Induced Gradients in Steady, Two-Dimensional Heat Conduction? Yes, But…

Abstract

A two-dimensional object conducts heat steadily between isothermal segments of its boundary that are at two different temperatures, with the heat flow occurring either through the object or through the region surrounding it. In classical potential theory, the isothermal surfaces are represented by source distributions, and the adiabatic surfaces that separate them are represented by dipole distributions. Sources or dipoles at one location can induce a temperature gradient at another location on an isothermal surface. This induced gradient adds to the gradient produced by a source at that location. In this paper, induced gradients are shown to produce zero net power in objects that have appropriate geometrical symmetry but not in objects that lack symmetry. Further, unpowered conductors within the domain (so-called floating conductors) are shown to have a nonzero induced source density that integrates to zero over the surface of the conductor. These results differ from those of a previous study of such configurations.