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Unsteady flows with a zero acceleration on the free boundary

Published online by Cambridge University Press:  04 August 2014

E. A. Karabut  and
E. N. Zhuravleva
E. A. Karabut*
Affiliation:
Lavrentyev Institute of Hydrodynamics, Akademika Lavrentyeva Prospekt 15, Novosibirsk, 630090, Russia
E. N. Zhuravleva
Affiliation:
Lavrentyev Institute of Hydrodynamics, Akademika Lavrentyeva Prospekt 15, Novosibirsk, 630090, Russia
*
†Email address for correspondence: eakarabut@gmail.com
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Abstract

A new approach to the construction of exact solutions of unsteady equations for plane flows of an ideal incompressible fluid with a free boundary is proposed. It is demonstrated that the problem is significantly simplified and reduces to solving the Hopf equation if the acceleration on the free surface is equal to zero. Some examples of exact solutions are given.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 754 , 10 September 2014 , pp. 308 - 331
Copyright
© 2014 Cambridge University Press 

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References

Dirichlet, G. L. 1860 Untersuchungen über ein Problem der Hydrodynamik. J. Reine Angew. Math. 58, 181–216.Google Scholar
Grant, M. A. 1973 The singularity at the crest of a finite amplitude progressive Stokes wave. J. Fluid Mech. 59, 257–262.Google Scholar
Karabut, E. A. 1996 Asymptotic expansions in the problem of a solitary wave. J. Fluid Mech. 319, 109–123.Google Scholar
Karabut, E. A. 1998 An approximation for the highest gravity waves on water of finite depth. J. Fluid Mech. 372, 45–70.CrossRefGoogle Scholar
Karabut, E. A. 2013 Exact solutions of the problem of free-boundary unsteady flows. C. R. Méc. 341, 533–537.Google Scholar
Kuznetsov, E. A., Spector, M. D. & Zakharov, V. E. 1994 Formation of singularities on the free surface of an ideal fluid. Phys. Rev. E 49 (2), 1283–1290.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lavrent’eva, O. M. 1980 Motion of a fluid ellipsoid. Dokl. Akad. Nauk SSSR 253 (4), 828–831.Google Scholar
Lavrent’eva, O. M. 1984 One class of motions of a fluid ellipsoid. J. Appl. Mech. Tech. Phys. 25 (4), 642–645.Google Scholar
Longuet-Higgins, M. S. 1972 A class of exact, time-dependent, free-surface flows. J. Fluid Mech. 55, 529–543.Google Scholar
Longuet-Higgins, M. S. 1980 On the forming of sharp corners at a free surface. Proc. R. Soc. Lond. A 371, 453–478.Google Scholar
Lukomsky, V. P. & Gandzha, I. S. 2003 Fractional Fourier approximations for potential gravity waves on deep water. Nonlinear Process. Geophys. 10, 599–614.CrossRefGoogle Scholar
Nalimov, V. I. & Pukhnachov, V. V.1975 Unsteady motions of an ideal fluid with a free boundary. Report. Novosibirsk State University, Novosibirsk.Google Scholar
Ovsyannikov, L. V. 1967 General equations and examples. In Problem of Unsteady Motion of a Fluid with a Free Boundary, pp. 5–75. Nauka.Google Scholar
Pukhnachov, V. V. 1978 Motion of a Fluid Ellipse, Dynamics of Continuous Media, vol. 33, pp. 68–75. Institute of Hydrodynamics.Google Scholar
Riemann, B. 1860 Ein Beitrag zu den Untersuchungen über die Bewegung eines flüssigen gleichartigen Ellipsoides. Abh. d. Köningl. Gesell. der Wiss. zur Göttingen 9, 3–36.Google Scholar
Taylor, G. I. 1960 Formation of thin flat sheets of water. Proc. R. Soc. Lond. A 259, 1–17.Google Scholar
Zakharov, V. E. & Dyachenko, A. I.2012 Free-surface hydrodynamics in the conformal variables. arXiv:1206.2046v1 [physics.flu-dyn].Google Scholar