A first-order Navier--Stokes treatment of the shape and instability of liquid metal ion sources

A three-dimensional axially symmetric model for a liquid ion source is used to obtain the shape and critical potential for breakdown of a conducting fluid at the onset of instability. A set of electromechanical equations is obtained for the first-order deformation of the fluid surface from the Taylor cone which is used as the zeroth-order configuration. The equations are evaluated in both the quasihydrostatic and hydrodynamic limits. These analyses yield the angle deformation ξ=(r/a)sε, for r<a, and the associated critical voltage VC=V(0)TC1 x[(a/r)s+C2]-0.5, where a is a boundary-matching parameter, r0 is the distance between the cone apex and the counterelectrode, and ε contains the time dependence of the first-order angle deformation. Here, V(0)T is the potential needed for formation of the purely hydrostatic Taylor cone and C1 and C2 are constants dependent upon boundary-matching conditions. The hydrodynamic analysis predicts that the shape of the surface is concave at the onset of instability. The results obtained in both limits are (i) that the instability is localized in the apex region, and (ii) the values of the critical voltages VC, are in good agreement with experiment for Ga