Dynamics of charged liquid surface using a shallow water approach

As is well known, a charged plane surface of a liquid conductor becomes instable with respect to spatially periodic perturbations when the electric field is higher than the critical value (the Tonks-Frenkel instability) [1]. As a result of the instability development, multiple cone-like structures form on the surface of the conducting liquid. The physical processes that govern the formation of these structures and the saturation of the instability are not completely understood now. To describe the dynamics of charged liquid metal in 3D, shallow water equations for the height of the liquid metal and its velocity were formulated [2]. The effects of the surface tension of the liquid and "negative pressure" due to the electric field were included in the model. In this work we extend the model [2] by calculating self-consistently the electric field on the surface of the liquid metal and analyze the importance of the field electron emission and metal heating in the process of the growth of several cone-like structures on the surface of the metal. The problem is of interest for studying explosive cathode emission, electrospray and physical processes in electron and ion sources with liquid electrodes.

References:

[1] L.D. Landau and E.M. Lifshitz, Electrodynamics of continuous media, Volume 8, Pergamon Press, 460 p. (1984).

[2] <span id="cke_bm_129C" style="display:none"> Zehua Liu,Kentaro Hara, and Mikhail N. Shneider, Dynamics of electrified liquid metal surface using shallow water model, Phys. Fluids 35, 042101 (2023).