A diffuse interface fractional time-stepping technique for incompressible two-phase flows with moving contact lines

Abner J. Salgado

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 3, page 743-769
  • ISSN: 0764-583X

Abstract

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For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.

How to cite

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Salgado, Abner J.. "A diffuse interface fractional time-stepping technique for incompressible two-phase flows with moving contact lines." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.3 (2013): 743-769. <http://eudml.org/doc/273294>.

@article{Salgado2013,
abstract = {For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.},
author = {Salgado, Abner J.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {Navier Stokes; Cahn Hilliard; multiphase flow; contact line; fractional time-stepping; Navier-Stokes; Cahn-Hilliard},
language = {eng},
number = {3},
pages = {743-769},
publisher = {EDP-Sciences},
title = {A diffuse interface fractional time-stepping technique for incompressible two-phase flows with moving contact lines},
url = {http://eudml.org/doc/273294},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Salgado, Abner J.
TI - A diffuse interface fractional time-stepping technique for incompressible two-phase flows with moving contact lines
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 3
SP - 743
EP - 769
AB - For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present several numerical illustrations of this scheme.
LA - eng
KW - Navier Stokes; Cahn Hilliard; multiphase flow; contact line; fractional time-stepping; Navier-Stokes; Cahn-Hilliard
UR - http://eudml.org/doc/273294
ER -

References

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  1. [1] H. Abels, H. Garcke and G. Grün, Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Mod. Methods Appl. Sci. 22 (2012) 1150013. Zbl1242.76342MR2890451
  2. [2] W. Bangerth, R. Hartmann and G. Kanschat, deal.II Differential Equations Analysis Library, Technical Reference. Available on http://www.dealii.org. 
  3. [3] W. Bangerth, R. Hartmann and G. Kanschat, deal.II — a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33 (2007). MR2404402
  4. [4] T.D. Blake, The physics of moving wetting lines. J. Coll. Interf. Sci.299 (2006) 1–13. 
  5. [5] T.D. Blake and Y.D. Shikhmurzaev, Dynamic wetting by liquids of different viscosity. J. Coll. Interf. Sci.253 (2002) 196–202. 
  6. [6] F. Boyer, C. Lapuerta, S. Minjeaud, B. Piar and M. Quintard, Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows. Transp. Porous Media82 (2010) 463–483. MR2646853
  7. [7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, NY (1991). Zbl0788.73002MR1115205
  8. [8] L.A. Caffarelli and N.E. Muler, An L∞ bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal.133 (1995) 129–144. Zbl0851.35010MR1367359
  9. [9] Antonio DeSimone, Natalie Grunewald and Felix Otto, A new model for contact angle hysteresis. Netw. Heterog. Media2 (2007) 211–225. Zbl1125.76011MR2291819
  10. [10] J.-B. Dupont and D. Legendre, Numerical simulation of static and sliding drop with contact angle hysteresis. J. Comput. Phys.229 (2010) 2453–2478. Zbl05680394MR2586196
  11. [11] J. Eggers and R. Evans, Comment on “dynamic wetting by liquids of different viscosity,” by t.d. blake and y.d. shikhmurzaev. J. Coll. Interf. Sci. 280 (2004) 537–538. 
  12. [12] A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences. Springer-Verlag, New York 159, 2004. Zbl1059.65103MR2050138
  13. [13] Xiaobing Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two-phase fluid flows. SIAM J. Numer. Anal.44 (2006) 1049–1072. Zbl05167765MR2231855
  14. [14] M. Gao and X.-P. Wang, A gradient stable scheme for a phase field model for the moving contact line problem. J. Comput. Phys.231 (2012) 1372–1386. Zbl06036056MR2876459
  15. [15] J.-F. Gerbeau and T. Lelièvre, Generalized Navier boundary condition and geometric conservation law for surface tension. Comput. Methods Appl. Mech. Engrg.198 (2009) 644–656. Zbl1229.76037MR2498521
  16. [16] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, Berlin, Germany (1986). Zbl0585.65077MR851383
  17. [17] J.-L. Guermond, P. Minev and J. Shen, Error analysis of pressure-correction schemes for the Navier-Stokes equations with open boundary conditions. SIAM J. Num. Anal.43 (2005) 239–258. Zbl1083.76044MR2177143
  18. [18] J.-L. Guermond and L. Quartapelle, A projection FEM for variable density incompressible flows. J. Comput. Phys.165 (2000) 167–188. Zbl0994.76051MR1795396
  19. [19] J.-L. Guermond and A. Salgado, A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys.228 (2009) 2834–2846. Zbl1159.76028MR2509298
  20. [20] Q. He, R. Glowinski and X.-P. Wang, A least-squares/finite element method for the numerical solution of the Navier-Stokes-Cahn-Hilliard system modeling the motion of the contact line. J. Comput. Phys.230 (2011) 4991–5009. Zbl05920272MR2795993
  21. [21] C. Huh and L.E. Scriven, Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Coll. Interf. Sci.35 (1971) 85–101. 
  22. [22] D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys.155 (1999) 96–127. Zbl0966.76060MR1716497
  23. [23] D. Kay, V. Styles and R. Welford, Finite element approximation of a Cahn-Hilliard-Navier-Stokes system. Interfaces Free Bound.10 (2008) 15–43. Zbl1144.35043MR2383535
  24. [24] D. Kay and R. Welford, Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in 2D. SIAM J. Sci. Comput.29 (2007) 2241–2257. Zbl1154.76033MR2357613
  25. [25] J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys.193 (2004) 511–543. Zbl1109.76348MR2030475
  26. [26] Z. Li, M.-C. Lai, G. He and H. Zhao, An augmented method for free boundary problems with moving contact lines. Comput. & Fluids 39 (2010) 1033–1040. Zbl1242.76047MR2644999
  27. [27] S. Manservisi and R. Scardovelli, A variational approach to the contact angle dynamics of spreading droplets. Comput. & Fluids 38 (2009) 406–424. Zbl1237.76145MR2645645
  28. [28] S. Minjeaud, An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model. Numer. Methods Partial Differ. Eqn. (2012). Zbl06143725MR3022900
  29. [29] R.H. Nochetto, A.J. Salgado and S.W. Walker, A diffuse interface model for electrowetting on dielectric with moving contact lines (2011). Submitted to M3AS. Zbl1280.35114
  30. [30] C.E. Norman and M.J. Miksis, Gas bubble with a moving contact line rising in an inclined channel at finite Reynolds number. Phys. D209 (2005) 191–204. Zbl1142.76371MR2167452
  31. [31] T. Qian, X.-P. Wang and P. Sheng, Generalized Navier boundary condition for the moving contact line. Commun. Math. Sci.1 (2003) 333–341. Zbl1160.76340MR1980479
  32. [32] T. Qian, X.-P. Wang and P. Sheng, Molecular hydrodynamics of the moving contact line in two-phase immiscible flows. Commun. Comput. Phys.1 (2006) 1–52. Zbl1115.76079
  33. [33] T. Qian, X.-P. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics. J. Fluid Mech.564 (2006) 333–360. Zbl1178.76296MR2261865
  34. [34] W. Ren and W.E, Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2007) 022101. Zbl1146.76513
  35. [35] J. Shen and X. Yang, Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows. Chin. Ann. Math. Ser. B31 (2010) 743–758. Zbl05816628MR2726065
  36. [36] J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst.28 (2010) 1669–1691. Zbl1201.65184MR2679727
  37. [37] J. Shen and X. Yang, A phase-field model and its numerical approximation for two-phase incompressible flows with different densities and viscosities. SIAM J. Sci. Comput32 (2010) 1159–1179. Zbl05880047MR2639233
  38. [38] Y.D. Shikhmurzaev, Capillary flows with forming interfaces. Chapman & Hall/CRC, Boca Raton, FL (2008). Zbl1165.76001MR2455379
  39. [39] Y.D. Shikhmurzaev and T.D. Blake, Response to the comment on [J. Colloid Interface Sci. 253 (2002) 196] by J. Eggers and R. Evans, J. Coll. Interf. Sci.280 (2004) 539–541. 
  40. [40] P.D.M. Spelt, A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. J. Comput. Phys.207 (2005) 389–404. Zbl1213.76127MR2144623
  41. [41] A. Turco, F. Alouges and A. DeSimone, Wetting on rough surfaces and contact angle hysteresis: numerical experiments based on a phase field model. ESAIM: M2AN 43 (2009) 1027–1044. Zbl05636845MR2588431
  42. [42] S.M. Wise, C. Wang and J.S. Lowengrub, An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal.47 (2009) 2269–2288. Zbl1201.35027MR2519603
  43. [43] E. Zeidler, Nonlinear functional analysis and its applications. I. Fixed-point theorems, Translated from the German by Peter R. Wadsack. Springer-Verlag, New York (1986). Zbl0583.47050MR816732

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