The dynamics of the liquid metal surface in strong electric fields is investigated. Two-dimensional numerical simulation of instability of a liquid conducting surface with axial symmetry in a strong electric field is carried out. To study the flow of a liquid with a free surface, the method of splitting by physical factors with the transformation of the computational domain to the canonical form was used. This approach made it possible to study the time dependence of the basic physical quantities in the nonlinear mode, when the emitter tip is formed. An example of such a calculation in comparison with the experiment is shown in Fig. 1. It is shown that this dependence has the character of collapse: there is a critical time t_c, near which the physical quantity either diverges or vanishes as ~ (t_c-t)^g. The critical parameters g for the electric field, the radius of curvature and the axial velocity at the tip are found and the relationship between them is shown.

experiment

theory

Fig. 1. Comparison of experimental observations of the development of EHD instability at the capillary cathode with the results of numerical simulation.

A two-dimensional numerical simulation of the development of electrohydrodynamic and thermal instability of a liquid conductive surface in a strong electric field is carried out. It is shown that the evolution of the free surface leads to the formation of a liquid-metal jet in the form of a "dynamic cone-shaped tip". In parallel with the solution of the Navier-Stokes equation, the problem of heating the resulting cone with an autoemission current was solved. The simultaneous self-consistent calculation of the energy release in the tip due to emission processes and Joule heating showed that the development of thermal instability leads to an avalanche-like increase in the temperature of the cone tip and, as a consequence, to explosive electron emission. It is shown that the Nottingham effect makes a significant contribution to the heating of the liquid, since the ratio between the surface temperature and the electric field on it during the evolution of the tip is always such that the surface energy source only heats it. The dependences on the time of the autoemission current density at the tip and the dynamics of the total current from its surface are calculated. It is found that the increase in the density of the autoemission current is mainly due not to the heat gain (as in the case of a solid-state cathode), but to the increase in the electric field during the growth of the tip.