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

Published online by Cambridge University Press:  29 March 2013

Abner J. Salgado
Abner J. Salgado*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, USA. abnersg@math.umd.edu
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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.

Keywords

Navier StokesCahn Hilliardmultiphase flowcontact linefractional time-stepping

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 743 - 769
Copyright
© EDP Sciences, SMAI, 2013

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References

Abels, H., Garcke, H. and Grün, G., Thermodynamically consistent, frame indifferent diffuse interface models for incompressible two-phase flows with different densities. Math. Mod. Methods Appl. Sci. 22 (2012) 1150013. Google Scholar
W. Bangerth, R. Hartmann and G. Kanschat, deal.II Differential Equations Analysis Library, Technical Reference. Available on http://www.dealii.org.
W. Bangerth, R. Hartmann and G. Kanschat, deal.II — a general-purpose object-oriented finite element library. ACM Trans. Math. Softw. 33 (2007).
Blake, T.D., The physics of moving wetting lines. J. Coll. Interf. Sci. 299 (2006) 1–13. Google ScholarPubMed
Blake, T.D. and Shikhmurzaev, Y.D., Dynamic wetting by liquids of different viscosity. J. Coll. Interf. Sci. 253 (2002) 196–202. Google ScholarPubMed
Boyer, F., Lapuerta, C., Minjeaud, S., Piar, B. and Quintard, M., Cahn-Hilliard/Navier-Stokes model for the simulation of three-phase flows. Transp. Porous Media 82 (2010) 463–483. Google Scholar
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York, NY (1991).
Caffarelli, L.A. and Muler, N.E., An L ∞ bound for solutions of the Cahn-Hilliard equation. Arch. Ration. Mech. Anal. 133 (1995) 129–144. Google Scholar
DeSimone, Antonio, Natalie Grunewald and Felix Otto, A new model for contact angle hysteresis. Netw. Heterog. Media 2 (2007) 211–225. Google Scholar
Dupont, J.-B. and Legendre, D., Numerical simulation of static and sliding drop with contact angle hysteresis. J. Comput. Phys. 229 (2010) 2453–2478. Google Scholar
Eggers, J. and Evans, R., 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. Google ScholarPubMed
A. Ern and J.-L. Guermond, Theory and practice of finite elements, Applied Mathematical Sciences. Springer-Verlag, New York 159, 2004.
Feng, Xiaobing, 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. Google Scholar
Gao, M. and Wang, X.-P., A gradient stable scheme for a phase field model for the moving contact line problem. J. Comput. Phys. 231 (2012) 1372–1386. Google Scholar
Gerbeau, J.-F. and Lelièvre, T., Generalized Navier boundary condition and geometric conservation law for surface tension. Comput. Methods Appl. Mech. Engrg. 198 (2009) 644–656. Google Scholar
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).
Guermond, J.-L., Minev, P. and Shen, J., Error analysis of pressure-correction schemes for the Navier-Stokes equations with open boundary conditions. SIAM J. Num. Anal. 43 (2005) 239–258. Google Scholar
Guermond, J.-L. and Quartapelle, L., A projection FEM for variable density incompressible flows. J. Comput. Phys. 165 (2000) 167–188. Google Scholar
Guermond, J.-L. and Salgado, A., A splitting method for incompressible flows with variable density based on a pressure Poisson equation. J. Comput. Phys. 228 (2009) 2834–2846. Google Scholar
He, Q., Glowinski, R. and Wang, X.-P., 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. Google Scholar
Huh, C. and Scriven, L.E., Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Coll. Interf. Sci. 35 (1971) 85–101. Google Scholar
Jacqmin, D., Calculation of two-phase Navier-Stokes flows using phase-field modeling. J. Comput. Phys. 155 (1999) 96–127. Google Scholar
Kay, D., Styles, V. and Welford, R., Finite element approximation of a Cahn-Hilliard-Navier-Stokes system. Interfaces Free Bound. 10 (2008) 15–43. Google Scholar
Kay, D. and Welford, R., Efficient numerical solution of Cahn-Hilliard-Navier-Stokes fluids in 2D. SIAM J. Sci. Comput. 29 (2007) 2241–2257. Google Scholar
Kim, J., Kang, K. and Lowengrub, J., Conservative multigrid methods for Cahn-Hilliard fluids. J. Comput. Phys. 193 (2004) 511–543. Google Scholar
Li, Z., Lai, M.-C., He, G. and Zhao, H., An augmented method for free boundary problems with moving contact lines. Comput. & Fluids 39 (2010) 1033–1040. Google Scholar
Manservisi, S. and Scardovelli, R., A variational approach to the contact angle dynamics of spreading droplets. Comput. & Fluids 38 (2009) 406–424. Google Scholar
S. Minjeaud, An unconditionally stable uncoupled scheme for a triphasic Cahn-Hilliard/Navier-Stokes model. Numer. Methods Partial Differ. Eqn. (2012).
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.
Norman, C.E. and Miksis, M.J., Gas bubble with a moving contact line rising in an inclined channel at finite Reynolds number. Phys. D 209 (2005) 191–204. Google Scholar
Qian, T., Wang, X.-P. and Sheng, P., Generalized Navier boundary condition for the moving contact line. Commun. Math. Sci. 1 (2003) 333–341. Google Scholar
Qian, T., Wang, X.-P. and Sheng, P., Molecular hydrodynamics of the moving contact line in two-phase immiscible flows. Commun. Comput. Phys. 1 (2006) 1–52. Google Scholar
Qian, T., Wang, X.-P. and Sheng, P., A variational approach to moving contact line hydrodynamics. J. Fluid Mech. 564 (2006) 333–360. Google Scholar
Ren, W. and W.E, Boundary conditions for the moving contact line problem. Phys. Fluids 19 (2007) 022101. Google Scholar
Shen, J. and Yang, X., Energy stable schemes for Cahn-Hilliard phase-field model of two-phase incompressible flows. Chin. Ann. Math. Ser. B 31 (2010) 743–758. Google Scholar
Shen, J. and Yang, X., Numerical approximations of Allen-Cahn and Cahn-Hilliard equations. Discrete Contin. Dyn. Syst. 28 (2010) 1669–1691. Google Scholar
Shen, J. and Yang, X., 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. Google Scholar
Y.D. Shikhmurzaev, Capillary flows with forming interfaces. Chapman & Hall/CRC, Boca Raton, FL (2008).
Shikhmurzaev, Y.D. and Blake, T.D., 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. Google Scholar
Spelt, P.D.M., A level-set approach for simulations of flows with multiple moving contact lines with hysteresis. J. Comput. Phys. 207 (2005) 389–404. Google Scholar
Turco, A., Alouges, F. and DeSimone, A., Wetting on rough surfaces and contact angle hysteresis: numerical experiments based on a phase field model. ESAIM: M2AN 43 (2009) 1027–1044. Google Scholar
Wise, S.M., Wang, C. and Lowengrub, J.S., An energy-stable and convergent finite-difference scheme for the phase field crystal equation. SIAM J. Numer. Anal. 47 (2009) 2269–2288. Google Scholar
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).