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On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework

Abstract

We provide a thermodynamic basis for the development of models that are usually referred to as “phase-field models” for compressible, incompressible, and quasi-incompressible fluids. Using the theory of mixtures as a starting point, we develop a framework within which we can derive “phase-field models” both for mixtures of two constituents and for mixtures of arbitrarily many fluids. In order to obtain the constitutive equations, we appeal to the requirement that among all admissible constitutive relations that which is appropriate maximizes the rate of entropy production (see Rajagopal and Srinivasa in Proc R Soc Lond A 460:631–651, 2004). The procedure has the advantage that the theory is based on prescribing the constitutive equations for only two scalars: the entropy and the entropy production. Unlike the assumption made in the case of the Navier–Stokes–Fourier fluids, we suppose that the entropy is not only a function of the internal energy and the density but also of gradients of the partial densities or the concentration gradients. The form for the rate of entropy production is the same as that for the Navier–Stokes–Fourier fluid. As observed earlier in Heida and Málek (Int J Eng Sci 48(11):1313–1324, 2010), it turns out that the dependence of the rate of entropy production on the thermodynamical fluxes is crucial. The resulting equations are of the Cahn–Hilliard–Navier–Stokes type and can be expressed both in terms of density gradients or concentration gradients. As particular cases, we will obtain the Cahn–Hilliard–Navier–Stokes system as well as the Korteweg equation. Compared to earlier approaches, our methodology has the advantage that it directly takes into account the rate of entropy production and can take into consideration any constitutive assumption for the internal energy (or entropy).

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References

  1. 1

    Atkin R.J., Craine R.E.: Continuum theories of mixtures: applications. J. Inst. Math. Appl. 17, 153–207 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Atkin R.J., Craine R.E.: Continuum theories of mixtures: basic theory and historical developments. Q. J. Mech. Appl. Math. 29, 209–244 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3

    Bowen R.M.: Theory of mixtures. In: Eringen, A.C. (ed.) Continuum Physics, vol. III, Academic Press, New York (1976)

    Google Scholar 

  4. 4

    Cahn J.W., Hilliard J.E.: Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28(2), 258–267 (1958)

    Article  Google Scholar 

  5. 5

    Callen H.: Thermodynamics and an Introduction to Thermostatics. Wiley, London (1985)

    Google Scholar 

  6. 6

    Darcy H.: Les Fontaines Publiques de La Ville de Dijon. Victor Dalmont, Paris (1856)

    Google Scholar 

  7. 7

    Fabrizio M., Giorgi C., Morro A.: A thermodynamic approach to non-isothermal phase-field evolution in continuum physics. Physica D: Nonlinear Phenomena 214(2), 144–156 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8

    Fick A.: Über diffusion. Ann. Phys. 94, 59–86 (1855)

    Google Scholar 

  9. 9

    Gibbs J.W.: Scientific Papers. Dover Publications, Dover (1961)

    Google Scholar 

  10. 10

    Goodman M.A., Cowin S.C.: A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44, 249–266 (1972)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Green A.E., Naghdi P.M.: On basic equations for mixtures. Q. J. Mech. Appl. Math. 22, 427–438 (1969)

    Article  MATH  Google Scholar 

  12. 12

    Hadamard J.: Lecons sur la propagation des ondes et les équations de l’hydrodynamique. Hermann, Paris (1903)

    MATH  Google Scholar 

  13. 13

    Heida M., Málek J.: On Korteweg-type compressible fluid-like materials. Int. J. Eng. Sci. 48(11), 1313–1324 (2010)

    Article  Google Scholar 

  14. 14

    Hills R.N., Loper D.E., Roberts P.H.: A thermodynamically consistent model of a mushy zone. Q. J. Mech. Appl. Math. 36, 505 (1983)

    Article  MATH  Google Scholar 

  15. 15

    Hills R.N., Roberts P.H.: On the formulation of diffusive mixture theories for two-phase regions. J. Eng. Math. 22(2), 93–106 (1988)

    MathSciNet  Article  Google Scholar 

  16. 16

    Hills R.N., Roberts P.H.: A macroscopic model of phase coarsening. Int. J. Non-Linear Mech. 25, 319 (1990)

    Article  MATH  Google Scholar 

  17. 17

    Hohenberg P.C., Halperin B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49(3), 435–479 (1977)

    Article  Google Scholar 

  18. 18

    Hutter K., Rajagopal K.R.: On flows of granular materials. Continuum Mech. Thermodyn. 6, 81–139 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19

    Kannan K., Rajagopal K.R.: A thermomechanical framework for the transition of a viscoelastic liquid to a viscoelastic solid. Math. Mech. Solids 9(1), 37–59 (2004) (Dedicated to Professor Ray W. Ogden)

    MathSciNet  MATH  Google Scholar 

  20. 20

    Korteweg D.J.: Sur la forme que prennent les équations du mouvement des fluides si l’on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l’hypothèse d’une variation continue de la densité. Archives Néerlandaises des sciences exactes et naturelles. Ser 2 6, 1–24 (1901)

    MATH  Google Scholar 

  21. 21

    Landau L.D.: Collected Papers. Gordon and Breach, New York (1967)

    Google Scholar 

  22. 22

    Lowengrub J., Truskinovsky L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 454(1978), 2617–2654 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23

    Málek J., Rajagopal K.: Compressible generalized Newtonian fluids. Z. Angew. Math. Phys. 61, 1097–1110 (2010). doi:10.1007/s00033-010-0061-8

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Málek J., Rajagopal K.R.: On the modeling of inhomogeneous incompressible fluid-like bodies. Mech. Mater. 38(3), 233–242 (2006)

    Article  Google Scholar 

  25. 25

    Morro A.: Phase-field models for fluid mixtures. Math. Comput. Model. 45(9–10), 1042–1052 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26

    Osher S., Sethian J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27

    Rajagopal K.R., Srinivasa A.R.: On the thermomechanics of materials that have multiple natural configurations. I. Viscoelasticity and classical plasticity. Z. Angew. Math. Phys. 55(5), 861–893 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28

    Rajagopal K.R., Srinivasa A.R.: On the thermomechanics of materials that have multiple natural configurations. II. Twinning and solid to solid phase transformation. Z. Angew. Math. Phys. 55(6), 1074–1093 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29

    Rajagopal K.R., Srinivasa A.R.: On thermomechanical restrictions of continua. Proc. R. Soc. Lond. A 460, 631–651 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30

    Rajagopal K.R., Srinivasa A.R.: On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys. 59(4), 715–729 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31

    Rajagopal K.R., Tao L.: Mechanics of mixtures, volume 35 of Series on Advances in Mathematics for Applied Sciences. World Scientific, River Edge, NJ (1995)

    Google Scholar 

  32. 32

    Rao I.J., Rajagopal K.R.: A thermodynamic framework for the study of crystallization in polymers. Z. Angew. Math. Phys. 53(3), 365–406 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33

    Rowlinson J.S., Widom B.: Molecular Theory of Capillarity. Dover Pubns, New York (2002)

    Google Scholar 

  34. 34

    Samohýl, I.: Thermodynamics of irreversible processes in fluid mixtures, volume 12 of Teubner-Texte zur Physik [Teubner Texts in Physics]. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1987. Approached by rational thermodynamics, With German, French, Spanish and Russian summaries

  35. 35

    Truesdell, C.: Rational Thermodynamics, vol. 53. Springer, Berlin, auflage: 2. korr. und erw. a. edition, (1985)

  36. 36

    Truesdell C.A.: Sulle basi della thermomeccanica. Rend. Lincei 22, 33–38 (1957)

    MathSciNet  MATH  Google Scholar 

  37. 37

    Truesdell C.A.: Sulle basi della thermomeccanica. Rend. Lincei 22, 158–166 (1957)

    MathSciNet  Google Scholar 

  38. 38

    Truesdell C.A.: Mechanical basis of diffusion. J. Chem. Phys. 37, 2336–2344 (1962)

    Article  Google Scholar 

  39. 39

    Van der Waals J.D.: Théorie thermodynamique de la capillarité, dans l’hypothèse d’une variation continue de la densité. Archives Néerlandaises des sciences exactes et naturelles XXVIII, 121–209 (1893)

    Google Scholar 

  40. 40

    Van der Waals J.D.: Thermodynamische Theorie der Kapillarität unter Voraussetzung stetiger Dichteänderung. Z. Phys. Chem. 13, 657–725 (1984)

    Google Scholar 

  41. 41

    Visintin, A.: Models of Phase Transitions. Progress in Nonlinear Differential Equations and their Applications. 28. Birkhäuser Boston Inc., Boston (1996)

  42. 42

    Young, T.: Cohesion. Miscellaneous Works of the Late Thomas Young, vol. 1, pp. 454–483 (1835)

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Correspondence to Josef Málek.

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This research has been partially performed during the stay of Martin Heida at the Charles University in Prague, the stay was supported by the Jindřich Nečas Center for Mathematical Modeling (the project LC06052 financed by MSMT). Josef Málek’s contribution is supported by GACR 201/09/0917. K. R. Rajagopal thanks the National Science Foundation for its support.

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Heida, M., Málek, J. & Rajagopal, K.R. On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework. Z. Angew. Math. Phys. 63, 145–169 (2012). https://doi.org/10.1007/s00033-011-0139-y

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Mathematics Subject Classification (2000)

  • Primary 76A02
  • 76T30
  • Secondary 00A71
  • 76A99