Derivation of the analytical solution of the fundamental equations for a rectilinearly channeled negative surface discharge

摘要

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 20 years when the second author began experiments with an exquisite mechanism be devised [1, 2]. The discharge phenomenon was like the famous Stephan problem, where two different phases of one and the same substance-ice and water-coexist at a free moving boundary between them [3]. But, unlike the Stephan problem, which is linear, the mathematical model of the discharge problem consists of a diffusion-type nonlinear partial differential equation with initial and boundary conditions. In particular, one of the boundaries is given a 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 usual methods. 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 novel 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 channeled negative surface discharge are discussed.