Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer

Abstract

We study a dynamic boundary condition problem in heat transfer which represents the interaction between a conducting solid enclosed by a conducting shell. Both the solid and the shell are thermally inhomogeneous and anisotropic. Interaction is modelled by considering the solid as a source of thermal energy to the shell. A constitutive equation proposed by Carslaw and Jaeger establishes a relation between temperature in the shell and the boundary value of temperature in the solid. This gives rise to a dynamic boundary condition problem that has not been studied in the recent literature. The system of equations so obtained is presented as an implicit evolution equation which involves a pair of unbounded linear operators that map between two different spaces. We extend the operators to a jointly closed pair for which the implicit equation makes sense. The solution of the initial value problem is constructed by means of a holomorphic family of solution operators. The class of admissible initial states is surprisingly large.