Time-dependent modelling of electrohydrodynamic effects on the surface of a liquid metal

Results of time-dependent modelling of electrohydrodynamic effects on the surface of a liquid metallic conductor are reported for a regime where no electron, ion or particle emission occurs. The Navier-Stokes equations, with free liquid boundaries subject to Maxwell field stress, surface-tension stress and viscous action, have been solved by a method that uses transformation of the interfaces into a rectangle; this overcomes a problem of surface oscillations that appeared using the Marker-and-Cell technique. The situation geometry is a deep unbounded surface with axial symmetry. With time, an almost flat surface evolves into a cone-like shape, which is in good agreement with experimental observations of this process. The calculations have also shown that, when the protrusion is formed, the time dependences of the surface radius of curvature, the electric field value at the protrusion apex, and the axial velocity of the liquid metal, exhibit a runaway behaviour: the physical values become very large for a short time. As a cusp evolves on a surface, the Maxwell stress acting outwards becomes very large and overtake the growth of both the surface tension and viscous stress acting inwards. The development of numerical methods using transformation of the interfaces appears very useful for the treatment of problems in which the cathode or the anode significantly change shape. This situation occurs, for example, when a liquid surface is covered by a metal plasma and evolution of the surface occurs in the context of a Langmuir shield.