Dynamic Taylor cone formation on liquid metal surface: numerical modelling

Abstract

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, with the angle of the cone depending on the initial shape of the surface. We describe this structure as a dynamic Taylor cone. The time-dependent profiles of the surface shape are 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 run-away 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 overtakes the growth of both the surface tension and viscous stress acting inwards. Analysis of the time dependences of physical values can strongly assist the development of analytical treatments of such phenomena, and give insight into the problem of the dynamic description of operating liquid metal ion source atomisers. 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 the evolution of the surface occurs in the context of a Langmuir shield.

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