Hydrodynamic Heat Transport Modeling: Continuum Models and the BTE*

The latest developments in microelectronics working at the nanometer scale, along with advancements in producing ultrapure monocrystalline materials and the discovery of the existence of second sound at 200 K in graphite, as reported by Ding et al. (Nature Communications 2021), indicates importance of hydrodynamic heat transport in technology. On a device scale in complex geometries, a direct solution of the Boltzmann Transport Equation may not be computationally feasible. Consequently, there is a need for models suitable for efficient computational methods, such as the Finite Element Method.

Early attempts to bypass the difficulty of directly solving the BTE included the development of the Guyer and Krumhansl (GK) equation. More recently, coarse-graining approaches to the BTE have been used to formulate the partial-differential Viscous Heat Equations (VHEs), and to predict the emergence of intriguing effects such as heat backflow and non-diffusive temperature profiles (Dragašević and Simoncelli, arXiv 2023). This work takes the next step in analyzing the VHEs and proposes an exact analytical solution for the Transient Thermal Grating geometry, allowing for a direct comparison with the recently developed semi-analytical approach to solving the full scattering matrix BTE. Furthermore, the BTE, VHEs, and GK equations are employed to solve different initial-boundary value problems, and their accuracies and regimes of applicability are systematically compared.