Thermomass Theory in the Framework of GENERIC

Abstract

Thermomass theory was developed to deal with the non-Fourier heat conduction phenomena involving the influence of heat inertia. However, its structure, derived from an analogy to fluid mechanics, requires further mathematical verification. In this paper, General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC) framework, which is a geometrical and mathematical structure in nonequilibrium thermodynamics, was employed to verify the thermomass theory. At first, the thermomass theory was introduced briefly; then, the GENERIC framework was applied in the thermomass gas system with state variables, thermomass gas density ρ h and thermomass momentum m h , and the time evolution equations obtained from GENERIC framework were compared with those in thermomass theory. It was demonstrated that the equations generated by GENERIC theory were the same as the continuity and momentum equations in thermomass theory with proper potentials and eta-function. Thermomass theory gives a physical interpretation to the GENERIC theory in non-Fourier heat conduction phenomena. By combining these two theories, it was found that the Hamiltonian energy in reversible process and the dissipation potential in irreversible process could be unified into one formulation, i.e., the thermomass energy. Furthermore, via the framework of GENERIC, thermomass theory could be extended to involve more state variables, such as internal source term and distortion matrix term. Numerical simulations investigated the influences of the convective term and distortion matrix term in the equations. It was found that the convective term changed the shape of thermal energy distribution and enhanced the spreading behaviors of thermal energy. The distortion matrix implies the elasticity and viscosity of the thermomass gas.Entropy 2020, 22, 227 2 of 24 the equation predicted negative temperature at some cases which is unphysical [22]. Afterwards, Phase lag models, such as Dual Phase Lag model [23,24], were proposed and thought that a time lag existed between heat flux and temperature gradient and it could reduce to Cattaneo-Vernotte equation via expansions. However, it was found that the solutions were not continually dependent on the initial condition [19], which resulted in instability, and it could not avoid the problem of negative temperature [25][26][27]. Another series of non-Fourier equations were derived from the Boltzmann equation [28][29][30]. Guyer-Krumhansl equation [28,29], typical for this kinds of equations led to the investigations into phonon hydrodynamics, which was named due to its similar structure with Navier-Stokes equation. However, it also met mathematical difficulties [31,32], such as the violation of maximum principle for stability and the failure in preserving variable non-negativity. Recently, to study the heat transfer in a more intrinsic way, thermomass theory [33-36] was developed. It was found that the heat was conserved during conduction, similar to mass. Then, the mass nature of heat was recognized in heat conducti...