J. Non-Equilib. Thermodyn. 2005  Vol. 30  pp. 375–383 J. Non-Equilib. Thermodyn.  2005  Vol. 30  No. 4 6 Copyright 2005 Walter de Gruyter  Berlin  New York. DOI 10.1515/JNETDY.2005.026 Close-to-Fourier heat conduction equation of solids of constant mass density Wolfgang Muschik*, Ju ¨ rgen Siebert, Heiko Herrmann and Gunnar Ru ¨ ckner Institut fu ¨ r Theoretische Physik, Technische Universita ¨ t Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany *Corresponding author (muschik@physik.tu-berlin.de) Communicated by K.-H. Hoffmann, Chemnitz, Germany Abstract Heat conduction close-to-Fourier means that we look for a minimal ex- tension of heat conduction theory using the usual Fourier expression of the heat flux density and modifying that of the internal energy as minimally as possible by choosing the minimal state space. Applying Liu’s procedure re- sults in the class of materials and a di¤erential equation both belonging to the close-to-Fourier case of heat conduction. 1. Introduction As is well-known, one of other shortcomings of the classical Fourier heat conduction theory is caused by a parabolic di¤erential equation which allows indefinite propagation of energy. This unphysical fact can be avoided if the parabolic heat conduction equation is replaced by an hyperbolic one [1–4]. Here the question is investigated whether a minimal change in Fourier heat conduction results in an hyperbolic di¤erential equation of first order in time. By use of Fourier’s expression for the heat flux density and the minimal state space spanned by the temperature and its gradient, Liu’s procedure is applied for exploiting the second law systematically. The Liu procedure results in coupled di¤erential equations for the specific internal energy, the specific en- tropy, the heat flux density, and the entropy flux density. One result is that the internal energy and the entropy depend on only one variable, which is a state function and which transforms to the thermostatic temperature in the Bereitgestellt von | Technische Universität Berlin Angemeldet Heruntergeladen am | 15.10.18 18:55