On the role of non-constant material parameters in the Guyer–Krumhansl heat equation

Abstract

Fourier’s law is widely used in engineering field to describe the heat conduction problems, but it leads to an infinite heat propagation velocity and ignores memory/inertial effects. These limitations reduce its accuracy in the modeling of fast or small-scale thermal processes. However, the Guyer–Krumhansl equation offers a more general and accurate framework, especially when complex or non-local heat transport phenomena are involved. Originally derived from phonon hydrodynamics at low temperatures, the Guyer–Krumhansl equation has since been successfully applied to heterogeneous materials and room-temperature problems using a continuum approach. This generalization is promising and could be the standard model in future engineering practice, but it would not be possible without a thorough investigation and understanding of its mathematical properties and the study of nonlinear effects resulting from non-constant material parameters. For this reason, this work investigates the nonlinear formulation of the Guyer–Krumhansl equation, focusing on the effects of temperature-dependent material parameters such as thermal conductivity and relaxation time. These nonlinearities are very essential for capturing real behaviors and ensuring thermodynamic consistency. The study is based on an analysis of the mathematical structure of the model and numerical results, illustrating the influence of non-constant material parameters. These insights contributes to extend the applicability of the Guyer–Krumhansl model in engineering field and guides future experimental validations.