Symmetric forms of the equations of heat transport in a rigid conductor of heat with internal state variabies. II. Alternative symmetric systems

Abstract

In Part I 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 equations for N unknowns, on which the "main dependency relation" (MDR) is imposed. In this paper (Part II), it is demonstrated how two families of symmetric 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 symmetric 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 symmetric systems are established, and it is shown that they are stronger than the conditions required for equivalence of symmetric 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.