Singularities of Capillary-Gravity Waves on Dielectric Fluid Under Normal Electric Fields

Abstract

SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 523-542, April 2024.
Abstract. As summarized by Papageorgiou [Annu. Rev. Fluid Mech., 51 (2019), pp. 155–187], a strong normal electric field can cause instability of the interface in a hydrodynamic system. In the present work, singularities arising in electrocapillary-gravity waves on a dielectric fluid of finite depth due to an electric field imposed in the direction perpendicular to the undisturbed free surface are investigated. In shallow water, for a small-amplitude periodic disturbance in the linearly unstable regime, the outcome of the system evolution is that the gas-liquid interface touches the solid bottom boundary, causing a rupture. A quasi-linear hyperbolic model is derived for the long-wave limit and used to study the formation of the touch-down singularity. The theoretical predictions are compared with the fully nonlinear computations by a time-dependent conformal mapping for the electrified Euler equations, showing good agreement. On the other hand, a nonlinear dispersive model system is derived for the deep-water scenario, which predicts the blowup singularity (i.e., the wave amplitude tends to infinity in a finite time). However, when the fluid thickness is significantly large, one can numerically show the self-intersection nonphysical wave structure or 2/3 power cusp singularity in the full Euler equations.