Analysis of the space charge singularity near the Taylor cone apex via simplified Eulerian model for electrospray beams in vacuum

A simplified Eulerian model for the description of steady state electric charge dispersion in vacuum, including space-charge effects, is introduced. The model is based on a potential velocity field for the charged particles, which is valid if all the particles are emitted with the same kinetic plus potential energy. Despite the high simplicity of the present model, the mathematical system still encompasses several types of singularities, which are analyzed here in full detail, thus enabling the numerical solution of the system. The model is applied to the description of electrospray emission in vacuum, in the limit of high electric conductivity liquids, assuming that all particles are emitted with zero initial velocity. Our main result in this regard is a relation existing between the cone semiangle α and the electrospray beam angular width. In particular, we show that the cone semiangle can only take values between a minimum angle ca. 21.89 deg, and a maximum value given by Taylor's angle (ca. 49.29 deg), with lower values of α corresponding to wider beams, while higher values of α correspond to narrower beams.