The Linear Instability of a Dielectric Liquid Contained in a Cylindrical Annulus and Subjected to Unipolar Charge Injection

An electrodynamic but hydrostatic equilibrium of dielectric liquid contained in the annular region between two concentric circular cylinders is shown to possess a mainstream and charge-diffusion boundary-layer character. Assuming autonomous injection of charge from the inner electrode a linear instability analysis of the diffusion-free mainstream solution is considered, attention being confined to two-dimensional perturbations. For weak injection it is shown that the Principle of Exchange of Stabilities is valid for disturbances independent of the axial coordinate. Moreover in the cases considered it is found that for arbitrary injection strength, and indeed for the space-charge limited current case, such perturbations are more susceptible to linear instability than those possessing axisymmetry.