Transient heat conduction from spheroids

We study the unsteady heat conduction from a spheroid (prolate or oblate) initially heated and then left to cool in an unbounded medium of constant temperature. We present two solutions of the problem. The first makes use of the spheroidal wave functions as basis. The second, which is numerical, is obtained by expanding the dimensionless temperature in terms of Legendre functions and then solving the resulting set of differential equations in the radial direction using an implicit finite difference scheme. The two solutions are further verified by comparing them to the limiting case of a sphere. We study the effect of the axis ratio on the time development of temperature inside the spheroid and the heat flux across the surface.