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Multi-gradient fluids

  • Henri Gouin
Article
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Abstract

An internal energy function of the mass density, the volumetric entropy and their gradients at n-order generates the representation of multi-gradient fluids. Thanks to Hamilton’s principle, we obtain a thermodynamical form of the equation of motion which generalizes the case of perfect compressible fluids. First integrals of flows are extended cases of perfect compressible fluids. The equation of motion and the equation of energy are written for dissipative cases, and are compatible with the second law of thermodynamics.

Keywords

Multi-gradient fluids Equation of motion Equation of energy First integrals Laws of thermodynamics 

Mathematics Subject Classification

76A02 76E30 76M30 

Notes

Acknowledgements

The results contained in the present paper have been partially presented in Wascom 2017.

References

  1. 1.
    Van der Waals, J.D.: The thermodynamic theory of capillarity under the hypothesis of continuous variation of density. Translation by J.S. Rowlinson. J. Stat. Phys. 20, 200–244 (1979)CrossRefGoogle Scholar
  2. 2.
    Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system, III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31, 688–699 (1959)CrossRefGoogle Scholar
  3. 3.
    Gouin, H., Gavrilyuk, S.: Wetting problem for multi-component fluid mixtures. Physica A 268, 291–308 (1999). arXiv:0803.0275 CrossRefGoogle Scholar
  4. 4.
    Gărăjeu, M., Gouin, H., Saccomandi, G.: Scaling Navier–Stokes equation in nanotubes. Phys. Fluids. 25, 082003 (2013). arXiv:1311.2484 CrossRefGoogle Scholar
  5. 5.
    Gouin, H.: Liquid nanofilms. A mechanical model for the disjoining pressure. Int. J. Eng. Sci. 47, 691–699 (2009). arXiv:0904.1809 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gouin, H.: The watering of tall trees—embolization and recovery. J. Theor. Biol. 369, 42–50 (2015). arXiv:1404.4343 CrossRefGoogle Scholar
  7. 7.
    Germain, P.: The method of the virtual power in continuum mechanics—part 2: microstructure. SIAM J. Appl. Math. 25, 556–575 (1973)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gouin, H., Ruggeri, T.: Mixtures of fluids involving entropy gradients and acceleration waves in interfacial layers. Eur. J. Mech. B Fluids 24, 596–613 (2005). arXiv:0801.2096 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eremeyev, V.A., Fischer, F.D.: On the phase transitions in deformable solids. Z. Angew. Math. Mech. 90, 535–536 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Rowlinson, J.S., Widom, B.: Molecular Theory of Capillarity. Clarendon Press, Oxford and Google books (2012)Google Scholar
  11. 11.
    Gouin, H., Seppecher, P.: Temperature profile in a liquid-vapour interface near the critical point. Proc. R. Soc. A 473, 20170229 (2017). arXiv:1703.07302
  12. 12.
    Casal, P., Gouin, H.: Equation of motion of thermocapillary fluids. C. R. Acad. Sci. Ser. II Mec. Phys. Chim. Sci. Terre Univ. 306, 99–104 (1988)zbMATHGoogle Scholar
  13. 13.
    Maitournam, M.H.: Entropy and temperature gradients thermomechanics: dissipation, heat conduction inequality and heat equation. C. R. Mec. 340, 434–443 (2012)CrossRefGoogle Scholar
  14. 14.
    Bertram, A., Forest, S.: The thermodynamics of gradient elastoplasticity. Contin. Mech. Thermodyn. 26, 269–286 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Serrin, J.: Mathematical principles of classical fluid mechanics. In: Flügge, S. (ed.) Encyclopedia of Physics VIII/1, pp. 125–263. Springer, Berlin (1960)Google Scholar
  16. 16.
    Gouin, H.: Noether’s theorem in fluid mechanics. Mech. Res. Commun. 3, 151–156 (1976)CrossRefzbMATHGoogle Scholar
  17. 17.
    Casal, P., Gouin, H.: Connexion between the energy equation and the motion equation in Korteweg’s theory of capillarity. C. R. Acad. Sci. Ser. II Mec. Phys. Chim. Sci. Terre Univ. 300, 231–234 (1985)zbMATHGoogle Scholar
  18. 18.
    Truesdell, C.: Introduction à la mécanique Rationnelle des Milieux Continus. Masson, Paris (1974)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Aix-Marseille Univ, CNRS, IUSTI UMR 7343MarseilleFrance

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