Visco-potential flows in electrohydrodynamics

• It is possible to obtain free surface profiles for weakly viscous flows in electrohydrodynamics.
• We show that the wavelengths are the same as that of inviscid flows.
• We show the increasing the viscosity decreases the amplitude of the waves.

In this study we consider the problem of the interface motion under the capillary–gravity and an external electric force. The infinitely deep fluid layer is assumed to be viscous, perfectly conducting and the flow to be incompressible. The weak viscous effects are introduced using the Helmholtz–Leray decomposition and the visco-potential flow approach. The electric charge distributions above and on the free surface are considered. Finally, we derive some linearized analytical solutions for the free surface elevation shape under the localized pressure distribution and the combined action of the forces mentioned herein above.