On rotational water waves with surface tension

2007 ◽  
Vol 365 (1858) ◽  
pp. 2215-2225 ◽  
Author(s):  
Erik Wahlén
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Bifurcation Theory ◽  
Two Dimensional ◽  
Flat Bottom ◽  
Recent Advances ◽  
Surface Profiles

The purpose of this paper is to present some recent advances in the study of water waves with vorticity and surface tension. These are periodic, two-dimensional waves over a flat bottom and the surface profiles are symmetric and monotone between crest and trough. The proofs rely on bifurcation theory.

scholarly journals Comment on “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” [Nonlinear Dyn,  doi:10.1007/s11071-017-3938-7]

2021 ◽  
Author(s):  
Piotr Rozmej ◽  
Anna Karczewska
Keyword(s):  
Surface Tension ◽  
Shallow Water ◽  
Water Waves ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Two Dimensional ◽  
Shallow Water Waves ◽  
Small Parameters ◽  
Fifth Order

AbstractThe authors of the paper “Two-dimensional third-and fifth-order nonlinear evolution equations for shallow water waves with surface tension” Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) claim that they derived the equation which generalizes the KdV equation to two space dimensions both in first and second order in small parameters. Moreover, they claim to obtain soliton solution to the derived first-order (2+1)-dimensional equation. The equation has been obtained by applying the perturbation method Burde (J Phys A: Math Theor 46:075501, 2013) for small parameters of the same order. The results, if correct, would be significant. In this comment, it is shown that the derivation presented in Fokou et al. (Nonlinear Dyn 91:1177–1189, 2018) is inconsistent because it violates fundamental properties of the velocity potential. Therefore, the results, particularly the new evolution equation and the dynamics that it describes, bear no relation to the problem under consideration.

Two-Phase Problem for Two-Dimensional Water Waves of Finite Depth

1997 ◽  
Vol 07 (06) ◽  
pp. 791-821
Author(s):  
Tatsuo Iguchi
Keyword(s):  
Water Waves ◽  
Finite Depth ◽  
Two Dimensional ◽  
Phase Problem ◽  
Flat Bottom ◽  
Two Phase ◽  
The Cauchy Problem ◽  
Lower Fluid ◽  
Quasi Linear System ◽  
Finite Smoothness

We consider the two-phase problem for two-dimensional and irrotational motion of incompressible ideal fluids in the case that the fluids are separated into the lower and the upper parts by an almost horizontal interface and that there is an almost flat bottom below the lower fluid. It is proved that the Cauchy problem is well-posed, locally in time, in a Sobolev space of finite smoothness, if the surface tension is taken into account and the initial data are suitably close to the equilibrium rest state. The main part of the proof is the reduction of the problem to a quasi-linear system of integro-differential equations for the function defining the interface and the horizontal component of the velocity of the lower fluid on the interface.

Water Waves in a Nonhomogeneous Incompressible Fluid

1977 ◽  
Vol 44 (4) ◽  
pp. 523-528 ◽  
Author(s):  
A. E. Green ◽  
P. M. Naghdi
Keyword(s):  
Surface Tension ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Incompressible Fluid ◽  
Water Waves ◽  
Nonlinear Differential Equations ◽  
Long Waves ◽  
Direct Approach ◽  
Hydraulic Jumps ◽  
Two Dimensional

After a brief discussion of some undesirable features of a number of different partial differential equations often employed in the existing literature on water waves, a relatively simple restricted theory is constructed by a direct approach which is particularly suited for applications to problems of fluid sheets. The rest of the paper is concerned with a derivation of a system of nonlinear differential equations (which may include the effects of gravity and surface tension) governing the two-dimensional motion of incompressible in-viscid fluids for propagation of fairly long waves in a nonhomogeneous stream of water of variable initial depth, as well as some new results pertaining to hydraulic jumps. The latter includes an additional class of possible solutions not noted previously.

scholarly journals Modified Babenko’s Equation For Periodic Gravity Waves On Water Of Finite Depth

2019 ◽  
Vol 72 (4) ◽  
pp. 415-428
Author(s):  
E Dinvay ◽  
N Kuznetsov
Keyword(s):  
Surface Tension ◽  
Gravity Waves ◽  
Water Waves ◽  
Surface Profile ◽  
Finite Depth ◽  
Parametric Representation ◽  
Two Dimensional ◽  
Free Surface Profile ◽  
The Mean ◽  
New Equation

Summary A new operator equation for periodic gravity waves on water of finite depth is derived and investigated; it is equivalent to Babenko’s equation considered in Kuznetsov and Dinvay (Water Waves, 1, 2019). Both operators in the proposed equation are nonlinear and depend on the parameter equal to the mean depth of water, whereas each solution defines a parametric representation for a symmetric free surface profile. The latter is a component of a solution of the two-dimensional, nonlinear problem describing steady waves propagating in the absence of surface tension. Bifurcation curves (including a branching one) are obtained numerically for solutions of the new equation; they are compared with known results.

On the evolution of packets of water waves

1979 ◽  
Vol 92 (4) ◽  
pp. 691-715 ◽  
Author(s):  
Mark J. Ablowitz ◽  
Harvey Segur
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Evolution Equations ◽  
Wave Packets ◽  
Short Wave ◽  
Similarity Solutions ◽  
Two Dimensions ◽  
Long Waves ◽  
Two Dimensional ◽  
One Dimensional

We consider the evolution of packets of water waves that travel predominantly in one direction, but in which the wave amplitudes are modulated slowly in both horizontal directions. Two separate models are discussed, depending on whether or not the waves are long in comparison with the fluid depth. These models are two-dimensional generalizations of the Korteweg-de Vries equation (for long waves) and the cubic nonlinear Schrödinger equation (for short waves). In either case, we find that the two-dimensional evolution of the wave packets depends fundamentally on the dimensionless surface tension and fluid depth. In particular, for the long waves, one-dimensional (KdV) solitons become unstable with respect to even longer transverse perturbations when the surface-tension parameter becomes large enough, i.e. in very thin sheets of water. Two-dimensional long waves (‘lumps’) that decay algebraically in all horizontal directions and interact like solitons exist only when the one-dimensional solitons are found to be unstable.The most dramatic consequence of surface tension and depth, however, occurs for capillary-type waves in sufficiently deep water. Here a packet of waves that are everywhere small (but not infinitesimal) and modulated in both horizontal dimensions can ‘focus’ in a finite time, producing a region in which the wave amplitudes are finite. This nonlinear instability should be stronger and more apparent than the linear instabilities examined to date; it should be readily observable.Another feature of the evolution of short wave packets in two dimensions is that all one-dimensional solitons are unstable with respect to long transverse perturbations. Finally, we identify some exact similarity solutions to the evolution equations.

Global bifurcation of capillary–gravity-stratified water waves

2014 ◽  
Vol 144 (4) ◽  
pp. 775-786 ◽  
Author(s):  
David Henry ◽  
Anca-Vocihita Matioc
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Bifurcation Theory ◽  
Global Bifurcation ◽  
Governing Equations ◽  
Stagnation Points ◽  
Global Bifurcation Theory

We study steady periodic water waves with variable vorticity and density, where we allow for stagnation points in the flow, and where we admit the capillarity effects of surface tension. Using global bifurcation theory, we extend a local curve of non-trivial solutions of the governing equations to a global continuum of solutions. Furthermore, we obtain a description of the behaviour of the solutions along this continuum.

On uniqueness in the problem of gravity–capillary water waves above submerged bodies

2009 ◽  
Vol 465 (2106) ◽  
pp. 1743-1761 ◽  
Author(s):  
Oleg V. Motygin ◽  
Philip McIver
Keyword(s):  
Surface Tension ◽  
Unique Solvability ◽  
Water Waves ◽  
Linear Problem ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Trapped Modes ◽  
Capillary Water ◽  
Simple Bounds

In this paper, we consider the two-dimensional linear problem of wave–body interaction with surface tension effects being taken into account. We suggest a criterion for unique solvability of the problem based on symmetrization of boundary integral equations. The criterion allows us to develop an algorithm for detecting non-uniqueness (finding trapped modes) for given geometries of bodies; examples of numerical computation of trapped modes are given. We also prove a uniqueness theorem that provides simple bounds for the possible non-uniqueness parameters.

scholarly journals An Existence Theory for Gravity–Capillary Solitary Water Waves

Water Waves ◽  
2021 ◽  
Author(s):  
M. D. Groves
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Applied Mathematics ◽  
Spatial Dynamics ◽  
Existence Theory ◽  
Water Wave Problem ◽  
Reduction Methods ◽  
Centre Manifold Reduction ◽  
Weak Surface ◽  
The One

AbstractIn the applied mathematics literature solitary gravity–capillary water waves are modelled by approximating the standard governing equations for water waves by a Korteweg-de Vries equation (for strong surface tension) or a nonlinear Schrödinger equation (for weak surface tension). These formal arguments have been justified by sophisticated techniques such as spatial dynamics and centre-manifold reduction methods on the one hand and variational methods on the other. This article presents a complete, self-contained account of an alternative, simpler approach in which one works directly with the Zakharov–Craig–Sulem formulation of the water-wave problem and uses only rudimentary fixed-point arguments and Fourier analysis.

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