Free Surface Flows in Electrohydrodynamics with a Constant Vorticity Distribution
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Free Surface Flows in Electrohydrodynamics with a Constant Vorticity Distribution M. J. Hunt1,2 · D. Dutykh3 Received: 15 October 2019 / Accepted: 10 September 2020 © The Author(s) 2020
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. Keywords Electrohydrodynamics · Free surface · Constant vorticity · Burn’s condition
1 Introduction 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. In other words, there has always been an implicit assumption of zero vorticity in the flow region. In numerous recent studies, this assumption started to be dropped and the assumption of constant vorticity in the flow region used. One of the pioneering
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M. J. Hunt [email protected]
1
Warwick Manufacturing Group, University of Warwick, Coventry CV4 7AL, UK
2
Warwick Mathematics Institute, University of Warwick, Zeeman Building, Coventry CV4 7AL, UK
3
Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France
M. J. Hunt, D. Dutykh
studies was made by Burns [6], and later, Da Silva and Peregrine [8] studied steep and steady waves on finite depth with constant vorticity. More recently, this problem was analyzed mathematically in some two-component systems [12]. The effect of the vorticity on travelling wave solutions (solitary and cnoidal) was investigated in the purely hydrodynamic context in [11] using the qualitative phase space analysis methods. and a Hamiltonian formulation has been reported in [20]. This problem in electrohydrodynamics seems to be still open to the best of our knowledge and the present study should be considered as a further attempt to fill in this gap in the literature. The current approach to examining flows with constant vorticity in two dimensions is via the use of a stream function, ψ, and its harmonic conjugate, the velocity potential, and ϕ, so u = ∇ϕ +∇ ⊥ ψ. This approach introduces two unnecessary functions which complicates the problem and has the limiting effect in being restricted to fully nonlinear and linear computations. There has been no attemp
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