A perturbation solution for the steady heat flow around a thin circular insulating annulus on the surface of a conducting half-space

Abstract

Abstract A constant temperature circular disc on the surface of a conducting half-space is surrounded by an insulating annulus. The remainder of the surface of the half space is maintained at zero temperature. The steady heat flow rate from the disc to the zero temperature surface is required. The mathematical problem, a three-part mixed boundary value problem can be reduced to integral equation form. Several alternative formulations are possible. The existing formulations do not readily yield solutions for the case in which the thickness-to-radius radio of the insulating annulus is small compared to unity. This case is considered here and a solution based on integral perturbation methods is obtained. An existing alternative integral equation formulation is also solved, by iteration, for cases in which the inner-to-outer radius ratio of the insulating annulus is small compared to unity. These two solutions are found to coalesce over an intermediate range of annulus thickness. Thus a composite solution is obtained which is valid for all cases. The solutions given here are also of practical interest in analogous problems in, e.g. the flow of fluids through porous media, and in elasticity theory.