Algebraically explicit analytical solutions of unsteady 3-D nonlinear non-Fourier (hyperbolic) heat conduction

Abstract

Analytical solutions of different nonlinear heat conduction equations are meaningful in heat transfer theory. In addition, they are very useful to computational heat transfer to verify numerical solutions and to develop numerical schemes, grid generation methods and so forth. However, none or very few explicit analytical solutions are known for nonlinear (thermal properties are functions of temperature) non-Fourier heat conduction equations. In this paper, some algebraically explicit analytical solutions of unsteady geometrical 3-D nonlinear non-Fourier heat conduction equation are derived. Some mathematical tools needed to extract such explicit exact solutions for complicated nonlinear partial differential equations are also discussed. For example, the little known and rarely used method of separating variables with addition is developed; matching relations between the functions of the thermal conductivity, of the freepath and of the volumetric specific heat are suggested. The main aim of this paper is to obtain some possibly explicit analytical solutions of the nonlinear and non-Fourier heat conduction equation as the benchmark solutions of computational heat transfer but not a specified solution for given initial and boundary conditions, therefore, the initial and boundary conditions are indeterminate before derivation and deduced from the solutions afterwards.