Abstract
There have been quite a number of theoretical as well as experimental studies concerning discharge phenomena, but the results (or partial results) are complicated mathematically as well as physically. The main subject of the present paper is traced back nearly twenty years ago when the second author began experiments with an exquisite mechanism he devised. The discharge phenomenon was like the famous Stephan problem, where two different phases of one and the same substance, i.e., ice and water, coexist at a free moving boundary between them. The mathematical model consists of a diffusion-type nonlinear partial differential equation with initial and boundary conditions. In particular, one of the boundaries is given fixed condition, whereas the other is moving with a little subtle physical assumption on the speed of the propagation of discharge. Therefore, the nonlinear diffusion equation is difficult to solve by a usual method. Assuming that the applied voltage is constant and that there exists a unique solution, the original initial- and boundary-value problem is reduced to the initial-value problem of an ordinary differential equation. This latter problem allows us to define a one-parameter family of new functions. Details of the derivation of the solution of the fundamental equations and the physical meaning of the propagation condition and the analytical solution for a rectilinearly channelled negative surface discharge are discussed.
