Investigating the whirling heat current density in the Guyer--Krumhansl equation

TL;DR

This paper explores the unique whirling heat current density in the Guyer--Krumhansl equation, revealing how it can cause localized temperature decreases due to the curl of heat flux in two-dimensional models.

Contribution

It introduces a novel analysis of the whirling heat flux in the Guyer--Krumhansl equation, including a new boundary condition extrapolation method and an analogy with acoustics.

Findings

Whirling heat flux can cause local temperature drops.

A new boundary condition extrapolation method was proposed.

Analogy with linearized acoustics of fluids was established.

Abstract

Among the numerous heat conduction models, the Guyer--Krumhansl equation has a special role. Besides its various application possibilities in nanotechnology, cryotechnology, and even in case of modeling heterogeneous materials, it poses additional mathematical challenges compared to the Fourier or Cattaneo {(a.k.a. Maxwell--Cattaneo--Vernotte)} equations. Furthermore, the Guyer--Krumhansl equation is the first heat conduction model, which includes the curl of the heat flux density in the evolution equation. In the present paper, we place our focus on the consequences of the existence of such whirling heat current density by solving the two-dimensional Guyer--Krumhansl equation with a space and time-dependent heat pulse boundary condition. The discretization poses further challenges in regard to the boundary condition for which we propose a particular extrapolation method. Furthermore, with the help of the Helmholtz decomposition, we show the analogy with the linearized acoustics of Newtonian fluids, which reveals how the heat flux density plays the role of the velocity field. Our solutions also reveal an unexpected temperature evolution caused by the whirling heat flux density, namely, the temperature can locally be decreased for a short time in a case when the curl of the heat flux density dominates the heat conduction process.