Abstract
In 1895, Korteweg and de Vries (Philos Mag 20:20, 1895) studied an equation describing the motion of waves using the assumptions of long wavelength and small amplitude. Two implicit assumptions which they also made were irrotational and inviscid fluids. Comparing experiment and observation seems to suggest that these two assumptions are well justified. This paper removes the assumption of irrotationality in the case of electrohydrodynamics with an assumption of globally constant vorticity in the fluid. A study of the effect of vorticity on wave profiles and amplitudes is made revealing some unusual features. The velocity potential is an important variable in irrotational flow; the vertical component of velocity takes place of this variable in our analysis. This allows the bypassing of the Burns condition and also demonstrates that waves exist even for negative values of the vorticity. The linear and weakly nonlinear models are derived.
Highlights
Water waves constitute a very classical problem in hydrodynamics [7]
There has always been an implicit assumption of zero vorticity in the flow region
This paper has not been the first to study the properties of the Benjamin equation, and previous works such as [1,16,18] where the emphasis was usually been on irrotational flow
Summary
Water waves constitute a very classical problem in hydrodynamics [7]. This problem is traditionally formulated in terms of the velocity potential to achieve some simplifications. There has always been an implicit assumption of zero vorticity in the flow region. This assumption started to be dropped and the assumption of constant vorticity in the flow region used. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
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