Free Surface Flows in Electrohydrodynamics with a Constant Vorticity Distribution

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.