Tangent law of refraction for heat conduction through an interface and underlying variational principle

Abstract

By considering heat flux as ‘‘rays,’’ a law of refraction for heat conduction through an interface is obtained from the boundary conditions. This law is analogous to but different from Snell’s law, with the tangents of the angles of incidence and refraction replacing the sines and the reciprocal of the thermal conductivity taking the place of the refractive index. The deviation of the refracted ray departs from that predicted by Snell’s law as the angle of incidence increases and is zero for both normal and grazing incidences. The tangent law of refraction is also derived from a minimal principle similar to Fermat’s. Finally, a variational principle of heat conduction is postulated and illustrated with a simple example.