Periodic travelling interfacial electrohydrodynamic waves: bifurcation and secondary bifurcation

In this paper, two-dimensional periodic capillary-gravity waves travelling under the effect of a vertical electric field are considered. The full system is a nonlinear, two-layered and free boundary problem. The interface dynamics arises from the coupling between the Euler equations for the lower fluid layer and an electric contribution from the upper gas layer. To investigate the electrohydrodynamic wave interactions, we first introduce the naive flattening technique to transform the free boundary problem into a fixed boundary problem. Then we prove the existence of the small-amplitude electrohydrodynamic waves with constant vorticity $\gamma$ by using local bifurcation theory. Moreover, we prove that these electrohydrodynamic waves are formally stable in linearized sense. Furthermore, we obtain a secondary bifurcation curve that emerges from the primary branch at a nonlaminar solution as $E_0$ being close to some special value. This secondary bifurcation curve consists of ripples solutions on the interface of a conducting fluid under normal electric fields. As far as we know, this new phenomenon in electrohydrodynamics (EHD) is first established mathematically. It is worth noting that the electric field $E_0$ plays a key role to control the shapes and types of waves on the interface.