An exact homogenization method in heat conduction

Abstract

We determine the macroscopic features of thermal transport in heterogeneous conductors by generalizing an exact, source-driven homogenization method originally developed for waves. The formulation accommodates random or periodic media of finite or infinite extent, with or without pores. Our homogenization shows that the effective heat flux and entropy are spatiotemporally nonlocal functions of both the effective temperature and its gradient, and that the emergent bianisotropic cross-couplings form an adjoint pair when the microscopic relations are self-adjoint. A spatially local approximation highlights how the homogenized diffusion equation can become hyperbolic due to temporal nonlocality, and that the medium’s thermal impedance can become direction-dependent as captured by the bianisotropic terms. In addition, we develop a retrieval method for one-dimensional deterministic composites, whose results reinforce our conclusions.