Giniyatullin, A. R. and Kurkin, A. A. and Kurkina, O. E. and Stepanyants, Y. A. ORCID: https://orcid.org/0000-0003-4546-0310
(2014)
*Generalized Korteweg–de Vries equation for internal waves in two-layer fluid.*
Fundamental and Applied Hydrophysics (in Russian), 7 (4).
pp. 16-28.
ISSN 2073-6673

## Abstract

The derivation of the fifth-order Korteweg—de Vries equation is presented for internal waves in two-layer fluid with surface tension on the interface between the layers. The fluid motion is not supposed to be potential, therefore similar derivation can be used for consideration of wave motion in viscous fluid, in rotated fluid or for the shear flows with nonzero vorticity. Explicit expressions are obtained for the coefficients of the equation depending on the parameters of the background medium: widths of the layers, densities of the fluids, coefficient of surface tension. It is shown that for some combinations of the parameters of background medium the coefficients of the quadratic nonlinear and lowest order dispersive terms in the derived generalized equation can vanish and change their signs. Especially interesting is the situation when these terms become small simultaneously, and the coefficients at the nonlinear dispersive terms are also small. This is possible when the widths of the layers are almost equal. In the vicinity of such a double critical point the derived equation reduces to the Gardner-Kawahara equation, which possesses solitary wave solutions with oscillating tails. Such a property makes this equation attractive theoretically and from the point of view of practical applications in the problems of flows in thin surface films of immiscible fluids. The characteristics of the flow in the presence of solitons significantly differ from those in the laminar flows, and this can lead to either negative or positive effects. On the base of the derived generalized equation and its solutions one can propose a method of control over a flow.

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Item Type: | Article (Commonwealth Reporting Category C) |
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Refereed: | Yes |

Item Status: | Live Archive |

Additional Information: | This publication is copyright. It may be reproduced in whole or in part for the purposes of study, research, or review, but is subject to the inclusion of an acknowledgment of the source. |

Faculty/School / Institute/Centre: | Historic - Faculty of Health, Engineering and Sciences - School of Agricultural, Computational and Environmental Sciences (1 Jul 2013 - 5 Sep 2019) |

Faculty/School / Institute/Centre: | Historic - Faculty of Health, Engineering and Sciences - School of Agricultural, Computational and Environmental Sciences (1 Jul 2013 - 5 Sep 2019) |

Date Deposited: | 20 Apr 2015 22:43 |

Last Modified: | 15 Feb 2021 23:33 |

Uncontrolled Keywords: | two-layer fluid; pycnocline; internal waves; nonpotential flow; surface tension; Korteweg—deVries equation |

Fields of Research (2008): | 02 Physical Sciences > 0203 Classical Physics > 020303 Fluid Physics 01 Mathematical Sciences > 0101 Pure Mathematics > 010109 Ordinary Differential Equations, Difference Equations and Dynamical Systems 01 Mathematical Sciences > 0102 Applied Mathematics > 010207 Theoretical and Applied Mechanics |

Fields of Research (2020): | 40 ENGINEERING > 4012 Fluid mechanics and thermal engineering > 401207 Fundamental and theoretical fluid dynamics 49 MATHEMATICAL SCIENCES > 4904 Pure mathematics > 490409 Ordinary differential equations, difference equations and dynamical systems 49 MATHEMATICAL SCIENCES > 4901 Applied mathematics > 490109 Theoretical and applied mechanics |

Socio-Economic Objectives (2008): | E Expanding Knowledge > 97 Expanding Knowledge > 970102 Expanding Knowledge in the Physical Sciences E Expanding Knowledge > 97 Expanding Knowledge > 970101 Expanding Knowledge in the Mathematical Sciences |

URI: | http://eprints.usq.edu.au/id/eprint/26982 |

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