Similarity solution for one-dimensional heat equation in spherical coordinates and its applications

Abstract

A similarity type of general solution for the one-dimensional heat equation in spherical coordinates is developed, and the solution is expressed by the Kummer functions. After introducing some useful properties of the general solution, the solution is then applied to establish analytical solutions of finite line source problems and moving boundary problems. For the finite line source problems, a continuous point source problem with a power-type source intensity is solved first, and then analytical solutions for a vertical finite line source problem and an inclined finite line source problem are established using the mirror image method and the solution of the continuous point source problem. For the moving boundary problems, a one-phase Stefan problem with power-type latent heat in spherical coordinates and a nonlinear heat conduction problem induced by a continuous point source are investigated, and analytical solutions are developed using the general solution. Computational examples are presented. For the finite line source problems, computational error caused by neglecting vertical heat transfer under the situation of a time-varying source intensity is investigated. For the moving boundary problems, an illustrative example of the one-phase Stefan problem is presented, and the computational results can be used to calibrate numerical solutions of Stefan problems.