On extremum principles for gaseous ionic movements with varying mobility coefficients

Variational principles for the movement of ions in gases are derived when the mobility coefficients that describe the motion of the ions in the prevailing electric field are functions of the independent variables of either position and time or of the electric field intensity. The work is in continuance of previously published results by the author on a fundamental analysis of ionic flow problems, with and without ionization, in which the mobility coefficients are treated as constants and the non-existence of certain duals is proved. As before, charge continuity is viewed both from a Eulerian and a Lagrangian perspective, and the appropriate functional dependencies are discussed. When mobility is a function of position and time, or of position or time separately, the standard extremizing principles are shown to be appropriate; but these are then shown not to hold in the situation when mobility depends on the electric field strength. In this second situation, an optimizing principle is then successfully derived by making a major modification to the integrand. This modelling with changing mobility coefficients for the charge movements has a distinct advantage over that using electric-field dependent charge transfer rates between the components of the ionic current, as the latter has been earlier proved not to have a variational principle. As previously shown, these variational methods have a physical basis in the rate of energy transfer from the electric field to the gaseous thermal motion, although a simple physical explanation of the integrand modification has not been found. The work is fundamental to the derivation of finite-element methods by numerical problem solvers in the fields of corona discharges, gas-insulated systems and ionization phenomena.