Advertisement

Journal of Mathematical Fluid Mechanics

, Volume 15, Issue 3, pp 453–480 | Cite as

Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

  • Helmut AbelsEmail author
  • Daniel Depner
  • Harald Garcke
Article

Abstract

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels et al. for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier–Stokes system with a modified convective term coupled to a Cahn–Hilliard system. The density of the mixture depends on an order parameter.

Mathematics Subject Classification (2010)

Primary 76T99 Secondary 35Q30 35Q35 76D03 76D05 76D27 76D45 

Keywords

Two-phase flow Navier–Stokes equation Diffuse interface model Mixtures of viscous fluids Cahn–Hilliard equation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abels, H.: Diffuse Interface Models for Two-Phase Flows of Viscous Incompressible Fluids. Habilitation Thesis, Leipzig (2007)Google Scholar
  2. 2.
    Abels H.: Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Comm. Math. Phys. 289, 45–73 (2009)MathSciNetADSzbMATHCrossRefGoogle Scholar
  3. 3.
    Abels H.: On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities. Arch. Rat. Mech. Anal. 194, 463–506 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Abels H.: Strong well-posedness of a diffuse interface model for a viscous, quasi-incompressible two-phase flow. SIAM J. Math. Anal. 44(1), 316–340 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent diffuse interface models for incompressible two-phase flows with different densities. preprint Nr. 20/2010 University Regensburg (2010)Google Scholar
  6. 6.
    Abels H., Garcke H., Grün G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Meth. Appl. Sci. 22(3), 1150013 (2011)CrossRefGoogle Scholar
  7. 7.
    Abels H., Röger M.: Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6), 2403–2424 (2009)ADSzbMATHCrossRefGoogle Scholar
  8. 8.
    Abels H., Wilke M.: Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy. Nonlin. Anal. 67, 3176–3193 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Amann H.: Linear and Quasilinear Parabolic Problems, vol. 1: Abstract Linear Theory. Birkhäuser, Basel (1995)zbMATHCrossRefGoogle Scholar
  10. 10.
    Anderson, D.-M., McFadden, G.B., Wheeler, A.A.: Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech., vol. 30, pp. 139–165. Annual Reviews, Paolo Alto (1998)Google Scholar
  11. 11.
    Bergh J., Löfström J.: Interpolation Spaces. Springer, Berlin (1976)zbMATHCrossRefGoogle Scholar
  12. 12.
    Boyer F.: Mathematical study of multi-phase flow under shear through order parameter formulation. Asymptot. Anal. 20(2), 175–212 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Boyer F.: Nonhomogeneous Cahn–Hilliard fluids. Ann. Inst. H. Poincar Anal. Non Linaire 18(2), 225–259 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  14. 14.
    Cahn J.W., Hilliard J.E.: Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys. 28(2), 258–267 (1958)ADSCrossRefGoogle Scholar
  15. 15.
    Diestel, J., Uhl, J.J., Jr.: Vector Measures. Am. Math. Soc., Providence (1977)Google Scholar
  16. 16.
    Ding H., Spelt P.D.M., Shu C.: Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comp. Phys. 22, 2078–2095 (2007)ADSCrossRefGoogle Scholar
  17. 17.
    Gurtin M.E., Polignone D., Viñals J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Meth. Appl. Sci. 6(6), 815–831 (1996)zbMATHCrossRefGoogle Scholar
  18. 18.
    Hohenberg P.C., Halperin B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479 (1977)ADSCrossRefGoogle Scholar
  19. 19.
    Lions J.-L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non linéaires. Dunod, Paris (1969)zbMATHGoogle Scholar
  20. 20.
    Lions P.-L.: Mathematical Topics in Fluid Mechanics, vol. 1, Incompressible Models. Clarendon Press, Oxford (1996)zbMATHGoogle Scholar
  21. 21.
    Liu C., Shen J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Phys. D 179(3–4), 211–228 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Lowengrub J., Truskinovsky L.: Quasi-incompressible Cahn–Hilliard fluids and topological transitions. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454, 2617–2654 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Roubíček T.: A generalization of the Lions-Temam compact embedding theorem. Časopis Pěst Mat. 115(4), 338–342 (1990)zbMATHGoogle Scholar
  24. 24.
    Showalter R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. AMS, Providence (1997)zbMATHGoogle Scholar
  25. 25.
    Simon J.: Compact sets in the space L p(0,T;B). Ann. Mat. Pura Appl. 146(4), 65–96 (1987)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Sohr, H.: The Navier–Stokes Equations. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Basel (2001)Google Scholar
  27. 27.
    Starovoĭtov, V.N.: On the motion of a two-component fluid in the presence of capillary forces. Mat. Zametki 62(2), 293–305 (1997) transl. in Math. Notes, 62(1–2), 244–254 (1997)Google Scholar
  28. 28.
    Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  29. 29.
    Triebel H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)Google Scholar
  30. 30.
    Zeidler E.: Nonlinear Functional Analysis and its Applications I. Springer, New York (1992)Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Universität RegensburgRegensburgGermany

Personalised recommendations