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.
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10 February 2021
A Correction to this paper has been published: https://doi.org/10.1007/s42985-020-00058-4
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Acknowledgements
The thoughts and insights of my students from years gone by, Nic van Rensburg (1982), Wessel Rossouw (1983) and Alna van der Merwe (1993), echo in this work.
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This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.
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Sauer, N. Dynamic boundary conditions and the Carslaw-Jaeger constitutive relation in heat transfer. SN Partial Differ. Equ. Appl. 1, 48 (2020). https://doi.org/10.1007/s42985-020-00050-y
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DOI: https://doi.org/10.1007/s42985-020-00050-y