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Diffuse Interface (D.I.) Model for Multiphase Flows

  • Andrea G. Lamorgese
  • Dafne Molin
  • Roberto Mauri
Chapter
  • 1.6k Downloads
Part of the CISM Courses and Lectures book series (CISM, volume 538)

Abstract

We review the 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 liquid binary mixtures. The equations of motion are derived, showing that the problem is well posed, as the rate of change of the total energy equals the energy dissipation. 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 convective motion observed in otherwise quiescent systems during phase change. Finally, the results of several numerical simulations are described, modeling, in particular, a) mixing, b) spinodal decomposition; c) nucleation; d) heat transfer; e) liquid-vapor phase separation.

Keywords

Free Energy Binary Mixture Multiphase Flow Peclet Number Spinodal Decomposition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. D.M. Anderson, G.B. McFadden, and A.A. Wheeler. Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics, 30:139–165, 1998.MathSciNetCrossRefGoogle Scholar
  2. L.K. Antanovskii. A phase field model of capillarity. Physics of Fluids, 7: 747–753, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  3. L.K. Antanovskii. Microscale theory of surface tension. Physical Review E, 54:6285–6290, 1996.CrossRefGoogle Scholar
  4. D. Beysens, Y. Garrabos, V. S. Nikolayev, C. Lecoutre-Chabot, J.-P. Delville, and J. Hegseth. Liquid-vapor phase separation in a thermocapillary force field. Europhysics Letters, 59(2):245–251, 2002.CrossRefGoogle Scholar
  5. R. Borcia and M. Bestehorn. Phase-field simulations for drops and bubbles. Physical Review E, 75:056309, 2007.CrossRefGoogle Scholar
  6. J.W. Cahn. On spinodal decomposition. Acta Metallurgica, 9:795–801, 1961.CrossRefGoogle Scholar
  7. J.W. Cahn. Critical point wetting. Journal of Chemical Physics, 66(8): 3667–3772, 1977.CrossRefGoogle Scholar
  8. J.W. Cahn and J.E. Hilliard. Free energy of a nonuniform system. I. Interfacial free energy. Journal of Chemical Physics, 28:258–267, 1958.CrossRefGoogle Scholar
  9. J.W. Cahn and J.E. Hilliard. Free energy of a nonuniform system. III. Nucleation in a two-component incompressible fluid. Journal of Chemical Physics, 31:688–699, 1959.CrossRefGoogle Scholar
  10. F. Califano and R. Mauri. Drop size evolution during the phase separation of liquid mixtures. Industrial & Engineering Chemistry Research, 43: 349–353, 2004.CrossRefGoogle Scholar
  11. E.L. Cussler. Diffusion. Cambridge University Press, 1982.Google Scholar
  12. H.T. Davis and L.E. Scriven. Stress and structure in fluid interfaces. Advances in Chemical Physics, 49:357–454, 1982.CrossRefGoogle Scholar
  13. P.G. de Gennes. Dynamics of fluctuations and spinodal decomposition in polymer blends. Journal of Chemical Physics, 72:4756–4763, 1980.MathSciNetzbMATHCrossRefGoogle Scholar
  14. S.R. de Groot and P. Mazur. Non-Equilibrium Thermodynamics. Dover, New York, 1984.Google Scholar
  15. B. U. Felderhof. Dynamics of the diffuse gas-liquid interface near the critical point. Physica, 48:541–560, 1970.CrossRefGoogle Scholar
  16. H. Furukawa. Role of inertia in the late stage of the phase separation of a fluid. Physica A, 204:237–245, 1994.CrossRefGoogle Scholar
  17. J.W. Gibbs. On the equilibrium of heterogeneous substances. Transactions of the Connecticut Academy of Arts and Sciences, 1876.Google Scholar
  18. J.D. Gunton. Homogeneous nucleation. Journal of Statistical Physics, 95: 903–923, 1999.zbMATHCrossRefGoogle Scholar
  19. R. Gupta, R. Mauri, and R. Shinnar. Liquid-liquid extraction using the composition induced phase separation process. Industrial & Engineering Chemistry Research, 35:2360–2368, 1996.CrossRefGoogle Scholar
  20. R. Gupta, R. Mauri, and R. Shinnar. Phase separation of liquid mixtures in the presence of surfactants. Industrial & Engineering Chemistry Research, 38:2418–2424, 1999.CrossRefGoogle Scholar
  21. P. C. Hohenberg and B. I. Halperin. Theory of dynamic critical phenomena. Reviews of Modern Physics, 49:435–479, 1977.CrossRefGoogle Scholar
  22. J.H. Israelachvili. Intermolecular and Surface Forces. Academic Press, 1992.Google Scholar
  23. D. Jacqmin. Contact-line dynamics of a diffuse fluid interface. Journal of Fluid Mechanics, 402:57, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  24. D. Jasnow and J. Viñals. Coarse-grained description of thermo-capillary flow. Physics of Fluids, 8:660–669, 1996.zbMATHCrossRefGoogle Scholar
  25. K. Kawasaki. Kinetic equations and time correlation functions of critical fluctuations. Annals of Physics, 61:1–56, 1970.CrossRefGoogle Scholar
  26. D.J. 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’hypothèse dúne variation continue de la densité. Archives Néerlandaises des Sciences Exactes et Naturelles. Series II, 6: 1–24, 1901.zbMATHGoogle Scholar
  27. A.G. Lamorgese and R. Mauri. Phase separation of liquid mixtures. In G. Continillo, S. Crescitelli, and M. Giona, editors, Nonlinear Dynamics and Control in Process Engineering: Recent Advances, pages 139–152. Springer, 2002.Google Scholar
  28. A.G. Lamorgese and R. Mauri. Nucleation and spinodal decomposition of liquid mixtures. Physics of Fluids, 17:034–107, 2005.Google Scholar
  29. A.G. Lamorgese and R. Mauri. Mixing of macroscopically quiescent liquid mixtures. Physics of Fluids, 18:044107, 2006.CrossRefGoogle Scholar
  30. A.G. Lamorgese and R. Mauri. Diffuse-interface modeling of phase segregation in liquid mixtures. International Journal of Multiphase Flow, 34: 987–995, 2008.CrossRefGoogle Scholar
  31. A.G. Lamorgese and R. Mauri. Diffuse-interface modeling of liquid-vapor phase separation in a van der Waals fluid. Physics of Fluids, 21:044107, 2009.CrossRefGoogle Scholar
  32. L.D. Landau and E.M. Lifshitz. Statistical Physics, Part I. Pergamon Press, 1980.Google Scholar
  33. M. Le Bellac. Quantum and Statistical Field Theory. Clarendon Press, 1991.Google Scholar
  34. S. K. Lele. Compact finite-difference schemes with spectral-like resolution. Journal of Computational Physics, 103:16–42, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  35. J. Lowengrub and L. Truskinovsky. Quasi-incompressible Cahn-Hilliard fluids and topological transitions. Proceedings of the Royal Society of London, Series A, 454:2617–2654, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  36. T.C. Lucretius. De Rerum Natura, Book I. 50 B.C.E. “Corpus inani distinctum, quoniam nec plenum naviter extat nec porro vacuum.” This is equivalent to one of the most basic principles of taoism, stating that nothing can be completely yin nor completely yang.Google Scholar
  37. S. Madruga and U. Thiele. Decomposition driven interface evolution for layers of binary mixtures: II. Influence of convective transport on linear stability. Physics of Fluids, 21:062104, 2009.CrossRefGoogle Scholar
  38. R. Mauri, R. Shinnar, and G. Triantafyllou. Spinodal decomposition in binary mixtures. Physical Review E, 53:2613–2623, 1996.CrossRefGoogle Scholar
  39. D. Molin and R. Mauri. Enhanced heat transport during phase separation of liquid binary mixtures. Physics of Fluids, 19:074102, 2007.CrossRefGoogle Scholar
  40. D. Molin, R. Mauri, and V. Tricoli. Experimental evidence of the motion of a single out-of-equilibrium drop. Langmuir, 23:7459–7461, 2007.CrossRefGoogle Scholar
  41. S. Nagarajan, S. K. Lele, and J. H. Ferziger. A robust high-order compact method for large-eddy simulation. Journal of Computational Physics, 191:392–419, 2003.MathSciNetzbMATHCrossRefGoogle Scholar
  42. S. Nagarajan, S. K. Lele, and J. H. Ferziger. Leading-edge effects in bypass transition. Journal of Fluid Mechanics, 572:471–504, 2007.zbMATHCrossRefGoogle Scholar
  43. E.B. Nauman and D.Q. He. Nonlinear diffusion and phase separation. Chemical Engineering Science, 56:1999–2018, 2001.CrossRefGoogle Scholar
  44. A. Onuki. Dynamic van der Waals theory. Physical Review E, 75:036304, 2007.MathSciNetCrossRefGoogle Scholar
  45. A. Oprisan, S. A. Oprisan, J. Hegseth, Y. Garrabos, C. Lecoutre-Chabot, and D. Beysens. Universality in early-stage growth of phase-separating domains near the critical point. Physical Review E, 77(5):051118, 2008.CrossRefGoogle Scholar
  46. L.M. Pismen. Nonlocal diffuse interface theory of thin films and moving contact line. Physical Review E, 64:021603, 2001.CrossRefGoogle Scholar
  47. L.M. Pismen and Y. Pomeau. Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics. Physical Review E, 62:2480–2492, 2000.MathSciNetCrossRefGoogle Scholar
  48. P. Poesio, G. Cominardi, A.M. Lezzi, R. Mauri, and G.P. Beretta. Effects of quenching rate and viscosity on spinodal decomposition. Physical Review E, 74:011507, 2006.CrossRefGoogle Scholar
  49. P. Poesio, A.M. Lezzi, and G.P. Beretta. Evidence of convective heat transfer enhancement induced by spinodal decomposition. Physical Review E, 75:066306, 2007.CrossRefGoogle Scholar
  50. P. Poesio, G.P. Beretta, and T. Thorsen. Dissolution of a liquid microdroplet in a nonideal liquid-liquid mixture far from thermodynamic equilibrium. Physical Review Letters, 103:064501, 2009.CrossRefGoogle Scholar
  51. S.D. Poisson. Nouvelle Theorie de l’Action Capillaire. Bachelier, 1831.Google Scholar
  52. Lord Rayleigh. On the theory of surface forces. II. Compressible fluids. Philosophical Magazine, 33:209–220, 1892.zbMATHCrossRefGoogle Scholar
  53. J.S. Rowlinson and B. Widom. Molecular Theory of Capillarity. Oxford University Press, 1982.Google Scholar
  54. I. S. Sandler. Chemical and Engineering Thermodynamics, 3rd Ed. Wiley, 1999. Ch. 7.Google Scholar
  55. G. M. Santonicola, R. Mauri, and R. Shinnar. Phase separation of initially non-homogeneous liquid mixtures. Industrial & Engineering Chemistry Research, 40:2004–2010, 2001.CrossRefGoogle Scholar
  56. E. Siggia. Late stages of spinodal decomposition in binary mixtures. Physical Review A, 20:595–605, 1979.CrossRefGoogle Scholar
  57. H. Tanaka. Coarsening mechanisms of droplet spinodal decomposition in binary fluid mixtures. Journal of Chemical Physics, 105:10099–10114, 1996.CrossRefGoogle Scholar
  58. H. Tanaka and T. Araki. Spontaneous double phase separation induced by rapid hydrodynamic coarsening in two-dimensional fluid mixtures. Physical Review Letters, 81:389–392, 1998.CrossRefGoogle Scholar
  59. U. Thiele, S. Madruga, and L. Frastia. Decomposition driven interface evolution for layers of binary mixtures: I. Model derivation and stratified base states. Physics of Fluids, 19:122106, 2007.CrossRefGoogle Scholar
  60. J.D. van der Waals. The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, 1893. Reprinted in Journal of Statistical Physics, 20:200–244 (1979).CrossRefGoogle Scholar
  61. N. Vladimirova and R. Mauri. Mixing of viscous liquid mixtures. Chemical Engineering Science, 59:2065–2069, 2004.CrossRefGoogle Scholar
  62. N. Vladimirova, A. Malagoli, and R. Mauri. Diffusion-driven phase separation of deeply quenched mixtures. Physical Review E, 58:7691–7699, 1998.CrossRefGoogle Scholar
  63. N. Vladimirova, A. Malagoli, and R. Mauri. Diffusio-phoresis of twodimensional liquid droplets in a phase separating system. Physical Review E, 60:2037–2044, 1999a.CrossRefGoogle Scholar
  64. N. Vladimirova, A. Malagoli, and R. Mauri. Two-dimensional model of phase segregation in liquid binary mixtures. Physical Review E, 60:6968–6977, 1999b.CrossRefGoogle Scholar
  65. N. Vladimirova, A. Malagoli, and R. Mauri. Two-dimensional model of phase segregation in liquid binary mixtures with an initial concentration gradient. Chemical Engineering Science, 55:6109–6118, 2000.CrossRefGoogle Scholar
  66. 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” that in 1751 Linnaeus (i.e. Carl von Linné) in Philosophia Botanica, Ch. 77, had referred to species evolution. B. Widom. Theory of phase equilibrium. Journal of Physical Chemistry, 100:13190–13199, 1996.CrossRefGoogle Scholar
  67. R. Yamamoto and K. Nakanishi. Computer simulation of vapor-liquid phase separation. Molecular Simulation, 16:119–126, 1996.CrossRefGoogle Scholar

Copyright information

© CISM, Udine 2012

Authors and Affiliations

  • Andrea G. Lamorgese
    • 1
  • Dafne Molin
    • 2
  • Roberto Mauri
    • 3
  1. 1.Department of Chemical EngineeringThe City College of CUNYNew YorkUSA
  2. 2.Department of Energy TechnologyUniversity of UdineUdineItaly
  3. 3.Department of Chemical Engineering, Industrial Chemistry and Material ScienceUniversity of PisaPisaItaly

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