Thermodynamics of Class I and Class II Classical Mixtures

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


In this chapter, two versions of mixtures of Boltzmann-type continua are subject to thermodynamic analyses for viscous fluids. Of the two forms of the Second Law that were introduced—the ClausiusDuhem inequality applied to open systems and the entropy principle of Müller—the latter principle is employed in the process of deduction of the implications revealed by the particular Second Law. The goal in the two parts of the chapter is to derive the ultimate forms of the governing equations, which describe the thermomechanical response of the postulated constitutive behavior without violation of the Second Law of thermodynamics. The versions of mixtures which are analyzed are
  • Diffusion of tracers in a classical fluid: The conceptual prerequisites of this type of processes are mixtures of class I, in which the major component is the bearer fluid within which a finite number of constituents with minute concentration are suspended or solved in the bearer fluid. The motion of these tracers is described by the difference of the constituent velocities relative to the barycentric velocity of the mixture as a whole. For the dissipative constitutive class applied to the entropy principle, the existence of the Kelvin temperature is proved, the form of the Gibbs relation could be determined as could the conditions of thermodynamic equilibrium and the constitutive behavior in its vicinity.

  • Thermodynamics of a saturated mixture of nonpolar solid–fluid constituents: Conceptually these systems are treated as classical mixtures of class II, in which the individual motions of the constituents are separately accounted for by their own balances of mass and momentum, but subject to a common temperature. The analysis of the dissipation inequality is performed subject to the assumption of constant true density of all constituents and the supposition of saturation of the mixture. The constitutive relations are postulated for a mixture of viscous heat conducting fluids. The explanation of the entropy principle is structurally analogous to that of the class I-diffusion theory, but is analytically much more complex. Unfortunately, intermediate ad hoc assumptions must be introduced to deduce concrete results that will lead to fully identifiable fluid dynamical equations, which are in conformity with the Second Law for the presented type of mixtures.


Diffusion of classical fluid mixtures Müller-type thermodynamics Thermodynamic equilibrium and nonequilibrium Classical saturated solid–fluid mixtures 


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Authors and Affiliations

  1. 1.c/o Laboratory of Hydraulics, Hydrology, GlaciologyETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringDarmstadt University of TechnologyDarmstadtGermany

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