Temperature Field Around Nearly Spherical Cavities in Uniform Heat Flow

Abstract

The problem of determining temperature distribution near a spherical cavity in an otherwise infinite medium which is under uniform heat flow is a classical problem of linear heat conduction theory. Extensive reviews of relevant work can be found in [1-5]. It is clear from these studies is that what has been achieved in this regard is largely related to cavities with highly canonical shapes, mostly spherical. Solutions of similar problems for cavities with non-canonical shapes are rare. In the present article, we consider the case of a cavity in an infinite solid in a uniform heat flow, whose shape deviates slightly from a perfectly spherical shape (hereafter called nearly spherical cavity). To the first order in the small parameter characterizing the boundary perturbation, we are able to derive closed-form expressions for the temperature field around the nearly spherical cavity for sufficiently smooth boundary perturbations that are arbitrary functions of the azimuthal and polar angles.