Approximate Analytical Method of Streamer Discharge: Mathematical Equation

The most straightforward way to describe the gas electrical discharge phenomenon is a set of partial differential equations coupled with the transport equation and Poisson equation, but the analytical solution cannot be obtained because of the problem's multivariable coupling and nonlinearity nature. In this article, this complex fluid model is transformed into a streamer head development model in a plasma background with an infinite volume. Based on the "coefficients decomposition and processes reorganization method", a spatial evolution equation of the electron density of the streamer head is established. The other physical quantities of the streamer head, including average positive ion density, average net charge density, space charge electric field strength, and streamer velocity, also satisfy the same equation and maintain a consistent variation pattern. More importantly, this equation also has a closed-form analytical solution, even if it contains a variety of nonlinear processes such as collision ionization, photoionization, recombination, and diffusion, which can not only be used to study the type and dynamic characteristics of streamer discharge, but also provide theoretical guidance for numerical simulation and engineering practice.