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.

Keywords: GENERIC; hyperbolic heat conduction; thermomass; thermomass energy.

Figure A1

Schematic diagram for the staggered grids.

Figure A2

Schematic diagram for the calculation volume of q_x.

Figure 1

Schematic diagram of the numerical regime for one-dimensional heat conduction problem.

Figure 2

Temperature distributions of Equation (76) at time t = 0.1, 0.3, 0.5 and 0.7. The solid line denotes the results without the convective term and the dash line denotes the results with the convective term.

Figure 3

Schematic diagram of the numerical regime for the two-dimensional heat conduction problem.

Figure 4

Energy density distribution patterns at time t = 0.1, 0.3 and 0.6. (a–c) show the figures e_n obtained from the equations without heat flux convective term, while (d–f) are the figures of the results e_c for the equations with heat flux convective term. The different color denotes the values of energy density e_n and e_c.

Figure 5

The energy difference patterns e_c-e_n at time t = 0.3 between those with a convective term and those without a convective term.

Figure 6

Energy distribution patterns at time t = 0.3 for the equations with distortion matrix and convective term e_D (a) and those equations without distortion matrix e_c (b).

Figure 7

Energy difference patterns e_D-e_c between those with distortion matrix e_D and those without distortion matrix e_c when the Hamiltonian energy H is the function of D_xx and D_yy.