Dynamical Governing Equations of Non-Fourier Heat Conduction

Abstract

Thermal energy has its corresponding equivalent mass according to Einstein’s mass–energy equivalence, which is termed as thermomass. The thermomass theory established the continuous governing equation for the non-Fourier heat conduction. The mass balance equation of thermomass gives the energy conservation equation while the momentum balance equation of thermomass gives the general heat conduction law. The microscopic foundation of the general heat conduction law based on the thermomass theory is investigated. The derivation based on the phonon Boltzmann equation shows that the second order expansion of phonon distribution function leads to the spatial inertia (or convective) term in the general heat conduction law, which makes the difference from the previous phonon hydrodynamic model. Limiting to the first order expansion will give the Cattaneo-Vernotte model, while the zeroth order expansion gives the classical Fourier’s law. Comparison with other derivations of phonon Boltzmann equation such as the Chapman–Enskog expansion and eigenvalue analysis is presented.