The Knudsen Layer in Modeling the Heat Transfer at Nanoscale: Bulk and Wall Contributions to the Local Heat Flux

Starting from the observation that the influence of the heat carriers' boundary scattering on the heat flux is mainly felt in the zone near the system's boundary, the characteristic dimension of which is of the order of the mean-free path of the heat carriers, in this paper, we introduce the concept of the Knudsen layer in the heat transport at nanoscale and regard the local heat flux as the final resultant of two different contributions: the bulk heat flux and the wall heat flux. In the framework of phonon hydrodynamics, we therefore, here, derive a theoretical model in agreement with the second law of thermodynamics that accounts for those two contributions. In steady states, we then predict both how the local heat flux behaves and how the thermal conductivity depends on the characteristic dimension of the system. This analysis is performed both in the case of a nanolayer and in the case of a nanowire.