Skip to main content
SpringerLink
Log in

Constitutive modeling of multiphase flows with moving interfaces and contact line

  • Original Article
  • Published:
Continuum Mechanics and Thermodynamics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Baehr H.D.: Thermodynamik. Springer, Heidelberg (2009)

    Book  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. Coleman B.D., Noll W.: The thermodynamics of elastic materials with heat condition and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ehlers, W.: Poröse Medien, ein kontinuumsmechanisches Modell auf der Basis der Mischungstheorie. Habilitation, Universität-Gesamthochschule Essen (1989)

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Grad H.: Principles of the Kinetic Theory. Handbuch der Physik, XII. Springer, Berlin (1958)

    Google Scholar 

  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)

  11. Hutter K., Jöhnk K., Svendsen B.: On interfacial transition conditions in two-phase gravity flow. Z. Angew. Math. Phys. 45, 746–762 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hutter K., Jöhnk K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  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)

  14. Kirchner N.: Thermodynamically consistent modelling of abrasive granular materials, Part I: non-equilibrium theory. Proc. R. Soc. Lond. A 458, 2153–2176 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    Article  ADS  MATH  Google Scholar 

  16. Liu I.-S.: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46, 131–148 (1972)

    MATH  Google Scholar 

  17. Liu I.-S., Müller I.: Thermodynamics of mixtures of fluids. In: Truesdell, C. (ed.) Rational Thermodynamics, pp. 264–285. Springer, New York (1984)

    Chapter  Google Scholar 

  18. Luca I., Fang C., Hutter K.: A thermodynamic model of turbulent motions in a granular material. Continuum Mech. Thermodyn. 16, 363–390 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  19. Müller I.: On the entropy inequality. Arch. Ration. Mech. Anal. 26, 118–141 (1967)

    Article  MATH  Google Scholar 

  20. Müller I.: Die Kältefunktion, eine universelle Funktion in der Thermodynamik viskoser wärmeleitender Flüssigkeiten. Arch. Ration. Mech. Anal. 40, 1–36 (1971)

    Article  MATH  Google Scholar 

  21. Müller I.: Thermodynamik—Grundlagen der Materialtheorie. Bertelsman Universitätsverlag, Düsseldorf (1972)

    Google Scholar 

  22. Müller I.: Thermodynamics. Pitman, London (1985)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Spurk J.H.: Fluid Mechanics. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  25. Svendsen B., Hutter K.: On the thermodynamics of a mixture of isotropic materials with constraints. Int. J. Eng. Sci. 33, 2021–2054 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  ADS  Google Scholar 

  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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  32. Wang Y., Oberlack M.: A thermodynamic model of multiphase flows with moving interfaces and contact line. Continuum Mech. Thermodyn. 23, 409–433 (2011)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  33. Wilmanski, K.: Continuum Thermodynamics, Part 1: Foundations. World Scientific Pub Co. (2009)

  34. Wood L.C.: The bogus axioms of continuum mechanics. Bull. Math. Appl. 17, 98–102 (1981)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chair of Fluid Dynamics, Department of Mechanical Engineering, Technische Universität Darmstadt, Petersenstrasse 30, 64287, Darmstadt, Germany

    Yongqi Wang, Martin Oberlack & Andreas Zieleniewicz

  2. Institute of Geotechnical Engineering, University of Natural Resources and Applied Life Sciences, Feistmantelstr. 4, 1180, Vienna, Austria

    Yongqi Wang

  3. Center of Smart Interfaces, Technische Universität Darmstadt, Petersenstrasse 32, 64287, Darmstadt, Germany

    Martin Oberlack

  4. Graduate School of Computational Engineering, Technische Universität Darmstadt, Dolivostrasse 15, 64293, Darmstadt, Germany

    Martin Oberlack

Authors
  1. Yongqi Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Oberlack
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreas Zieleniewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongqi Wang.

Additional information

Communicated by Andreas Öchsner.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Y., Oberlack, M. & Zieleniewicz, A. Constitutive modeling of multiphase flows with moving interfaces and contact line. Continuum Mech. Thermodyn. 25, 705–725 (2013). https://doi.org/10.1007/s00161-012-0269-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00161-012-0269-2

Keywords

Search

Navigation