Analytical solution of steady state heat conduction equations in irregular domains with various BCs by use of Schwarz-Christoffel conformal mapping

Abstract

Undoubtedly, analytical solutions of conduction heat transfer problems play an important role in formation of both qualitative and quantitative understanding of the heat transfer phenomenon and other phenomena related to it. Analytical methods for solving this class of problems, especially in irregular domains, contain much complexity, making the solution process extremely difficult. This paper demonstrates, in a general manner, the use of conformal mapping, specifically the Schwarz-Christoffel (SC) mapping, to solve the steady-state conduction heat transfer equations (namely Laplace and Poisson) and to obtain the analytical exact solution. The exact temperature distribution can then be obtained for irregular domains (i.e. domains with irregular boundaries) for homogeneous and non-homogeneous conditions. To get to this goal and to clarify the method more, for a specific case, namely a triangular plate with a right angle on the origin and constant-temperature boundary conditions, the steady-state heat conduction equation has been analytically solved using the conformal mapping method. In the analytical solution process, the importance and capabilities of complex variables and their applications, especially the SC mapping, is highlighted. Finally, in order to validate the exact solution, the problem has been solved using the finite-element software COMSOL Multi-physics 5.0 and the results have been compared to the analytical solution. Good agreement between two solutions has confirmed the accuracy of the analytical solution.