Thermodynamics—Field Formulation

  • Kolumban HutterEmail author
  • Yongqi Wang
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)


This chapter extends and applies the concepts of Chap.  17 to continuous material systems. The Second Law is written in global form as a balance law of entropy with flux, supply and production quantities, which can be written in local form as a differential statement. The particular form of the Second Law then depends upon, which postulates the individual terms in the entropy balance are subjected to. When the entropy flux equals heat flux divided by absolute temperature and the entropy-production-rate density is requested to be non-negative, the entropy balance law appears as the ClausiusDuhem inequality and its exploitation follows the axiomatic procedure of open systems thermodynamics as introduced by Bernard Coleman and Walter Noll. When the entropy flux is left arbitrary but is of the same function class as the other constitutive relations and the entropy supply rate density is identically zero, then the entropy inequality appears in the form of Müller. In both cases, the second law is expressed by the requirement that the entropy-production-rate density is non-negative, but details of the exploitation of the second law in the two cases are subtly different from one another. For standard media such as elastic and/or viscous fluids the results are the same. However, for complex media they may well differ from one another. Examples illustrate the procedures and results.


Entropy balance law Thermodynamic process—equilibrium ClausiusDuhem inequality Müller’s entropy principle Exploitation of the entropy principle Absolute/empirical temperature Specific heats of ideal gasses Hyperbolic heat conduction equation 


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.c/o Versuchsanstalt für Wasserbau, Hydrologie und GlaziologieETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany

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