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Evolution of nonlinear perturbations for a fluid flow with a free boundary. Exact results

Published online by Cambridge University Press:  02 December 2022

E.A. Karabut Open the ORCID record for E.A. Karabut [Opens in a new window] ,
E.N. Zhuravleva Open the ORCID record for E.N. Zhuravleva [Opens in a new window] ,
N.M. Zubarev Open the ORCID record for N.M. Zubarev [Opens in a new window]  and
O.V. Zubareva Open the ORCID record for O.V. Zubareva [Opens in a new window]
E.A. Karabut*
Affiliation:
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk 630090, Russia Novosibirsk State University, Novosibirsk 630090, Russia
E.N. Zhuravleva
Affiliation:
Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk 630090, Russia Novosibirsk State University, Novosibirsk 630090, Russia
N.M. Zubarev
Affiliation:
P. N. Lebedev Physical Institute, Russian Academy of Sciences, Moscow 119333, Russia Institute of Electrophysics, Ural Branch, Russian Academy of Sciences, Ekaterinburg 620016, Russia
O.V. Zubareva
Affiliation:
Institute of Electrophysics, Ural Branch, Russian Academy of Sciences, Ekaterinburg 620016, Russia
*
†Email address for correspondence: eakarabut@gmail.com
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Abstract

The problem of a plane unsteady potential flow of an ideal incompressible fluid bounded by free boundary segments with a constant pressure and by solid walls moving in accordance with a known law is considered. External forces are absent, and capillary forces are neglected. An approach to constructing exact solutions for this type of problem is proposed. The corresponding solutions can be treated as nonlinear perturbations of a certain base flow. As an example of the application of this approach, nonlinear perturbations in a known problem of a fluid flow with a linear velocity field in the region bounded by a straight-line free boundary and parallel approaching or receding solid walls are considered. It is demonstrated that perturbations grow, which leads to variants of the formation of singularities on the free surface of the fluid within a finite time: formation of droplets, bubbles or cusps. A solution describing the collapse of a bubble in a fluid layer bounded by two approaching solid walls has also been found and studied. Thus, a new method of studying nonlinear stability of complicated unsteady fluid flows with combined boundary conditions is proposed and tested.

JFM classification

Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 953 , 25 December 2022 , A1
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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References

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