Journal of Elasticity

, Volume 104, Issue 1–2, pp 115–131 | Cite as

On the Thermodynamics of Korteweg Fluids with Heat Conduction and Viscosity

  • V. A. Cimmelli
  • F. Oliveri
  • A. R. Pace


A model of third-grade Korteweg fluid with heat conduction and viscosity is developed. The restrictions placed by the Dissipation Principle are investigated by applying two different methods which generalize the classical Coleman-Noll and Liu procedures. Compatibility with thermodynamics is achieved for arbitrary form of the energy and entropy fluxes. In the one-dimensional case a particular solution of the system of thermodynamic restrictions is provided.


Weakly nonlocal thermodynamics Third-grade Korteweg type fluids Extended Liu procedure Generalized Coleman–Noll procedure Interstitial work flux Nonlocal entropy 

Mathematics Subject Classification (2000)

74A15 74A20 76A05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Korteweg, D.J.: Sur la forme qui prennent les équations du mouvement des fluids si l’on tient compte des forces capillaires par des variations de densité. Arch. Neerl. Sci. Exactes Nat., Ser. II 6, 1–24 (1901) MATHGoogle Scholar
  2. 2.
    Sansone, E.: Deduzione della teoria dei fluidi maxwelliani dalla termodinamica dei sistemi continui. Rend. Semin. Mat. Univ. Padova 5, 39–52 (1977) MathSciNetGoogle Scholar
  3. 3.
    Truesdell, C., Noll, W.: The nonlinear field theories of mechanics. In: Encyclopedia of Physics, vol. III/3. Springer, Berlin (1965) Google Scholar
  4. 4.
    Dunn, J.E., Serrin, J.: On the thermomechanics of the interstitial working. Arch. Ration. Mech. Anal. 88, 95–133 (1985) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Gurtin, M.E., Vianello, M., Williams, W.O.: On fluids of grade n. Meccanica 21, 179–183 (1986) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Toupin, R.A.: Elastic materials with couple stress. Arch. Ration. Mech. Anal. 11, 385–414 (1962) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Toupin, R.A.: Theories of elasticity with couple stress. Arch. Ration. Mech. Anal. 17, 85–112 (1964) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Aifantis, E.C., Serrin, J.: The mechanical theory of fluid interfaces and Maxwell’s rule. J. Colloid Interface Sci. 96, 517–529 (1983) CrossRefGoogle Scholar
  9. 9.
    Aifantis, E.C., Serrin, J.: Equilibrium solutions in the mechanical theory of fluid microstructures. J. Colloid Interface Sci. 96, 530–547 (1983) CrossRefGoogle Scholar
  10. 10.
    Slemrod, M.: Admissibility criteria for propagating phase boundaries in a van der Waals fluid. Arch. Ration. Mech. Anal. 81, 301–315 (1983) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Slemrod, M.: Dynamic phase transitions in a van der Waals fluid. J. Differential Equations 52, 1–23 (1984) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Fried, E., Gurtin, M.E.: Continuum theory of thermally induced phase transitions based on an order parameter. Physica D 68, 326–343 (1993) MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    Fried, E., Gurtin, M.E.: Dynamic solid-solid transitions with phase characterized by an order parameter. Physica D 72, 287–308 (1994) MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Gurtin, M.E.: Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D 92, 178–192 (1996) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Fabrizio, M., Giorgi, C., Morro, A.: A thermodynamic approach to nonisothermal phase-field evolution in continuum physics. Physica D 214, 144–156 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  16. 16.
    Morro, A.: Non-isothermal phase-field models and evolution equation. Arch. Mech. 58, 207–221 (2006) MathSciNetGoogle Scholar
  17. 17.
    Morro, A.: Unified approach to evolution equations for non-isothermal phase transitions. Appl. Math. Sci. 1, 339–353 (2007) MathSciNetMATHGoogle Scholar
  18. 18.
    Frémond, M.: Non-smooth Thermomechanics. Springer, Berlin (2001) Google Scholar
  19. 19.
    Aifantis, E.C.: Pattern formation in plasticity. Int. J. Eng. Sci. 33, 2161–2178 (1995) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Aifantis, E.C.: Gradient deformation models at nano, micro, and macroscales. J. Eng. Mater. Technol. 121, 189–202 (1999) CrossRefGoogle Scholar
  21. 21.
    Aifantis, E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 229–314 (1999) CrossRefGoogle Scholar
  22. 22.
    Askes, H., Aifantis, E.C.: Gradient elasticity and flexural wave dispersion in carbon nanotubes. Phys. Rev. B 80, 195412 (2009) (8 pages) ADSCrossRefGoogle Scholar
  23. 23.
    Dunn, J.E.: Interstitial working and a nonclassical continuum thermodynamics. In: Serrin, J. (ed.) New Perspectives in Thermodynamics, pp. 187–222. Springer, Berlin (1986) CrossRefGoogle Scholar
  24. 24.
    Müller, I.: On the entropy inequality. Arch. Ration. Mech. Anal. 26, 118–141 (1967) MATHCrossRefGoogle Scholar
  25. 25.
    Coleman, B.D., Noll, W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Ration. Mech. Anal. 13, 167–178 (1963) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Jou, D., Casas-Vázquez, J., Lebon, G.: Extended Irreversible Thermodynamics, 4th edn. Springer, Berlin (2010) MATHCrossRefGoogle Scholar
  27. 27.
    Lebon, G., Jou, D., Casas-Vázquez, J., Muschik, W.: Weakly nonlocal and nonlinear heat transport in rigid solids. J. Non-Equilib. Thermodyn. 23, 176–191 (1998) ADSMATHCrossRefGoogle Scholar
  28. 28.
    Coleman, B.D., Mizel, V.J.: Thermodynamics and departures from Fourier’s law of heat conduction. Arch. Ration. Mech. Anal. 13, 245–260 (1963) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Liu, I-Shih: Method of Lagrange multipliers for exploitation of the entropy principle. Arch. Ration. Mech. Anal. 46, 131–148 (1972) MATHGoogle Scholar
  30. 30.
    Müller, I., Ruggeri, T.: Rational Extended Thermodynamics, 2nd edn. Springer, New York (1998) MATHCrossRefGoogle Scholar
  31. 31.
    Coleman, B.D., Fabrizio, M., Owen, D.R.: The thermodynamics of second sound in crystals. Arch. Ration. Mech. Anal. 80, 135–158 (1982) MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Cimmelli, V.A., Sellitto, A., Triani, V.: A generalized Coleman-Noll procedure for the exploitation of the entropy principle. Proc. R. Soc. A 466, 911–925 (2010) MathSciNetADSMATHCrossRefGoogle Scholar
  33. 33.
    Cimmelli, V.A., Sellitto, A., Triani, V.: A new perspective on the form of first and second laws in rational thermodynamics: Korteweg fluids as an example. J. Non-Equilib. Thermodyn. 35, 251–265 (2010) ADSMATHCrossRefGoogle Scholar
  34. 34.
    Cimmelli, V.A., Sellitto, A., Triani, V.: A new thermodynamic framework for second-grade Korteweg-type viscous fluids. J. Math. Phys. 50, 053101 (2009) (16 pages) MathSciNetADSCrossRefGoogle Scholar
  35. 35.
    Cimmelli, V.A.: An extension of Liu procedure in weakly nonlocal thermodynamics. J. Math. Phys. 48, 113510 (2007) (13 pages) MathSciNetADSCrossRefGoogle Scholar
  36. 36.
    Muschik, W., Papenfuss, C., Triani, V.: Exploitation of the entropy inequality, if some balances are missing. J. Mech. Mater. Struct. 3, 1125–1133 (2008) CrossRefGoogle Scholar
  37. 37.
    Cimmelli, V.A., Oliveri, F., Triani, V.: Exploitation of the entropy principle: proof of Liu Theorem if the gradients of the governing equations are considered as constraints. J. Math. Phys. 52 (2011, in press) Google Scholar
  38. 38.
    Truesdell, C.: A First Course in Rational Continuum Mechanics, 2nd edn. Academic Press, San Diego (1991) MATHGoogle Scholar
  39. 39.
    Triani, V., Papenfuss, C., Cimmelli, V.A., Muschik, W.: Exploitation of the second law: Coleman–Noll and Liu procedure in comparison. J. Non-Equilib. Thermodyn. 33, 47–60 (2008) ADSMATHCrossRefGoogle Scholar
  40. 40.
    Antanovskii, L.K.: A phase-field model of capillarity. Phys. Fluids 7, 747–753 (1995) MathSciNetADSMATHCrossRefGoogle Scholar
  41. 41.
    Anderson, D.M., McFadden, G.B., Wheeler, A.A.: Diffuse–interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139–165 (1998) MathSciNetADSCrossRefGoogle Scholar
  42. 42.
    Verhás, J.: Thermodynamics and Rheology. Kluwer Academic, Dordrecht (1997) MATHGoogle Scholar
  43. 43.
    Gurtin, M.E.: An Introduction to Continuum Mechanics. Academic Press, New York (1981) MATHGoogle Scholar
  44. 44.
    Speziale, C.G.: A review of material frame-indifference in mechanics. Appl. Mech. Rev. 51, 489–504 (1998) ADSCrossRefGoogle Scholar
  45. 45.
    Svendsen, B., Bertram, A.: On frame-indifference and form-invariance in constitutive theory. Acta Mech. 132, 195–207 (1999) MathSciNetCrossRefGoogle Scholar
  46. 46.
    Muschik, W., Restuccia, L.: Systematic remarks on objectivity and frame-indifference, liquid crystal theory as an example. Arch. Appl. Mech. 78, 837–854 (2008) ADSMATHCrossRefGoogle Scholar
  47. 47.
    Murdoch, A.I.: Objectivity in classical continuum physics: a rationale for discarding the ‘principle of invariance under superposed rigid body motions’ in favour of purely objective considerations. Contin. Mech. Thermodyn. 15, 309–320 (2003) MathSciNetADSMATHCrossRefGoogle Scholar
  48. 48.
    Murdoch, A.I.: On criticism of the nature of objectivity in classical continuum physics. Contin. Mech. Thermodyn. 17, 135–148 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  49. 49.
    Liu, I-Shih: On Euclidean objectivity and the principle of material frame-indifference. Contin. Mech. Thermodyn. 16, 177–183 (2004) MathSciNetADSMATHCrossRefGoogle Scholar
  50. 50.
    Liu, I-Shih: Further remarks on Euclidean objectivity and the principle of material frame indifference. Contin. Mech. Thermodyn. 17, 125–133 (2005) MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of BasilicataPotenzaItaly
  2. 2.Department of MathematicsUniversity of MessinaMessinaItaly

Personalised recommendations