A diffusion problem with spherical symmetry: Temperature distribution for different initial conditions

Extract

The present paper extends some earlier work [1] on heat flow in a composite system, in which a central, high-temperature region loses heat to a surrounding medium. An example of the type of situation in mind is the intrusion of igneous rock into a mass of cooler sedimentary material. As an idealization, spherical symmetry is assumed and the outer region is taken to be infinite in extent. In the earlier work, the central region and the surrounding medium were each taken to be at a constant temperature initially. The present paper gives solutions for a number of alternative situations. The temperature in the outer region is still taken to be constant initially but the temperature in the central region is represented by functions of type rⁿ or (1/r) sin k^r, where r is the distance from the centre of the system. As the system of equations for the temperature is linear, and any continuous function can be approximated arbitrarily closely by a polynomial or a Fourier series, the solutions given here can be superposed to give a solution for any continuous initial temperature distribution in the central region.