Given a composite material composed of two materials each having different thermal conductivities and heat flux vectors. This leads us to write the two properties as position dependent. One thing I don't understand is taking the temperatures of both materials to be equal at the interface when writing the boundary condition. And also, the normal heat flux vectors to be equal at the interface between the two materials. The writer in the paper An Introduction to Periodic Homogenization mentions that it's for the sake of continuity. It isn't clear to me.
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$\begingroup$ Given temperature as a function of x, you cannot get two temperatures out of the function for the same x. If not equal, where does the heat go? That all said, sometimes it makes more sense to model the interfaces between two materials as a third 'material' $\endgroup$– AbelOct 5, 2022 at 13:43
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Temperatures must be equal at the interface, otherwise the gradient $\frac{\partial u}{\partial x}$ would be infinite which would mean infinite heat flux. Regarding heat flux: the heat that goes from one material through the interface must end up in the other material (interface by itself cannot generate or consume any heat).
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$\begingroup$ Thank you. So, the interface allows only for passing the heat flux. But, why is the heat flux leaving material 1 equal to the heat flux leaving material 2? $\endgroup$ Oct 7, 2022 at 7:58
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1$\begingroup$ The heat fluxes are not equal, the equality is put between heat fluxes multiplied by respective outward pointing unit vectors. These vectors have the opposite direction, so they basically change one sign. In 2D, with $q$ in the $x$ direction and perpendicular boundary, $\vec{q}_1\cdot \vec{n}_1 = \vec{q}_2\cdot \vec{n}_2$ could be written as $[q_{1,x}, q_{1,y}]\cdot [1, 0]^T = [q_{2,x}, q_{2,y}]\cdot [-1, 0]^T$, which is $q_{1,x} = -q_{2,x}$. $\endgroup$ Oct 7, 2022 at 16:44