Two-dimensional steady heat conduction shows a remarkable equality of the heat transfer rate around the inside and the outside of closed curves, despite the asymmetry and singularity of the boundary heat flux distributions. Steady solutions can exist outside closed boundaries on which two isothermal segments at different temperatures are separated by a pair of adiabatic segments. In this study, classical potential theory is used to find solutions as a combination of simple-layer and double-layer potentials, including the relationships of heat fluxes and temperatures inside and outside the boundary curve. Isothermal boundaries exhibit an induced heat flux that varies from point-to-point on the boundary. The induced flux integrates to zero over each isothermal edge. Singularities of the heat flux are identified and resolved. Computations that validate the theory are provided for mixed boundary conditions on a disk and a square. Numerical fits to both the simple-layer and double-layer densities are given for the disk and the square. The analysis explains why the interior and exterior conduction shape factors are equal despite wildly differing heat flux distributions, and the results are compared to a previous study of this configuration.

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