Existence, uniqueness and construction of the solution of the energy transfer problem in a rigid and non-convex blackbody with temperature-dependent thermal conductivity

Abstract

In this paper, we study the steady-state (coupled) conduction–radiation heat transfer phenomenon in a non-convex opaque blackbody with temperature-dependent thermal conductivity. The mathematical description consists of a nonlinear partial differential equation subjected to a nonlinear boundary condition involving an integral operator that is inherently associated with the non-convexity of the body. The unknown is the absolute temperature distribution. The problem is rewritten with the aid of a Kirchhoff transformation, giving rise to linear partial differential equation and a new unknown. An iterative procedure is proposed for constructing the solution of the problem by means of a sequence of problems, each of them with an equivalent minimum principle. Proofs of convergence as well as existence and uniqueness of the solution are presented. An error estimate, for each element of the sequence, is presented too.