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Extensions of Classical Hydrodynamics

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Abstract

An abstract dynamical system is formulated from three features extracted from classical hydrodynamics. One its particular realization is then the classical hydrodynamics, other possible realizations are extensions of the classical hydrodynamics. The three features entering the formulation of the abstract dynamical system are the conservation laws, the compatibility with equilibrium thermodynamics, and the compatibility with classical mechanics in the limit of no dissipation. The particular extensions on which we illustrate the process of constructing different realizations are those arising when dealing with fluids in the vicinity of gas-liquid phase transitions (i.e. fluids involving large spatial inhomogeneities and large fluctuations).

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References

  1. Grmela, M.: Weakly nonlocal hydrodynamics. Phys. Rev. E 47, 351 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  2. Grmela, M.: Bracket formulation of the diffusion-convection equation. Phys. D 21, 179–212 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  3. Grmela, M.: Hamiltonian dynamics of complex fluids. J. Phys. A Math. Gen. 22, 342–348 (1989)

    Article  Google Scholar 

  4. Grmela, M., Ottinger, H.C.: Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E 56, 6620–6632 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  5. Ottinger, H.C., Grmela, M.: Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys. Rev. E 56, 6633–6655 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  6. Grmela, M.: Reciprocity relations in thermodynamics. Phys. A 309, 304–328 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Grmela, M.: Geometry of mesoscopic dynamics and thermodynamics. J. Non-Newton. Fluid Mech. 120, 137–147 (2004)

    Article  MATH  Google Scholar 

  8. Clebsch, A.: Ueber die Integration der hydrodynamischen Gleichungen. J. Reine Angew. Math. 56, 1–10 (1895)

    Google Scholar 

  9. Arnold, V.I.: Ann. Inst. Fourier 16, 319 (1966)

    Google Scholar 

  10. Grmela, M.: Coupling between microscopic and macroscopic dynamics in NEMD. Phys. Lett. A 174, 59 (1993)

    Article  ADS  Google Scholar 

  11. Grmela, M.: Complex fluids subjected to external influences. J. Non-Newton. Fluid Mech. 96, 221 (2001)

    Article  MATH  Google Scholar 

  12. Mueller, I., Ruggeri, T.: Rational Extended Thermodynamics. Springer, New York (1998)

    MATH  Google Scholar 

  13. Jou, D., Casas-Vazquez, J., Lebon, G.: Extended Irreversible Thermodynamics, 3rd edn. Springer, New York (2001)

    MATH  Google Scholar 

  14. Bird, R.B., Hassager, O., Armstrong, R.C., Curtiss, C.F.: Dynamics of Polymeric Fluids, vol. 2. Wiley, New York (1987)

    Google Scholar 

  15. Smoluchowski, M.V.: Ann. Physik 48, 1103 (1915)

    ADS  Google Scholar 

  16. Zmievski, V., Grmela, M., Bousmina, M.: Phys. A 376, 51–74 (2007)

    Article  Google Scholar 

  17. Landau, L.D., Lifschitz, E.M.: Fluid Mechanics. Pergamon, Oxford (1987)

    MATH  Google Scholar 

  18. van Kampen, N.G.: Stochastic Processes in Physics and Chemistry, 3rd edn. Elsevier, Amsterdam (2007)

    Google Scholar 

  19. Rubi, J.M., Mazur, P.: Nonequilibrium thermodynamics and hydrodynamic fluctuations. Phys. A 276, 477–488 (2000)

    Article  MathSciNet  Google Scholar 

  20. Anderson, D.M., McFadden, B., Wheeler, A.A.: Diffuse-interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30, 139 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  21. Bedeaux, D., Johannessen, E., Rosjorde, A.: The nonequilibrium van der Waals square gradient model. (I). The model and its numerical solution. Phys. A 330, 329–353 (2003)

    Article  MathSciNet  Google Scholar 

  22. Onuki, A.: Dynamic van der Waals theory. Phys. Rev. E 75, 036304 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  23. Espanol, P.: Thermohydrodynamics for the van der Waals fluid. J. Chem. Phys. 115, 5392 (2001)

    Article  ADS  Google Scholar 

  24. van Kampen, N.: Phys. Rev. 135, A362 (1964)

    Article  Google Scholar 

  25. Grmela, M., Teichmann, J.: Lagrangean formulation of the Maxwell-Cattaneo hydrodynamics. J. Eng. Sci. 21, 297–313 (1983)

    Article  MATH  Google Scholar 

  26. Sieniutycz, S., Berry, R.S.: Open Syst. Inf. Dyn. 4, 15 (1997)

    Article  MATH  Google Scholar 

  27. Callen, H.B.: Thermodynamics. Wiley, New York (1960)

    MATH  Google Scholar 

  28. Cahn, J.W., Hilliard, J.E.: J. Chem. Phys. 28, 258 (1958)

    Article  ADS  Google Scholar 

  29. Marsden, J.E., Weinstein, A.: Phys. D 7, 305 (1983)

    Article  MathSciNet  Google Scholar 

  30. Bilicki, Z., Badur, J.: J. Nonequilib. Thermodyn. 28, 311–340 (2003)

    Article  MATH  Google Scholar 

  31. Liu, L.S.: Method of Lagrange multipliers for exploiting the entropy principle. Arch. Rat. Mech. Anal. 46, 131–148 (1972)

    MATH  Google Scholar 

  32. Friedrichs, K.O., Lax, P.D.: Systems of conservation equations with a convex extension. Proc. Nat. Acad. Sci. USA 68, 1686–1688 (1971)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Grmela, M.: Kinetic approach to phase transitions. J. Stat. Phys. 3, 347–364 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  34. Karkheck, J., Stell, G.: Kinetic mean-field theories. J. Chem. Phys. 75, 1475 (1981)

    Article  ADS  Google Scholar 

  35. Kazmierczak, B., Piechor, K.: Phase boundary solutions to model kinetic equations via the Conley Index Theory Part. I. Math. Comput. Modeling 31, 77 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  36. Bastea, S., Esposito, R., Lebowitz, J.L., Marra, R.: Binary fluids with long range segregating interactions. I: Derivation of kinetic and hydrodynamic equations. J. Stat. Phys. 101, 1087 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  37. Frezzotti, A., Gibelli, L., Lorenzan, S.: Mean field kinetic theory description of evaporation of a fluid into vacuum. Phys. Fluids 17, 012102 (2005)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Ecole Polytechnique de Montreal, C.P.6079 suc. Centre-ville, Montreal, H3C 3A7, Quebec, Canada

    Miroslav Grmela

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  1. Miroslav Grmela
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Correspondence to Miroslav Grmela.

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Grmela, M. Extensions of Classical Hydrodynamics. J Stat Phys 132, 581–602 (2008). https://doi.org/10.1007/s10955-008-9558-3

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  • DOI: https://doi.org/10.1007/s10955-008-9558-3

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