On the global variational principles for corona discharges with particular reference to the active glow region

This is the authors' abstract. We don't add key points for this paper.

This is a sequel to a recently published companion paper (Jones J E 1999 J. Phys. D: Appl. Phys. 32 1243) on variational principles for the passive inter-electrode regions of all corona discharges. The earlier work considered extremal principles for the ionic-flow regions and the outer Laplacian domains of these phenomena, but did not apply these methods to the ionization zones that give rise to the charge flow. It is with the search for integral forms whose optimization corresponds to the charge-separation and creation terms in the field equations that the present paper is concerned. The work is fundamental to the theoretical analysis of the practical and engineering uses of corona discharges. It relates to any electrode geometry, e.g. point-plane, wire-to-plane, cylindrical and ducted flows; and also to dc, ac and other time-dependent glows. Although they are usually physically relatively small, the active regions are a significant part of the overall high-voltage manifestation. In particular, if current-creation calculations rather than measurements are being used in an analysis, the modelling of these terms is crucial to the overall success or failure of the theory. The power of the variational calculus would be a considerable asset if it can be shown to work in these analyses. The preceding paper showed that there is only one approach to simultaneous variational methods for the potential and charge distribution of the non-active regions, and that there are additional difficulties associated with the location of free boundaries between domains. The present work shows that current-creation terms can be incorporated into simultaneous variational methods for the glow region, but only if the electric-field dependency of these terms is ignored. A proof that integrands giving field-dependent separation rates do not exist is given, so that other methods must be used for the modelling of these effects. The results are of primal importance to the construction of global finite-element methods in electrical discharge problems.