Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Entropy Principle Exploited by Lagrange Multipliers

Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_60-1

Synonyms

Definition

A proper thermodynamic theory of mixtures avoids the so-called paradox of diffusion, by which disturbances in concentration propagate at infinite speeds. A theory with equations of balance of momenta for the constituents does that. Constraints on the constitutive functions are implied by the entropy principle which is exploited by use of Lagrange multipliers.

Introduction

The entropy principle provides restrictions on the form of the constitutive functions in thermodynamics. And Lagrange multipliers are a convenient tool for the exploitation of such restrictions. The method is illustrated here for a mixture of inviscid fluids – a paradigmatic case.

The basis for a systematic theory of mixtures was laid down by Truesdell (1957), and Müller has developed its thermodynamics in several papers (Müller, 1968, 1973; Müller and Villaggio, 1976), the latter in collaboration with Villaggio. Those were...

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References

  1. de Groot SR, Mazur P (1962) Non-equilibrium thermodynamics. North Holland, AmsterdamGoogle Scholar
  2. Liu I-S (1972) Method of lagrange multipliers for the exploitation of the entropy inequality. Arch Ration Mech Anal 46:131Google Scholar
  3. Liu I-S, Müller I (1984) Thermodynamics of mixtures of fluids. In: Truesdell C (ed) Rational thermodynamics, 2nd edn. Appendix 5B. Springer, Heidelberg, p 264Google Scholar
  4. Meixner J, Reik HG (1959) Die Thermodynamik der irreversiblen Prozesse in kontinuierlichen Medien mit inneren Umwandlungen. p. 471ff of Flügge’s Handbuch der Physik, vol III/2. Springer, HeidelbergGoogle Scholar
  5. Müller I (1968) A thermodynamic theory of mixtures of fluids. Arch Ration Mech Anal 28:1MathSciNetCrossRefMATHGoogle Scholar
  6. Müller I (1970) A new systematic approach to non-equilibrium thermodynamics. Pure Appl Chem 22:335CrossRefGoogle Scholar
  7. Müller I (1973) A new approach to thermodynamics of simple mixtures. Zeitschrift für Naturforschung 28a:1801Google Scholar
  8. Müller I, Villaggio P (1976) Conditions for stability and wave speeds for fluid mixtures. Meccánica 11:191CrossRefMATHGoogle Scholar
  9. Truesdell C (1957) Sulle basi della termodinamica. Accademia Nazionale dei Lincei. Rendiconti della Classe di Scienze Fisiche, Matematiche e Naturali 22:33Google Scholar

Copyright information

© Springer-Verlag GmbH Germany 2018

Authors and Affiliations

  1. 1.Technical University BerlinBerlinGermany

Section editors and affiliations

  • Elena Ivanova
    • 1
  1. 1.Department of Theoretical Mechanics, Institute of Applied Mathematics and MechanicsPeter the Great St. Petersburg Polytechnic UniversitySt. PetersburgRussia