Large-Amplitude Steady Electrohydrodynamic Solitary Waves with Constant Vorticity

TL;DR

This study explores steady solitary water waves with constant vorticity influenced by an electric field, revealing conditions for wave behavior and bifurcation phenomena through a coupled nonlinear free boundary model.

Contribution

It introduces a new analytical framework for coupled electrohydrodynamic waves with constant vorticity, establishing nodal properties and bifurcation scenarios.

Findings

Wave remains a symmetric elevation profile along the global branch.

Four possible outcomes along the bifurcation curve: stagnation, degeneration, flow stagnation, or unbounded wave speed.

New nodal properties ensure the wave's symmetry despite electric field interactions.

Abstract

This paper investigates solitary water waves propagating along the surface of a two-dimensional dielectric fluid with constant vorticity in the presence of an external electric field. We formulate the system as a nonlinear free boundary problem where the Euler equations and electric potential equations are strongly coupled at the interface. A major challenge in such setting is the loss of standard monotonicity arguments due to the interaction between the velocity and electric fields. We overcome this difficulty by establishing new nodal properties for the combined system, ensuring the wave remains a symmetric elevation profile along the global branch. Moreover, along the global bifurcation curve, one of the following case must occur: (i) the formation of an equilibrium stagnation point, (ii) the degeneration of the conformal mapping, (iii) the onset of flow stagnation, or (iv) an unbounded increase in the dimensionless wave speed.