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Phonon hydrodynamics module for COMSOL Multiphysics

Enrico Di Lucente, EPFL
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This user guide presents the Phonon hydrodynamics module (V1.0) for COMSOL Multiphysics, enabling simulation of viscous heat transport in high-conductivity materials. The module implements the viscous heat equations (VHE), a mesoscopic framework that extends Fourier’s law by coupling temperature evolution with phonon drift velocity dynamics, capturing hydrodynamic phenomena such as Poiseuille flow, second sound, heat backflow, and thermal vortices. The guide provides detailed instructions for module installation via the COMSOL Physics Builder, including importing the builder file and adding the viscous heat equations interface to user models. Users are guided on applying temperature, heat flux, drift velocity, slip, and no-slip boundary conditions, supporting both steady-state and time-dependent simulations. Example cases illustrate classical versus hydrodynamic heat transport, incompressible-flow limits, vortex formation, and oscillatory boundary-driven dynamics. Default parameters are provided for materials with first-principles transport coefficients, such as diamond, silicon, graphite, and hexagonal boron nitride. By bridging full Boltzmann transport calculations and continuum diffusion theory, the module offers a practical, efficient approach for exploring non-diffusive, hydrodynamic heat transport in complex geometries. Complementary COMSOL files accompany the guide, allowing users to replicate and extend the examples.

Figure source: E. Di Lucente, F. Libbi, and N. Marzari, "Vortices and backflow in hydrodynamic heat transport", arXiv preprint arXiv:2501.16580 (2025).

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