Acta Mechanica

, Volume 225, Issue 11, pp 3157–3186 | Cite as

On the natural structure of thermodynamic potentials and fluxes in the theory of chemically non-reacting binary mixtures

  • Ondřej Souček
  • Vít Průša
  • Josef Málek
  • K. R. Rajagopal


A theory describing the behavior of chemically non-reacting binary mixtures can be based on a detailed formulation of the governing equations for the individual components of the mixture or on treating the mixture as a single homogenized continuous medium. We argue that if we accept that both approaches can be used to describe the behavior of the given mixture, then the requirement on the equivalence of these approaches places restrictions on the possible structure of the internal energy, entropy, Helmholtz potential, and also of the diffusive, energy, and entropy fluxes. (The equivalence of the approaches is understood in the sense that the quantities used in one approach can be interpreted in terms of the quantities used in the other approach and vice versa. Further, both approaches must lead to the same predictions concerning the evolution of the physical system under consideration). In the case of a general chemically non-reacting binary mixture of components at the same temperature, we show that these restrictions can indeed be obtained by purely algebraic manipulations. An important outcome of this analysis is, for example, a general form of the evolution equation for the diffusive flux. The restrictions can be further exploited in the specification of thermodynamically consistent constitutive relations for quantities such as the interaction (drag) force or the Cauchy stress tensor. As an example of the application of the current framework, we derive, among others, a generalization of Fick’s law and we recover several non-trivial results obtained by other techniques. The qualitative features of the derived generalization of Fick’s law are demonstrated by a numerical experiment.


Entropy Internal Energy Constitutive Relation Entropy Production Cauchy Stress Tensor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bowen R.M.: Continuum Physics, vol. 3, Chap. Theory of Mixtures, pp. 1–127. Academic Press, New York (1976)Google Scholar
  2. 2.
    Cussler E.L.: Diffusion: Mass Transfer in Fluid Systems. Cambridge Series in Chemical Engineering, 3rd edn. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
  3. 3.
    Depireux N., Lebon G.: An extended thermodynamics modeling of non-Fickian diffusion. J. Non-Newton. Fluid Mech. 96(1-2), 105–117 (2001). doi: 10.1016/S0377-0257(00)00134-8 CrossRefzbMATHGoogle Scholar
  4. 4.
    de Groot S.R., Mazur P.: Non-equilibrium thermodynamics. Series in Physics. North-Holland, Amsterdam (1962)Google Scholar
  5. 5.
    Elafif A., Grmela M.: Non-Fickian mass transport in polymers. J. Rheol. 46(3), 591–628 (2002). doi: 10.1122/1.1470520 CrossRefGoogle Scholar
  6. 6.
    Elafif A., Grmela M., Lebon G.: Rheology and diffusion in simple and complex fluids. J. Non-Newtonian Fluid Mech. 86(1–2), 253–275 (1999). doi: 10.1016/S0377-0257(98)00211-0 CrossRefzbMATHGoogle Scholar
  7. 7.
    Fick A.: On liquid diffusion. Philos. Mag. 10(63), 30–39 (1855). doi: 10.1080/14786445508641925 Google Scholar
  8. 8.
    Grmela M., Elafif A.G.L.: Isothermal nonstandard diffusion in a two-component fluid mixture: a Hamiltonian approach. J. Non-Equilib. Thermodyn. 23, 312–327 (1988). doi: 10.1515/jnet.1998.23.4.376 Google Scholar
  9. 9.
    Grmela M., Öttinger H.C.: Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E 56(6), 6620–6632 (1997). doi: 10.1103/PhysRevE.56.6620 MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hutter K., Jöhnk K.: Continuum methods of physical modeling: continuum mechanics, dimensional analysis, turbulence. Springer, Berlin (2004)CrossRefGoogle Scholar
  11. 11.
    Joy D., Casas-Vázquez J., Lebon F.: Extended irreversible thermodynamics, 4th edn. Springer, New York (2010). doi: 10.1007/978-90-481-3074-0 CrossRefGoogle Scholar
  12. 12.
    Liu I.S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Rational Mech. Anal. 46, 131–148 (1972)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Liu Q., Kee D.De: Modeling of diffusion through polymeric membranes. Rheol. Acta 44(3), 287–294 (2005). doi: 10.1007/s00397-004-0410-7 CrossRefGoogle Scholar
  14. 14.
    Massoudi, M.: Boundary conditions in mixture theory and in CFD applications of higher order models. Recent advances in non-linear mechanics. Comput. Math. Appl. 53(2), 156–167 (2007). doi: 10.1016/j.camwa.2006.02.016
  15. 15.
    Mauri R.: Non-Equilibrium Thermodynamics in Multiphase Flows. Soft and Biological Matter, 4th edn. Springer, Berlin (2013)CrossRefGoogle Scholar
  16. 16.
    Maxwell J.C.: Illustrations of the dynamical theory of gases. Philos. Mag. 19, 19–32 (1860)Google Scholar
  17. 17.
    Maxwell J.C.: Illustrations of the dynamical theory of gases. Philos. Mag. 20, 21–37 (1860)Google Scholar
  18. 18.
    Müller I.: Thermodynamics. Interaction of Mechanics and Mathematics. Pitman Publishing Limited, London (1985)Google Scholar
  19. 19.
    Müller I.: Thermodynamics of mixtures and phase field theory. Int. J. Solids Struct. 38(6–7), 1105–1113 (2001). doi: 10.1016/S0020-7683(00)00076-7 CrossRefzbMATHGoogle Scholar
  20. 20.
    Prasad S.C., Rajagopal K.R.: On the diffusion of fluids through solids undergoing large deformations. Math. Mech. Solids 11(3), 291–305 (2006). doi: 10.1177/1081286504046484 MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rajagopal, K.R., Tao, L.: Mechanics of mixtures. Series on Advances in Mathematics for Applied Sciences, vol. 35. World Scientific. River Edge, NJ (1995)Google Scholar
  22. 22.
    Rajagopal K.R., Srinivasa A.R.: On thermomechanical restrictions of continua. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 460(2042), 631–651 (2004). doi: 10.1098/rspa.2002.1111 MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rajagopal K.R., Wineman A.S., Gandhi M.: On boundary conditions for a certain class of problems in mixture theory. Int. J. Eng. Sci. 24(8), 1453–1463 (1986). doi: 10.1016/0020-7225(86)90074-1 MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ricard Y., Bercovici D., Schubert G.: A two-phase model for compaction and damage: 1. General theory. J. Geophys. Res. 106(B5), 8907–8924 (2001). doi: 10.1029/2000JB900430 CrossRefGoogle Scholar
  25. 25.
    Samohýl I.: Thermodynamics of irreversible processes in fluid mixtures, Teubner-Texte zur Physik [Teubner Texts in Physics], vol. 12. Teubner, Leipzig (1987)Google Scholar
  26. 26.
    Shi J.J.J., Rajagopal K.R., Wineman A.S.: Applications of the theory of interacting continua to the diffusion of a fluid through a non-linear elastic media. Int. J. Eng. Sci. 19(6), 871–889 (1981)CrossRefzbMATHGoogle Scholar
  27. 27.
    Truesdell, C.: Rational thermodynamics, 2nd edn. Springer, New York (1984). doi: 10.1007/978-1-4612-5206-1. With an appendix by C. C. Wang, and additional appendices by 23 contributors.

Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  • Ondřej Souček
    • 1
  • Vít Průša
    • 1
  • Josef Málek
    • 1
  • K. R. Rajagopal
    • 2
  1. 1.Mathematical Institute, Faculty of Mathematics and PhysicsCharles University in PraguePraha 8, KarlínCzech Republic
  2. 2.Department of Mechanical EngineeringTexas A&M UniversityCollege StationUSA

Personalised recommendations