A Temperature Total Fourier Series Solution For a Hollow Sphere

Abstract

In the following pages, we exhibit an analytical solution of a two-dimensional temperature field in a hollow sphere under total periodic boundary condition. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Till now periodic boundary condition was derived with a harmonic vibration, whereas there is a noticeable difference in the practical conditions with harmonic vibration. In this essay, by means of Fourier analysis, we imagine the outside total periodic boundary condition, as aggregate of harmonic vibrations . To solve the problem, first we imagine the boundary condition as constant values and with separation of variables; we can obtain temperature distribution in the  sphere. Then Duhamel's theorem is used to calculate temperature field under fully periodic boundary condition. For confirmation of accurate solution, we can compare the result for a harmonic vibration and those reported by others. Also, solutions for a hollow sphere were compared with other present references. At last we can obtain thermal stresses which is caused by temperature field in the hollow sphere.