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Symmetric form of the equations of heat transport in a rigid conductor of heat with internal state variables. [Cz.] 1, Analysis of the model and thermodynamic restrictions via the "main dependency relation"

Larecki W.

Archives of Mechanics

1998

Vol. 50, nr 2

143-173

The objective of this series of two papers is twofold: to analyse the phenomenological model of a rigid conductor of heat with (vector) internal state variable, and to promote the application of the "main dependency relation" (MDR) as a tool for derivation of the restrictions on constitutive functions implied by the entropy inequality as well as a tool for direct derivation of alternative symmetric systems of field eguations. In this paper (Part I), the analysis of the model of a rigid conductor of heat with (vector) internal state variable is focused on two aspects, namely, on the form of the respective field equations and on the relation to other phenomenological models proposed in the literature, with the emphasis put on those models which have been succesfully adjusted to experimental data on heat transport at finite speeds. The relation to the model of a rigid conductor of heat with scalar internal state variable, called "semi-empirical temperature" is demonstrated. It is proved that, for the system of N corservation equations, consistency with the entropy inequality (in the form of first-order unirateral differential constrains) is equivalent to the requirement that the corresponding system of N + 1 conservation equations satisfies the "main depency relation" (MDR). For the model of a rigid conductor of heat with conservative evolution equation for internal state variable, the procedure of derivation of thermodynamic restrictions via the MDR is demonstrated.

Symmetric form of the equations of heat transport in a rigid conductor of heat with internal state variables. [Cz.] 2, Alternative symmetric systems

Larecki W.

Archives of Mechanics

1998

Vol. 50, nr 2

175-205

In part 1 of this series, it has been shown that the field equations corresponding to the model of a rigid conductor of heat with (vector) internal state variable subject to the entropy inequality can be represented as the respective system of N + 1 conservation eqations for N unknowns, on which the "main dependency relation" (MDR) is imposed. In this paper (part 2), it is demonstrated how two families of symetric systems corresponding to the consistent system of N conservation equations (family of symmetric systems for original unknowns and the family of N + 1 symmetric conservative systems for transformed unknowns) can be directly derived with the aid of the MDR. The condition of equivalence of simmetric systems to the original system of conservation equations is analysed and alternatively formulated. For the considered model of a rigid conductor of heat, the conditions on free energy that assure symmetric hyperbolicity of simmetric systems are established, and it is shown that they are stronger than the conditions required for equivalence of simmetric systems to the original system of conservation equations. Two alternative symmetric conservative systems are derived for the considered model of a rigid conductor of heat and the conditions of symmetric hyperbolicity for those systems are established with the aid of the relation between convexity (concavity) of the respective generating potentials, and with the aid of the relation between symmetric hyperbolicity of the symmetric systems for original unknowns and symmetric conservative system for the transformed unknowns.