Skip to main content
SpringerLink
Log in

Phase Field Approach to Multiphase Flow Modeling

  • Published:
Milan Journal of Mathematics Aims and scope Submit manuscript

Abstract

We review the phase field (otherwise called diffuse interface) model for fluid flows, where all quantities, such as density and composition, are assumed to vary continuously in space. This approach is the natural extension of van der Waals’ theory of critical phenomena both for one-component, two-phase fluids and for partially miscible liquid mixtures. The equations of motion are derived, assuming a simple expression for the pairwise interaction potential. In particular, we see that a non-equilibrium, reversible body force appears in the Navier-Stokes equation, that is proportional to the gradient of the generalized chemical potential. This, so called Korteweg, force is responsible for the convection that is observed in otherwise quiescent systems during phase change. In addition, in binary mixtures, the diffusive flux is modeled using a Cahn-Hilliard constitutive law with a composition-dependent diffusivity, showing that it reduces to Fick’s law in the dilute limit case. Finally, the results of several numerical simulations are described, modeling, in particular, a) mixing, b) spinodal decomposition, c) nucleation, d) enhanced heat transport, e) liquid-vapor phase separation.

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.

Institutional subscriptions

Similar content being viewed by others

References

  1. Anderson D., McFadden G., Wheeler A.: Diffuse-interface methods in fluid mechanics. Ann. Rev. Fluid Mech. 30, 139–165 (1998)

    Article  MathSciNet  Google Scholar 

  2. Antanovskii L.: Microscale theory of surface tension. Phys. Rev. E 54, 6285–6290 (1996)

    Article  Google Scholar 

  3. Cahn J.: On spinodal decomposition. Acta Metall. Mater 9, 795–801 (1961)

    Article  Google Scholar 

  4. Cahn J.: Critical point wetting. J. Chem. Phys. 66, 3667–3772 (1977)

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Cahn J., Hilliard J.: Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. J. Chem. Phys. 31, 688–699 (1959)

    Google Scholar 

  7. Califano F., Mauri R.: Drop size evolution during the phase separation of liquid mixtures. Ind. Eng. Chem. Res. 43, 349–353 (2004)

    Article  Google Scholar 

  8. Cussler E., Diffusion, Cambridge University Press, (1982)

  9. Davis H., Scriven L.: Stress and structure in fluid interfaces. Adv. Chem. Phys. 49, 357–454 (1982)

    Article  Google Scholar 

  10. de Groot S., Mazur P.: Non-Equilibrium Thermodynamics. Dover, New York (1984)

    Google Scholar 

  11. de Gennes P.G.: Dynamics of fluctuations and spinodal decomposition in polymer blends. J. Chem. Phys. 72, 4756–4763 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  12. Felderhof B.U.: Dynamics of the diffuse gas-liquid interface near the critical point. Physica 48, 541–560 (1970)

    Article  Google Scholar 

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

    Article  Google Scholar 

  14. Furukawa H.: Role of inertia in the late stage of the phase separation of a fluid. Physica A 204, 237–245 (1994)

    Article  Google Scholar 

  15. J. Gibbs, On the equilibrium of heterogeneous substances, 1876; reprinted in Scientific Papers by J. Willard Gibbs, Vol. 1, Dover, New York (1961).

  16. Gunton J.: Homogeneous nucleation. J. Stat. Phys. 95, 903–923 (1999)

    Article  MATH  Google Scholar 

  17. Gupta R., Mauri R., Shinnar R.: Liquid-liquid extraction using the composition induced phase separation process. Ind. Eng. Chem. Res. 35, 2360–2368 (1996)

    Article  Google Scholar 

  18. Gupta R., Mauri R., Shinnar R.: Phase separation of liquid mixtures in the presence of surfactants. Ind. Eng. Chem. Res. 38, 2418–2424 (1999)

    Article  Google Scholar 

  19. Herivel J.W.: The derivation of the equations of motion of an ideal fluid by hamilton’s principle. Proc. Cam. Phil. Soc., 51, 344–353 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hohenberg P.C., Halperin B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–480 (1977)

    Article  Google Scholar 

  21. J.H. Israelachvili, Intermolecular and Surface Forces, Academic Press, 1992.

  22. Jacqmin D.: Contact-line dynamics of a diffuse fluid interface. J. Fluid Mech. 402, 57–88 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jasnow D., Viñals J.: Coarse-grained description of thermo-capillary flow. Phys. Fluids 8, 660–669 (1996)

    Article  MATH  Google Scholar 

  24. Kawasaki K.: Kinetic equations and time correlation functions of critical fluctuations. Ann. Phys. 61, 1–56 (1970)

    Article  Google Scholar 

  25. D. Korteweg, Sur la forme que prennent les équations du mouvements 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’hypoth‘ese d’une variation continue de la densité, Archives Néerlandaises des Sciences Exactes et Naturelles. Series II, 6 (1901), 1–24.

  26. A. Lamorgese, and R.Mauri, Phase separation of liquid mixtures, in Nonlinear Dynamics and Control in Process Engineering: Recent Advances, G. Continillo, S. Crescitelli, and M. Giona, eds., Springer, 2002, pp. 139–152.

  27. Lamorgese A., Mauri R.: Nucleation and spinodal decomposition of liquid mixtures. Phys. Fluids 17, 034107 (2005)

    Article  Google Scholar 

  28. Lamorgese A., Mauri R.: Mixing of macroscopically quiescent liquid mixtures. Phys. Fluids 18, 044107 (2006)

    Article  Google Scholar 

  29. Lamorgese A., Mauri R.: Diffuse-interface modeling of phase segregation in liquid mixtures. Int. J. Multiphase Flow 34, 987–995 (2008)

    Article  Google Scholar 

  30. Lamorgese A., Mauri R.: Diffuse-interface modeling of liquid-vapor phase separation in a van der Waals fluid. Phys. Fluids 21, 044107 (2009)

    Article  Google Scholar 

  31. L. Landau, and E. Lifshitz, Statistical Physics, Part I, Pergamon Press, 1980.

  32. Lele S.K.: Compact finite-difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  33. G.W.F. Von Leibnitz, Nouveaux Essais sur l’Entendement Humain, book II, Ch. IV., 1765. Here Leibnitz applied to the natural world the statement ”Natura non facit saltus” (Nature does not jump) that in 1751 Lineus (i.e. C. von Linn) in Philosophia Botanica, Ch. 77, had referred to species evolution.

  34. Lowengrub J., Truskinovsky L.: Quasi-incompressible Cahn-Hilliard fluids and topological transitions. P. R. Soc. London A 454, 2617–2654 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  35. Titus Carus Lucretius, De Rerum Natura, Book I, 50 B.C.E. ”Corpus inani distinctum, quoniam nec plenum naviter extat nec porro vacuum.”

  36. Mauri R., Shinnar R., Triantafyllou G.: Spinodal decomposition in binary mixtures. Phys. Rev. E 53, 2613–2623 (1996)

    Article  Google Scholar 

  37. Maxwell J.C., Capillary action, 1876; reprinted in Scientific Paper of James Clerk Maxwell, Vol. 2, Dover, New York, 1952, pp. 541-591

  38. Molin D., Mauri R.: Enhanced heat transport during phase separation of liquid binary mixtures. Phys. Fluids 19, 074102 (2007)

    Article  Google Scholar 

  39. Molin D., Mauri R., Tricoli V.: Experimental evidence of the motion of a single out-of-equilibrium drop. Langmuir 23, 7459–7461 (2007)

    Article  Google Scholar 

  40. Nagarajan S., Lele S.K., Ferziger J.H.: A robust high-order compact method for large-eddy simulation. J. Comp. Phys. 191, 392–419 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nagarajan S., Lele S.K., Ferziger J.H.: Leading-edge effects in bypass transition. J. Fluid Mech. 572, 471–504 (2007)

    Article  MATH  Google Scholar 

  42. Nauman E., He D.: Nonlinear diffusion and phase separation. Chem. Eng. Sci. 56, 1999–2018 (2001)

    Article  Google Scholar 

  43. Pismen L.: Nonlocal diffuse interface theory of thin films and moving contact line. Phys. Rev. E 64, 021603 (2001)

    Article  Google Scholar 

  44. Pismen L., Pomeau Y.: Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics. Phys. Rev. E 62, 2480–2492 (2000)

    Article  MathSciNet  Google Scholar 

  45. Poesio P., Beretta G., Thorsen T.: Dissolution of a liquid microdroplet in a nonideal liquid-liquid mixture far from thermodynamic equilibrium. Phys. Rev. Lett. 103, 064501 (2009)

    Article  Google Scholar 

  46. Poesio P., Cominardi G., Lezzi A., Mauri R., Beretta G.: Effects of quenching rate and viscosity on spinodal decomposition. Phys. Rev. E 74, 011507 (2006)

    Article  Google Scholar 

  47. Poesio P., Lezzi A., Beretta G.: Evidence of convective heat transfer enhancement induced by spinodal decomposition. Phys. Rev. E 75, 066306 (2007)

    Article  Google Scholar 

  48. S.D. Poisson, Nouvelle Theorie de l’Action Capillaire, Bachelier, Paris, 1831.

  49. Lord Rayleigh, (1892) On the theory of surface forces. II. Compressible fluids. Philosophical Magazine 33: 209–220

  50. Rohde C.: On local and non-local avier-Stokes.Korteweg systems for liquid-vapour phase transition. ZAMM 85, 839–857 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  51. J. Rowlinson, and B. Widom, Molecular Theory of Capillarity, Oxford University Press, 1982.

  52. I.S. Sandler, Chemical and Engineering Thermodynamics, 3rd Ed., Wiley, 1999, Ch. 7.

  53. Santonicola G.M., Mauri R., Shinnar R.: Phase separation of initially nonhomogeneous liquid mixtures. Ind. Eng. Chem. Res. 40, 2004–2010 (2001)

    Article  Google Scholar 

  54. J. Serrin, Mathematical principles of classical fluid mechanics, in Encyclopedia of Physics, Vol. VIII-1, S. Flügge, ed., Springer, 1959, pp. 125–263.

  55. Siggia E.: Late stages of spinodal decomposition in binary mixtures. Phys. Rev. A 20, 595–605 (1979)

    Article  Google Scholar 

  56. Tanaka H.: Coarsening mechanisms of droplet spinodal decomposition in binary fluid mixtures. J. Chem. Phys. 105, 10099–10114 (1996)

    Article  Google Scholar 

  57. Tanaka H., Araki T.: Spontaneous double phase separation induced by rapid hydrodynamic coarsening in two-dimensional fluid mixtures. Phys. Rev. Lett. 81, 389–392 (1998)

    Article  Google Scholar 

  58. Thiele U., Madruga S., Frastia L.: Decomposition driven interface evolution for layers of binary mixtures: I. Model derivation and stratified base states. Phys. Fluids 19, 122106 (2007)

    Google Scholar 

  59. van der Waals J., (1979) The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, 1893; reprinted in Journal of Statistical Physics 20: 200–244

  60. Vladimirova N., Malagoli A., Mauri R.: Diffusion-driven phase separation of deeply quenched mixtures. Phys. Rev. E 58, 7691–7699 (1998)

    Article  Google Scholar 

  61. Vladimirova N., Malagoli A., Mauri R.: Diffusio-phoresis of two-dimensional liquid droplets in a phase separating system. Phys. Rev. E 60, 2037–2044 (1999a)

    Article  Google Scholar 

  62. Vladimirova N., Malagoli A., Mauri R.: Two-dimensional model of phase segregation in liquid binary mixtures. Phys. Rev. E 60, 6968–6977 (1999b)

    Article  Google Scholar 

  63. Vladimirova N., Malagoli A., Mauri R.: Two-dimensional model of phase segregation in liquid binary mixtures with an initial concentration gradient. Chem. Eng. Sci. 55, 6109–6118 (2000)

    Article  Google Scholar 

  64. Vladimirova N., Mauri R.: Mixing of viscous liquid mixtures. Chem. Eng. Sci. 59, 2065–2069 (2004)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, The City College of CUNY, 10031, New York, U.S.A.

    Andrea G. Lamorgese

  2. Department of Mechanical Engineering, University of Brescia, Via Branze 38, 25123, Brescia, Italy

    Dafne Molin

  3. Department of Chemical Engineering, Industrial Chemistry and Material Science, University of Pisa, Via Diotisalvi 2, 56126, Pisa, Italy

    Roberto Mauri

Authors
  1. Andrea G. Lamorgese
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dafne Molin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Roberto Mauri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roberto Mauri.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lamorgese, A.G., Molin, D. & Mauri, R. Phase Field Approach to Multiphase Flow Modeling. Milan J. Math. 79, 597–642 (2011). https://doi.org/10.1007/s00032-011-0171-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00032-011-0171-6

Mathematics Subject Classification (2010)

Keywords

Search

Navigation