. & Kuznetsov, N. 2014 No steady water waves of small amplitude are supported by a shear flow with a still free surface. J. Fluid Mech. 717, 523–534.CrossRefGoogle Scholar
Kozlov, V. & Kuznetsov, N. 2014 Dispersion equation for water waves with vorticity and Stokes waves on flows with counter-currents. Arch. Rat. Mech. Anal. 214, 971–1018.Google Scholar
Strauss, W. 2010 Steady water waves. Bull. Am. 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
Varvaruca, E. & Zarnescu, A. 2012 Equivalence of weak formulations of the steady water waves equations. Phil. Trans. R. Soc. Lond. A 370, 1703–1719.Google Scholar
Wheeler, M. H.2014, The Froude number for solitary water waves with vorticity. Preprint available online at arXiv:1405.1083.Google Scholar