Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-30T23:26:23.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical Simulation for Moving Contact Line with Continuous Finite Element Schemes

Part of: Two-phase and multiphase flows Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  03 July 2015

Yongyue Jiang ,
Ping Lin ,
Zhenlin Guo  and
Shuangling Dong
Yongyue Jiang*
Affiliation:
School of Mathematics and Physics, University of Science and Technology, Beijing 100083, P.R. China
Ping Lin
Affiliation:
Department of Mathematics, University of Dundee, Dundee DD1 4HN, Scotland, United Kingdom
Zhenlin Guo
Affiliation:
Department of mathematics, University of California Irvine, CA 92697-3875, 540H Rowland Hall, Irvine, USA
Shuangling Dong
Affiliation:
School of Aerospace Engineering, Tsinghua University, Beijing 100084, P.R. China
*
*Corresponding author. Email addresses: [email protected] (Y. Jiang), [email protected] (P. Lin)
Get access

Abstract

In this paper, we compute a phase field (diffuse interface) model of Cahn-Hilliard type for moving contact line problems governing the motion of isothermal multiphase incompressible fluids. The generalized Navier boundary condition proposed by Qian et al. [1] is adopted here. We discretize model equations using a continuous finite element method in space and a modified midpoint scheme in time. We apply a penalty formulation to the continuity equation which may increase the stability in the pressure variable. Two kinds of immiscible fluids in a pipe and droplet displacement with a moving contact line under the effect of pressure driven shear flow are studied using a relatively coarse grid. We also derive the discrete energy law for the droplet displacement case, which is slightly different due to the boundary conditions. The accuracy and stability of the scheme are validated by examples, results and estimate order.

Keywords

Two-phase flowgeneralized Navier boundary conditioncontinuous finite elementsmoving contact line

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 76T10: Liquid-gas two-phase flows, bubbly flows
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 1 , July 2015 , pp. 180 - 202
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Qian, T., Wang, X. P., Sheng, P., Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.Google Scholar
[2]Zhang, L., Aoki, J. and Thomas, B. G., Inclusion removal by bubble floation in a continuous casting mold, Metall. Mater. Trans. B, 37B (2006), 361–379.Google Scholar
[3]Norman, C. E. and Miksis, M. J.Gas bubble with a moving contact line rising in an inclined channel at finite Reynolds number, Physica D, 209 (2005), 191–204.Google Scholar
[4]Luo, X., Wang, X. P., Qian, T. and Sheng, P., Moving contact line over undulating surfaces, Solid State Commun., 139 (2006), 623–629.Google Scholar
[5]Chen, Y., Mertz, R., Kulenovic, R., Numerical simulation of bubble formation on orifice plates with a moving contact line, Int. J. Multiphase Flow, 35 (2009), 66–77.Google Scholar
[6]Fuentes, J. and Cerro, R. L.Surface forces and inertial effects on moving contact lines, Chem. Eng. Sci., 62 (2007), 3231–3241.Google Scholar
[7]Nikolayev, V. S., Gavrilyuk, S. L. and Gouin, H., Modeling of the moving deformed triple contact line influence of the fluid inertia, J. Colloid Interface Sci., 302 (2006), No. 2, pp. 605–612.Google Scholar
[8]Zahedi, S., Gustavsson, K. and Kreiss, G., A conservative level set method for contact line dynamics, J. Comput. Phys., 228 (2009), No. 17, pp. 6361–6375.Google Scholar
[9]Gerbeau, J. F. and Lelivre, T., Generalized Navier boundary condition and geometric conservation law for surface tension, Comput. Methods Appl. Mech. Engrg., 198 (2009), 644–656.Google Scholar
[10]Xu, X. M. and Wang, X. P.Derivation of the wenzel and cassie equation from a phase field model for two phase flow on rough surface, SIAM J. APPL. MATH. 70 (2010), 2929–2941.Google Scholar
[11]Jacqmin, D., Contact line dynamics of a diffuse fluid interface, J. Fluid Mech., 402 (2000), 57–88.Google Scholar
[12]Seppecher, P., Moving contact lines in the Cahn-Hilliard theory, Int. J. Eng. Sci., 34 (1996), 977–992.Google Scholar
[13]Li, Z., Lai, M. C., He, G. and Zhao, H., An augmented method for free boundary problems with moving contact lines, Fluids Comput., 39 (2010), 1033–1040.Google Scholar
[14]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.CrossRefGoogle Scholar
[15]Renardy, M., Renardy, Y. and Li, J., Numerical simulation of moving contact line problems using a volume-of-fluid method, J. Comput. Phys., 171 (2001), 243–263.Google Scholar
[16]Hadjiconstantinou, N. G., Hybrid atomistic-continuum formulations and the moving contact-line problem, J. Comput. Phys., 154 (1999), 245–265.Google Scholar
[17]Shen, J. and Yang, X., An efficient moving mesh spectral method for the phase-field model of two-phase flows, J. Comput. Phys., 228 (2009), 2978–2992.Google Scholar
[18]Hecht, F., Pironneau, O., Hyaric, A. Le and Ohtsuka, K., FreeFem++ (Version 2.17-1), 2007 (http://www.freefem.org/ff++/ftp/freefem++doc.pdf).Google Scholar
[19]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
[20]Hua, J., Lin, P., Liu, C. and Wang, Q., Energy law preserving C 0 finite element schemes for phase field models in two-phase flow computations, J. Comput. Phys., 230 (2011), 7115–7131.Google Scholar
[21]Qian, T., Wang, X. P. and Sheng, P., A variational approach to the moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333–360.Google Scholar
[22]Lin, P. and Liu, C., Simulation of singularity dynamics in liquid crystal flows: A C 0 finite element approach, J. Comput. Phys., 215 (2006), 348–362.Google Scholar
[23]Lin, P., Liu, C. and Zhang, H., An energy law preserving C 0 finite element scheme for simulating the kinematic effects in liquid crystal flow dynamics, J. Comput. Phys., 227 (2007), 1411–1427.Google Scholar
[24]Guo, Z. L., Lin, P. and Wang, Y. F., Continuous finite element schemes for a phase field model in two-layer fluid Benark-Marangoni convection computations, Comput. Phys. Comm., 185 (2014), 63–78.CrossRefGoogle Scholar
[25]Bao, K., Shi, Y., Sun, S. and Wang, X. P.. A finite element method for the numerical solution of the coupled Cahn-Hilliard and Navier-Stokes system for moving contact line problems, J. Comput. Phys., 231 (2012), 8083–8099.Google Scholar
[26]Spelt, P., Shear flow past two-dimensional droplets pinned or moving on an adhering channel wall ar moderate reynolds numbers: A numerical study, J. Fluid Mech., 561 (2006), 439–463.Google Scholar
[27]Schleizer, A. D. and Bonnecaze, R. T.. Displacement of a two-dimensional immiscible droplet adhering to a wall in shear and pressure driven flows, J. Fluid Mech., 383 (1999), 29–54.Google Scholar
[28]Yang, X. F., Feng, J. J., Liu, C. and Shen, J., Numerical simulations of jet pinching-off and drop formation using an energetic variational phase-field method, J. Comput. Phys., 218 (2006), 417–428.Google Scholar
[29]Lin, P., Liu, C. and Zhang, H., An energy law preserving C 0 finite element scheme for simulating the kinematic effects in liquid crystal dynamics, J. Comput. Phys., 227 (2007), 1411–1427.Google Scholar
[30]Altmann, R. and Heiland, J., Finite element decomposition and minimal extension for flow equations, Preprint 2013–11, Technische Universität Berlin, Germany, 2013.Google Scholar
[31]Lin, P., A sequantial regularization method for time-dependent incompressible Navier-Stokes equations, SIAM J. NUMER. ANAL., 34 (1997), 1051–1071.Google Scholar
[32]Arnold, M., Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic systems of index 2, BIT Numer. Math., 38 (1998), 415–438.Google Scholar
[33]Arnold, M., Strehmel, K. and Weiner, R., Half-explicit Runge-Kutta methods with explicit stages for differential-algebraic equations of index 1, Numer. Math., 64 (1993), 409–431.Google Scholar
[34]Yue, P., Zhou, C. and Feng, J. J., Sharp-interface limit of the Cahn-Hilliard model for moving contact lines, J. Fluid Mech., 645 (2010), 279–294.Google Scholar
[35]Yue, P. and Feng, J. J.Wall energy rexlation in the Cahn-Hilliard model for moving contact lines, Fluids Phys., 23 (2011), 012106: 1–8.Google Scholar
[36]Sui, Y. and Spelt, P. D. M., An efficient computational model for macroscale simulations of moving contact lines, J. Comput. Phys., 242 (2013), 37–52.Google Scholar
[37]Sui, Y. and M, P. D.. Spelt, Validation and modification of asymptotic analysis of slow and rapid droplet spreading by numerical simulation, J. Fluid Mech., 715 (2013), 283–313.CrossRefGoogle Scholar
[38]Sui, Y., Ding, H. and Spelt, P. D. M., Numerical Simulations of Flows with Moving Contact Lines, Annu. Rev. Fluid Mech., 46 (2014), 97–119.CrossRefGoogle Scholar