Continuum Mechanics and Thermodynamics

, Volume 25, Issue 6, pp 705–725 | Cite as

Constitutive modeling of multiphase flows with moving interfaces and contact line

  • Yongqi Wang
  • Martin Oberlack
  • Andreas Zieleniewicz
Original Article


A continuum description of multiphase flows, in which excess physical quantities associated with phase interfaces and the three-phase contact line are incorporated, is briefly presented. A thermodynamic analysis, based on the Müller–Liu thermodynamic approach of the second law of thermodynamics, is performed to derive the expressions of the constitutive variables in thermodynamic equilibrium. Non-equilibrium responses are proposed by use of a quasi-linear theory. A set of constitutive equations for the surface and line constitutive quantities is postulated. Some restrictions for the emerging material parameters are derived by means of the minimum conditions of the surface and line entropy productions in thermodynamic equilibrium. Hence, a complete continuum mechanical model to describe excess surface and line physical quantities is formulated.


Multiphase flow Entropy principle Constitutive equations 


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  1. 1.
    Baehr H.D.: Thermodynamik. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  2. 2.
    Bargmann S., Steinmann P.: Classical results for a non-classical theory: remarks on thermodynamic relations in Green–Naghdi thermo-hyperelasticity. Continuum Mech. Thermodyn. 19(1–2), 59–66 (2007)MathSciNetADSCrossRefMATHGoogle Scholar
  3. 3.
    Coleman B.D., Noll W.: The thermodynamics of elastic materials with heat condition and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ehlers, W.: Poröse Medien, ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie. Habilitation, Universität-Gesamthochschule Essen (1989)Google Scholar
  5. 5.
    Fang C., Wang Y., Hutter K.: Shearing flows of a dry granular material—hypoplastic constitutive theory and numerical simulations. Int. J. Numer. Anal. Methods. Geomech. 30, 1409–1437 (2006)CrossRefMATHGoogle Scholar
  6. 6.
    Fang C., Wang Y., Hutter K.: A thermo-mechanical continuum theory with internal length for cohesionless granular materials. Part I: a class of constitutive models. Continuum Mech. Thermodyn. 17(8), 545–576 (2006)MathSciNetADSCrossRefMATHGoogle Scholar
  7. 7.
    Fang C., Wang Y., Hutter K.: A thermo-mechanical continuum theory with internal length for cohesionless granular materials. Part II: Non-equilibrium postulates and numerical simulations of simple shear, plane Poiseuille and gravity driven problems. Continuum Mech. Thermodyn. 17(8), 577–607 (2006)MathSciNetADSCrossRefMATHGoogle Scholar
  8. 8.
    Fang C., Wang Y., Hutter K.: A unified evolution equation for the Cauchy stress tensor of an isotropic elasto-visco-plastic material. I. On thermodynamically consistent evolution. Continuumn. Mech. Thermodyn. 19, 423–440 (2008)MathSciNetADSCrossRefMATHGoogle Scholar
  9. 9.
    Grad H.: Principles of the Kinetic Theory. Handbuch der Physik, XII. Springer, Berlin (1958)Google Scholar
  10. 10.
    Hutter, K.: The physics of ice-water phase change surfaces. In: Kosinski, W., Murdoch, A.I. (eds.) Modelling Macroscopic Phenomena at Liquid Boundaries. CISM Course 318, Springer, Vienna (1991)Google Scholar
  11. 11.
    Hutter K., Jöhnk K., Svendsen B.: On interfacial transition conditions in two-phase gravity flow. Z. Angew. Math. Phys. 45, 746–762 (1994)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Hutter K., Jöhnk K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004)CrossRefMATHGoogle Scholar
  13. 13.
    Hutter, K., Wang, Y.: Phenomenological thermodynamics and entropy principles. In: Greven, A., Keller, G., Warnecke, G. (eds.) Entropy. Princeton University Press, Princeton-Oxford, pp. 57–78. ISBN 0-691-11338-6 (2003)Google Scholar
  14. 14.
    Kirchner N.: Thermodynamically consistent modelling of abrasive granular materials, Part I: non-equilibrium theory. Proc. R. Soc. Lond. A 458, 2153–2176 (2002)MathSciNetADSCrossRefMATHGoogle Scholar
  15. 15.
    Kirchner N., Teufel A.: Thermodynamically consistent modelling of abrasive granular materials Part II: thermodynamic equilibrium and applications to steady shear flows. Proc. R. Soc. Lond. A 458, 3053–3077 (2002)ADSCrossRefMATHGoogle Scholar
  16. 16.
    Liu I.-S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46, 131–148 (1972)MATHGoogle Scholar
  17. 17.
    Liu I.-S., Müller I.: Thermodynamics of mixtures of fluids. In: Truesdell, C. (ed.) Rational Thermodynamics, pp. 264–285. Springer, New York (1984)CrossRefGoogle Scholar
  18. 18.
    Luca I., Fang C., Hutter K.: A thermodynamic model of turbulent motions in a granular material. Continuum Mech. Thermodyn. 16, 363–390 (2004)MathSciNetADSCrossRefMATHGoogle Scholar
  19. 19.
    Müller I.: On the entropy inequality. Arch. Ration. Mech. Anal. 26, 118–141 (1967)CrossRefMATHGoogle Scholar
  20. 20.
    Müller I.: Die Kältefunktion, eine universelle Funktion in der Thermodynamik viskoser wärmeleitender Flüssigkeiten. Arch. Ration. Mech. Anal. 40, 1–36 (1971)CrossRefMATHGoogle Scholar
  21. 21.
    Müller I.: Thermodynamik—Grundlagen der Materialtheorie. Bertelsman Universitätsverlag, Düsseldorf (1972)Google Scholar
  22. 22.
    Müller I.: Thermodynamics. Pitman, London (1985)MATHGoogle Scholar
  23. 23.
    Sadiki A., Bauer W., Hutter K.: Thermodynamically consistent coefficient calibration in nonlinear and anisotropic closure models for turbulence. Continuum Mech. Thermodyn. 12, 131–149 (2000)MathSciNetADSCrossRefMATHGoogle Scholar
  24. 24.
    Spurk J.H.: Fluid Mechanics. Springer, Berlin (1997)CrossRefMATHGoogle Scholar
  25. 25.
    Svendsen B., Hutter K.: On the thermodynamics of a mixture of isotropic materials with constraints. Int. J. Eng. Sci. 33, 2021–2054 (1995)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Svendsen B., Hutter K., Laloui L.: Constitutive models for granular materials including quasi-static frictional behaviour: toward a thermodynamic theory of plasticity. Continuum Mech. Thermodyn. 4, 263–275 (1999)MathSciNetADSCrossRefGoogle Scholar
  27. 27.
    Svendsen B., Chanda T.: Continuum thermodynamic formulation of models for electromagnetic thermoinelastic solids with application in electromagnetic metal forming. Continuum Mech. Thermodyn. 17, 1–16 (2005)MathSciNetADSCrossRefMATHGoogle Scholar
  28. 28.
    Wang Y., Hutter K.: Comparison of two entropy principles and their applications in granular flows with/without fluid. Arch. Mech. 51, 605–632 (1999)MathSciNetMATHGoogle Scholar
  29. 29.
    Wang Y., Hutter K.: Shearing flows in a Goodman-Cowin type granular material—theory and numerical results. Part. Sci. Technol. 17, 97–124 (1999)ADSCrossRefGoogle Scholar
  30. 30.
    Wang Y., Hutter K.: A constitutive model for multi-phase mixtures and its application in shearing flows of saturated soil-fluid mixtures. Granul. Matter 1, 163–181 (1999)CrossRefGoogle Scholar
  31. 31.
    Wang Y., Hutter K.: A constitutive theory of fluid-saturated granular materials and its application in gravitational flows. Rheol. Acta 38, 214–223 (1999)CrossRefGoogle Scholar
  32. 32.
    Wang Y., Oberlack M.: A thermodynamic model of multiphase flows with moving interfaces and contact line. Continuum Mech. Thermodyn. 23, 409–433 (2011)MathSciNetADSCrossRefMATHGoogle Scholar
  33. 33.
    Wilmanski, K.: Continuum Thermodynamics, Part 1: Foundations. World Scientific Pub Co. (2009)Google Scholar
  34. 34.
    Wood L.C.: The bogus axioms of continuum mechanics. Bull. Math. Appl. 17, 98–102 (1981)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Yongqi Wang
    • 1
    • 2
  • Martin Oberlack
    • 1
    • 3
    • 4
  • Andreas Zieleniewicz
    • 1
  1. 1.Chair of Fluid Dynamics, Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Institute of Geotechnical EngineeringUniversity of Natural Resources and Applied Life SciencesViennaAustria
  3. 3.Center of Smart InterfacesTechnische Universität DarmstadtDarmstadtGermany
  4. 4.Graduate School of Computational EngineeringTechnische Universität DarmstadtDarmstadtGermany

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