The Nonlinear Evolution of Vortex Sheets with Surface Tension in Axisymmetric Flows

2001 ◽  
Vol 174 (1) ◽  
pp. 438-459 ◽  
Author(s):  
Qing Nie
Keyword(s):  
Surface Tension ◽  
Nonlinear Evolution ◽  
Vortex Sheets

scholarly journals Traveling waves from the arclength parameterization: Vortex sheets with surface tension

2013 ◽  
Vol 15 (3) ◽  
pp. 359-380 ◽  
Author(s):  
Benjamin Akers ◽  
David Ambrose ◽  
J. Douglas Wright
Keyword(s):  
Surface Tension ◽  
Traveling Waves ◽  
Vortex Sheets

scholarly journals Well-posedness of 3D vortex sheets with surface tension

2007 ◽  
Vol 5 (2) ◽  
pp. 391-430 ◽  
Author(s):  
David M. Ambrose ◽  
Nader Masmoudi
Keyword(s):  
Surface Tension ◽  
Well Posedness ◽  
Vortex Sheets

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.

Nonlinear instability of the wake behind a flat plate placed parallel to a uniform stream

1988 ◽  
Vol 419 (1856) ◽  
pp. 1-28 ◽  
Keyword(s):  
Surface Tension ◽  
Flat Plate ◽  
Euler Equations ◽  
Theoretical Study ◽  
Nonlinear Evolution ◽  
Nonlinear Instability ◽  
Pressure Jump ◽  
Long Wavelength ◽  
Evolution With Time ◽  
Uniform Stream

A theoretical study is presented on nonlinear aspects in the instability of the wake behind an aligned flat plate. This is done in the long-wavelength régime by asymptotic solutions of the unsteady Euler equations, in a rational fashion. In particular, equations are obtained that describe the nonlinear evolution with time and space of the vortex-layer interface formed just downstream of the trailing edge. The instability is found to be driven by a pressure jump across this layer, which has the same form as that arising from surface-tension effects but with a negative (destabilizing) sign. Finite-time breakdown of the solutions is conjectured and analysed. The usefully simplified case of a bounded wake is formulated, analysed and computed numerically, and comparisons are made with experiments, showing qualitative agreement.

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