Heat conduction in a fine scale mixture of two conductors is examined in the presence of a contact resistance between phases. The problem is studied rigorously in the context of periodic homogenization. Unlike the case of perfect heat transmission between phases, the temperature gradients converge weakly as Radon measures. The strict ellipticity of the homogenized transport equation depends upon the geometry of the interface. The effective conductivity associated with the overall heat dissipation rate inside a composite cube is considered. It is shown that this property exhibits a size effect under rescaling.
Publication Source (Journal or Book title)
SIAM Journal on Applied Mathematics
Lipton, R. (1998). Heat conduction in fine scale mixtures with interfacial contact resistance. SIAM Journal on Applied Mathematics, 58 (1), 55-72. https://doi.org/10.1137/S0036139995295153