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

Salgado, Abner J.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 743-769.

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.

[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. | MR | Zbl

[2] W. Bangerth, R. Hartmann and G. Kanschat, deal.II Differential Equations Analysis Library, Technical Reference. Available on http://www.dealii.org.

[3] W. Bangerth, R. Hartmann and G. Kanschat, deal.II - a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33 (2007). | MR

[4] T.D. Blake, The physics of moving wetting lines. J. Coll. Interf. Sci. 299 (2006) 1-13.

[5] T.D. Blake and Y.D. Shikhmurzaev, Dynamic wetting by liquids of different viscosity. J. Coll. Interf. Sci. 253 (2002) 196-202.

[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 Media 82 (2010) 463-483. | MR

[7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, NY (1991). | MR | Zbl

[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. | MR | Zbl

[9] Antonio Desimone, Natalie Grunewald and Felix Otto, A new model for contact angle hysteresis. Netw. Heterog. Media 2 (2007) 211-225. | MR | Zbl

[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. | MR

[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] A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences. Springer-Verlag, New York 159, 2004. | MR | Zbl

[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. | MR

[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. | MR

[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. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[18] J.-L. Guermond and L. Quartapelle, A projection FEM for variable density incompressible flows. J. Comput. Phys. 165 (2000) 167-188. | MR | Zbl

[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. | MR | Zbl

[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. | MR

[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] D. Jacqmin, Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1999) 96-127. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[25] J. Kim, K. Kang and J. Lowengrub, Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. 193 (2004) 511-543. | MR | Zbl

[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. | MR | Zbl

[27] S. Manservisi and R. Scardovelli, A variational approach to the contact angle dynamics of spreading droplets. Comput. & Fluids 38 (2009) 406-424. | MR | Zbl

[28] S. Minjeaud, An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model. Numer. Methods Partial Differ. Eqn. (2012). | MR

[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. | Zbl

[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. D 209 (2005) 191-204. | MR | Zbl

[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. | MR | Zbl

[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. | Zbl

[33] T. Qian, X.-P. Wang and P. Sheng, A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564 (2006) 333-360. | MR | Zbl

[34] W. Ren and W.E, Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2007) 022101. | Zbl

[35] J. Shen and X. Yang, Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows. Chin. Ann. Math. Ser. B 31 (2010) 743-758. | MR

[36] J. Shen and X. Yang, Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28 (2010) 1669-1691. | MR | Zbl

[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. Comput 32 (2010) 1159-1179. | MR

[38] Y.D. Shikhmurzaev, Capillary flows with forming interfaces. Chapman & Hall/CRC, Boca Raton, FL (2008). | MR | Zbl

[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] 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. | MR | Zbl

[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. | Numdam | MR

[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. | MR | Zbl

[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). | MR | Zbl