Nonlinear Guyer–Krumhansl equation and its boundary conditions in nanolayers

Abstract

This paper deals with nonlinear effects in the Guyer-Krumhansl equation for nonlocal heat transport, both from the perspective of the boundary conditions (phonon-wall collisions) and of the bulk equation (phonon-phonon collisions) and explores their consequences on the effective thermal conductivity of nanosystems between two parallel layers or in two-dimensional ribbons. The nonlinearity arises from a dependence of the respective mean free paths on the values of the heat flux. The boundary conditions refer to slip heat flow along the limiting walls of the system, analogous to velocity slip flow along the walls in rarefied fluid dynamics. The effective thermal conductivity turns out to depend on the Knudsen numbers related to both mean free paths and on the temperature gradient.