Unsteady flows with a zero acceleration on the free boundary

2014 ◽  
Vol 754 ◽  
pp. 308-331 ◽  
Author(s):  
E. A. Karabut ◽  
E. N. Zhuravleva
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Unsteady Flows ◽  
Ideal Incompressible Fluid ◽  
Hopf Equation ◽  
New Approach

AbstractA 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.

Instability of unsteady flows or configurations Part 1. Instability of a horizontal liquid layer on an oscillating plane

1968 ◽  
Vol 31 (4) ◽  
pp. 737-751 ◽  
Author(s):  
Chia-Shun Yih
Keyword(s):  
Free Surface ◽  
Space Variable ◽  
Unsteady Flows ◽  
Lower Boundary ◽  
Time Variable ◽  
Long Waves ◽  
Time Dependent ◽  
Primary Flow ◽  
New Approach ◽  
The Stability

A layer of viscous liquid with a free surface is set in motion by the lower boundary moving simple-harmonically in its own plane. The stability of this motion is investigated. Since the primary flow is time-dependent, the time variable cannot be separated from at least one space variable, and a new approach must be used to investigate the problem. In this paper the stability of long waves is studied by a perturbation method which has not been applied before to problems of stability of unsteady flows, and it is found that the flow under consideration can be unstable for long waves.

On some exact solutions in magnetohydrodynamics with astrophysical applications

1972 ◽  
Vol 51 (1) ◽  
pp. 33-38 ◽  
Author(s):  
C. Sozou
Keyword(s):  
Free Surface ◽  
Angular Velocity ◽  
Incompressible Fluid ◽  
Exact Solutions ◽  
Equilibrium Configuration ◽  
Constant Angular Velocity ◽  
Magnetohydrodynamic Equations ◽  
The Stability

Some exact solutions of the steady magnetohydrodynamic equations for a perfectly conducting inviscid self-gravitating incompressible fluid are discussed. It is shown that there exist solutions for which the free surface of the liquid is that of a planetary ellipsoid and rotates with constant angular velocity about its axis. The stability of the equilibrium configuration is not investigated.

Conformal mapping, Padé approximants, and an example of flow with a significant deformation of the free boundary

2014 ◽  
Vol 25 (6) ◽  
pp. 729-747 ◽  
Author(s):  
E. A. KARABUT ◽  
A. A. KUZHUGET
Keyword(s):  
Velocity Field ◽  
Free Boundary ◽  
Power Series ◽  
Conformal Mapping ◽  
Incompressible Fluid ◽  
Padé Approximants ◽  
Ideal Incompressible Fluid ◽  
Inertial Motion ◽  
Pade Approximants ◽  
Summation Of Series

A problem of plane inertial motion of an ideal incompressible fluid with a free boundary, which initially has a quadratic velocity field, is studied by semi-analytical methods. A conformal mapping of the domain occupied by the fluid onto a unit circle is sought in the form of a power series with respect to time. Summation of series is performed by using Padé approximants.

Bound Coherent Structures Propagating on the Free Surface of Deep Water

2021 ◽  
Author(s):  
Sergey Dremov ◽  
Dmitry Kachulin ◽  
Alexander Dyachenko
Keyword(s):  
Free Surface ◽  
Incompressible Fluid ◽  
Deep Water ◽  
Coherent Structures ◽  
Water Waves ◽  
Initial Conditions ◽  
Soliton Solutions ◽  
Ideal Incompressible Fluid ◽  
System Of Nonlinear Equations ◽  
Computational Domain

<p><span>               The work presents the results of studying the bound coherent structures propagating on the free surface of ideal incompressible fluid of infinite depth. Examples of such structures are bi-solitons which are exact solutions of the known approximate model for deep water waves — the nonlinear Schrödinger equation (NLSE). Recently, when studying multiple breathers collisions, the occurrence of such objects was found in a more accurate model of the supercompact equation for unidirectional water waves [1]. The aim of this work is obtaining and further studying such structures with different parameters in the supercompact equation and in the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. </span><span>The algorithm used for finding the bound coherent objects was similar to the one described in [2]. As the initial conditions for obtaining such structures in the framework of the above models, the NLSE bi-soliton solutions were used, as well as two single breathers numerically found by the Petviashvili method and placed in a same point of the computational domain. During the evolution calculation the initial structures emitted incoherent waves which were filtered at the boundaries of the domain using the damping procedure. It is shown that after switching off the filtering of radiation, periodically oscillating coherent objects remain on the surface of the liquid, propagate stably during one hundred thousand characteristic wave periods and do not lose energy. The profiles of such structures at different parameters are compared.</span></p><p><span>This work was supported by RSF grant </span><span>19-72-30028</span><span> and </span><span>RFBR grant </span><span>20-31-90093</span><span>.</span></p><p><span>[1] Kachulin D., Dyachenko A., Dremov S. Multiple Soliton Interactions on the Surface of Deep Water //Fluids. – 2020. – Т. 5. – №. 2. – С. 65.</span></p><p><span>[2] Dyachenko A. I., Zakharov V. E. On the formation of freak waves on the surface of deep water //JETP letters. – 2008. – Т. 88. – №. 5. – С. 307.</span></p>

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