Skip to main content
Springer Nature Link
Log in

Decoupled Energy Stable Schemes for a Phase-Field Model of Two-Phase Incompressible Flows with Variable Density

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We consider in this paper numerical approximations of two-phase incompressible flows with different densities and viscosities. We present a variational derivation for a thermodynamically consistent phase-field model that admits an energy law. Two decoupled time discretization schemes for the coupled nonlinear phase-field model are constructed and shown to be energy stable. Numerical experiments are carried out to validate the model and the schemes for problems with large density and viscosity ratios.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. After we derived the model independently, we learned that the identical model was already derived in [2].

References

  1. Abels, H.: Existence of weak solutions for a diffuse interface model for viscous, incompressible fluids with general densities. Commun. Math. Phys. 289(1), 45–73 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  2. Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent diffuse interface models for incompressible two-phase flows with different densities. arXiv preprint arXiv:1011.0528 (2010)

  3. Abels, H., Garcke, H., Grün, G.: Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Models Methods Appl. Sci. 22, 1150013 (2012)

  4. Abraham, R., Marsden, J.E.: Foundations of Mechanics. Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, MA (1978). Second edition, revised and enlarged, With the assistance of Tudor Ratiu and Richard Cushman

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

  6. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989). Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition

  7. Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1997)

    Google Scholar 

  8. Becker, R., Feng, X., Prohl, A.: Finite element approximations of the Ericksen–Leslie model for nematic liquid crystal flow. SIAM J. Numer. Anal. 46(4), 1704–1731 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  9. Boyer, F. Nonhomogeneous Cahn–Hilliard fluids. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(2), 225–259 (2001)

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

    Article  Google Scholar 

  11. Chaikin, P.M., Lubensky, T.C.: Principles of Condensed Matter Physics. Cambridge University Press, Cambridge (1995)

  12. Chen, F., Shen, J.: Efficient spectral-Galerkin methods for systems of coupled second-order equations and their applications. J. Comput. Phys. 231(15), 5016–5028 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  13. Chen, W., Conde, S., Wang, C., Wang, X., Wise, S.: A linear energy stable scheme for a thin film model without slope selection. Preprint (2011)

  14. Ding, H., Spelt, P.D.M., Shu, C.: Calculation of two-phase Navier–Stokes flows using phase-field modeling. J. Comput. Phys. 226(2), 2078–2095 (2007)

    Article  MATH  Google Scholar 

  15. Weinan, E., Liu, J.-G.: Gauge method for viscous incompressible flows. Commun. Math. Sci. 1(2), 317–332 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Egorov, Y.V., Shubin, M.A.: Foundations of the Classical Theory of Partial Differential Equations. Springer, Berlin (1998). Translated from the 1988 Russian original by R. Cooke, Reprint of the original English edition from the series Encyclopaedia of Mathematical Sciences [ıt Partial differential equations. I, Encyclopaedia Math. Sci., 30, Springer, Berlin, 1992; MR1141630 (93a:35004b)]

  17. Eyre, D.J.: Unconditionally gradient stable time marching the Cahn–Hilliard equation. In: Computational and Mathematical Models of Microstructural Evolution (San Francisco, CA, 1998), Mater. Res. Soc. Sympos. Proc., Warrendale, PA, vol. 529, pp. 39–46, (1998)

  18. Feng, X., He, Y., Liu, C.: Analysis of finite element approximations of a phase field model for two-phase fluids. Math. Comput. 76(258), 539–571 (2007). (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fjordholm, U.S., Mishra, S., Tadmor, E.: Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230, 5587–5609 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition

  21. Guermond, J.L., Minev, P., Shen, J.: An overview of projection methods for incompressible flows. Comput. Methods Appl. Mech. Eng. 195, 6011–6045 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Guermond, J.-L., Quartapelle, L.: A projection FEM for variable density incompressible flows. J. Comput. Phys. 165(1), 167–188 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Guermond, J.-L., Salgado, A.: A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228(8), 2834–2846 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  24. Gurtin, M.E., Polignone, D., Vinals, J.: Two-phase binary fluids and immiscible fluids described by an order parameter. Math. Models Methods Appl. Sci. 6(6), 815–831 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hirsch, F., Lacombe, G.: Elements of functional analysis, volume 192 of Graduate Texts in Mathematics. Springer, New York (1999). Translated from the 1997 French original by Silvio Levy

  26. Jacqmin, D.: Diffuse interface model for incompressible two-phase flows with large density ratios. J. Comput. Phys. 155(1), 96–127 (2007)

    Article  MathSciNet  Google Scholar 

  27. Kim, J.: Phase-field models for multi-component fluid flows. Commun. Comput. Phys. 12(3), 613–661 (2012)

    MathSciNet  Google Scholar 

  28. Kubo, R.: The fluctuation-dissipation theorem. Reports on Progress in Physics (1966)

  29. Lin, P., Liu, C., Zhang, H.: An energy law preserving \(C^0\) finite element scheme for simulating the kinematic effects in liquid crystal dynamics. J. Comput. Phys. 227(2), 1411–1427 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  30. Liu, C., Shen, J.: A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D 179(3–4), 211–228 (2003)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Nochetto, R., Pyo, J.-H.: The gauge-Uzawa finite element method part i: the Navier–Stokes equations. SIAM J. Numer. Anal. 43, 1043–1068 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  33. Onsager, L.: Reciprocal relations in irreversible processes. I. Phys. Rev. 37, 405 (1931)

    Article  Google Scholar 

  34. Onsager, L.: Reciprocal relations in irreversible processes. II. Phys. Rev. 38, 2265 (1931)

    Article  MATH  Google Scholar 

  35. Prohl, A.: Projection and quasi-compressibility methods for solving the incompressible Navier–Stokes equations. In: Advances in Numerical Mathematics. B. G. Teubner, Stuttgart (1997)

  36. Pyo, J., Shen, J.: Gauge-Uzawa methods for incompressible flows with variable density. J. Comput. Phys. 221, 181–197 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  37. Rannacher, R.: On Chorin’s projection method for the incompressible Navier–Stokes equations. In: Lecture Notes in Mathematics, vol. 1530 (1991)

  38. Rayleigh, L.: Some general theorems relating to vibrations. Proc. Lond. Math. Soc. 4, 357–368 (1873)

    Google Scholar 

  39. Rayleigh, L.: On the theory of surface forces II. Philols. Mag. 33, 209 (1892)

    Article  MATH  Google Scholar 

  40. Shen, J.: Efficient spectral-Galerkin method I. Direct solvers for second- and fourth-order equations by using Legendre polynomials. SIAM J. Sci. Comput. 15, 1489–1505 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  41. Shen, J.: On error estimates of projection methods for the Navier–Stokes equations: second-order schemes. Math. Comput. 65, 1039–1065 (1996)

    Article  MATH  Google Scholar 

  42. Shen, J., Yang, X.: An efficient moving mesh spectral method for the phase-field model of two-phase flows. J. Comput. Phys. 228, 2978–2992 (2009)

  43. Shen, J., Yang, X.: Energy stable schemes for Cahn–Hilliard phase-field model of two-phase incompressible flows. Chin. Ann. Math. Ser. B 31, 743–758 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  44. Shen, J., Yang, X.: A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscositites. SIAM J. Sci. Comput. 32, 1159–1179 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  45. Shen, J., Yang, X.: Decoupled, energy stable schemes for phase-field models of two-phase incompressible flows. Submitted (2013)

  46. Shen, J. Modeling and numerical approximation of two-phase incompressible ows by a phase-eld approach. In: Bao, W., Du, Q. (eds.) Multiscale Modeling and Analysis for Materials Simulation, Lecture Note Series, vol. 9. IMS, National University of Singapore, pp. 147–196 (2011)

  47. Shen, J., Yang, X., Wang, Q.: On mass conservation in phase field models for binary fluids. Commun. Comput. Phys. 13, 1045–1065 (2013)

  48. Tadmor, E., Zhong, W.: Energy-preserving and stable approximations for the two-dimensional shallow water equations. “Mathematics and Computation—A Contemporary View”, Proceedings of the Third Abel Symposium, vol. 3. Springer, pp. 67–94 (2008)

  49. van der Waals, J.: The thermodynamic theory of capillarity under the hypothesis of a continuous density variation. J. Stat. Phys. 20, 197–244 (1893)

    Google Scholar 

  50. Wang, C., Liu, J.-G.: Convergence of gauge method for incompressible flow. Math. Comput. 69, 1385–1407 (2000)

    Article  MATH  Google Scholar 

  51. Wang, C., Wise, S.M.: An energy stable and convergent finite-difference scheme for the modified phase field crystal equation. SIAM J. Numer. Anal. 49, 945–969 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  52. Yamaleev, N.K., Carpenter, M.H.: A systematic methodology for constructing high-order energy stable weno schemes. J. Comput. Phys. 228(11), 4248–4272 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  53. Yang, X., Feng, J.J., Liu, C., Shen, J.: Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method. J. Comput. Phys. 218, 417–428 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  54. Yue, P., Feng, J.J., Liu, C., Shen, J.: A diffuse-interface method for simulating two-phase flows of complex fluids. J. Fluid Mech. 515, 293–317 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The research of C. Liu is partially supported by NSF DMS-1109107 and DMS-1216938. The research of J. Shen is partially supported in part by NSF DMS-1217066 and AFOSR FA9550-11-1-0328. The work of X. Yang is partially supported by SC EPSCOR GEAR program, AFOSR FA9550-12-1-0178 and NSF DMS-1200487.

Author information

Authors and Affiliations

  1. Department of Mathematics, Penn State University, University Park, PA , 16802, USA

    Chun Liu

  2. Department of Mathematics, Purdue University, West Lafayette, IN , 47907, USA

    Jie Shen

  3. Department of Mathematics, University of South Carolina, Columbia, SC , 29208, USA

    Xiaofeng Yang

Authors
  1. Chun Liu
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Jie Shen
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Xiaofeng Yang
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Jie Shen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, C., Shen, J. & Yang, X. Decoupled Energy Stable Schemes for a Phase-Field Model of Two-Phase Incompressible Flows with Variable Density. J Sci Comput 62, 601–622 (2015). https://doi.org/10.1007/s10915-014-9867-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9867-4

Keywords