Universality in heat conduction theory: weakly nonlocal thermodynamics

Abstract

A linear irreversible thermodynamic framework of heat conduction in rigid conductors is introduced. The deviation from local equilibrium is characterized by a single internal variable and a current multiplier. A general constitutive evolution equation of the current density of the internal energy is derived by introducing a linear relationship between the thermodynamic forces and fluxes. The well-known Fourier, Maxwell-Cattaneo-Vernotte, Guyer-Krumhansl, Jeffreys-type, and Green-Naghdi-type equations of heat conduction are obtained as special cases. The universal character of the approach is demonstrated by two examples. Solutions illustrating the properties of the equation with jump boundary conditions are given.

Theories and heat pulse experiments of non-Fourier heat conduction

“…It is remarkable that the Green-Naghdi model III and II cannot be obtained here, however it is a valid special case if a general vectorial internal variable is introduced instead of the heat flux [69]). …”

Thermodynamic framework for a generalized heat transport equation

“…(6) is a particular hypothesis concerning the form of the entropy flux, which turns out to be far more general than the classical expression J s =q/T . Analyses of generalized expressions of entropy flux may be found in [21,22,[35][36][37][38][39][40][41][42]. Substitution of Eq.…”

“…Later, classical linear irreversible thermodynamics [2] was developed by Onsager [3,4], Eckart [5,6], Meixner [7] and Prigogine [8] for the near-equilibrium heat transport described by Fourier law. In recent years, generalized laws of heat transport in micro-and nanoscale systems [9][10][11][12][13][14] have been again the stimulus for further developments of compatible irreversible thermodynamics and wider formulations of the second law, as in diverse branches of rational thermodynamics [15], rational extended thermodynamics [16][17][18][19], extended irreversible thermodynamics [20,21], weakly nonlocal thermodynamics [22,23] and GENERIC [24][25][26]. In the present work, we will illustrate the close connection between generalized heat transport equations and generalized forms of the second law in the framework of extended irreversible thermodynamics.…”

Entropy Principle and Recent Results in Non-Equilibrium Theories

“…In such a case the balance equations are difficult to solve, and sometime the well posedness of the relevant Cauchy problem is not guaranteed [146]. Another possibility, used both in RT [147][148][149] and in CIT [150] is to introduce internal state variables [123,126]. Such an approach leads to more simple systems of equations.…”