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A note on the relation between the hypothesis of local equilibrium and the Clausius-Duhem inequality. (English) Zbl 0719.76004

The paper is devoted to different interpretations of Clausius-Duhem inequality (DCI) with regard to the domain on which constitutive equations are defined: The “physical space” \((x\in {\mathbb{R}}^ 3)\) is rigorously distinguished from the “state space” which is spanned by all equilibrium states of the system. By this distinction (a fiber bundle \({\mathbb{R}}^ 3\) consisting of the nonequilibrium states of the system is not introduced) the possibility of two interpretations of the CDI arises: On the “physical space” we get the one containing nonequilibrium contact quantities [W. Muschik, ibid. 8, No.3, 219-228 (1983)], on the “state space” we get the usual CDI of irreversible thermodynamics including a nonnegative entropy production, an unexpected fact because the “state space” was defined only to include equilibrium states. The solution is that the “state space” includes the internal variables as independent variables, and therefore it can not be the equilibrium subspace [W. Muschik, ibid. 15, No.2, 127-137 (1990)]. Consequently the accompanying processes introduced by a projection of the “physical space” onto the “state space” are not reversible, and the existence of the assumed equation \(dS=dQ^ 0/T\) on the “state space” is doubtful. These difficulties are avoidable by introducing a comprehensive nonequilibrium state space and an embedding axiom [J. Bataille and the author, ibid. 4, No.4, 228-229 (1979)]. Finally a distance between equilibrium and nonequilibrium states is introduced by the Deborah number.
Reviewer: W.Muschik (Berlin)

MSC:

76A02 Foundations of fluid mechanics
80A05 Foundations of thermodynamics and heat transfer
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References:

[1] Born M., Phys. Zeitschr. 22 pp 218– (1921)
[2] DOI: 10.1007/BF00248902 · doi:10.1007/BF00248902
[3] DOI: 10.1016/0020-7225(80)90096-8 · Zbl 0457.73003 · doi:10.1016/0020-7225(80)90096-8
[4] DOI: 10.1515/jnet.1979.4.4.229 · doi:10.1515/jnet.1979.4.4.229
[5] DOI: 10.1088/0034-4885/51/8/002 · doi:10.1088/0034-4885/51/8/002
[6] DOI: 10.1007/BF01392753 · doi:10.1007/BF01392753
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