Homogenization of a conductive, convective, and radiative heat transfer problem in a heterogeneous domain

We are interested in the homogenization of heat transfer in periodic porous media where the fluid part is made of long thin parallel cylinders, the diameter of which is of the same order as the period. The heat is transported by conduction in the solid part of the domain and by conduction, convection, and radiative transfer in the fluid part (the cylinders). A nonlocal boundary condition models the radiative heat transfer on the cylinder walls. To obtain the homogenized problem we first use a formal two-scale asymptotic expansion method. The resulting effective model is a convection-diffusion equation posed in a homogeneous domain with homogenized coefficients evaluated by solving so-called cell problems where radiative transfer is taken into account. In a second step we rigorously justify the homogenization process by using the notion of two-scale convergence. One feature of this work is that it combines homogenization with a three dimensional (3D) to two dimensional (2D) asymptotic analysis since the radiative transfer in the limit cell problem is purely 2D. Eventually, we provide some 3D numerical results in order to show the convergence and the computational advantages of our homogenization method.