Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 80 / 1 / i

Take $a_1,a_2>0$ and $p_1+p_2=1$. Write $g^{(k)}=\nabla u$ in phase $k$ and $h^{(k)}=a_k{g}^{(k)}$. Here $h$ is the positive conductivity flux; physical heat flux is $-h$. Continuity of temperature on a planar interface implies continuity of its tangential derivatives, while conservation gives continuity of normal heat flux. With interface normal $e_1$, these conditions areThe mean gradient is $E=p_1{g}^{(1)}+p_2{g}^{(2)}$. ThereforeBy the definition of effective conductivity, $⟨h⟩=a^{\ast }E$. ThusThe laminate conductivity is a weighted harmonic mean across the layers and a weighted arithmetic mean along them. This also explains why isotropic constituents can yield an anisotropic thermal conductivity tensor: flux must cross each resistance in sequence in the normal direction, whereas tangential transport occurs in parallel.