# American Institute of Mathematical Sciences

August  2013, 7(3): 947-959. doi: 10.3934/ipi.2013.7.947

## 3D adaptive finite element method for a phase field model for the moving contact line problems

 1 Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 2 Division of Computer, Electrical and Mathematical Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal 23955-6900, Kingdom of Saudi Arabia 3 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Received  July 2012 Revised  December 2012 Published  September 2013

In this paper, we propose an adaptive finite element method for simulating the moving contact line problems in three dimensions. The model that we used is the coupled Cahn-Hilliard Navier-Stokes equations with the generalized Navier boundary condition(GNBC) proposed in [18]. In our algorithm, to improve the efficiency of the simulation, we use the residual type adaptive finite element algorithm. It is well known that the phase variable decays much faster away from the interface than the velocity variables. Therefore we use an adaptive strategy that will take into account of such difference. Numerical experiments show that our algorithm is both efficient and reliable.
Citation: Yi Shi, Kai Bao, Xiao-Ping Wang. 3D adaptive finite element method for a phase field model for the moving contact line problems. Inverse Problems and Imaging, 2013, 7 (3) : 947-959. doi: 10.3934/ipi.2013.7.947
##### References:
 [1] H. D. Ceniceros, Rudimar L. Nos and Alexandre M. Roma, Theree-dimensional, fully adaptive simulations of phase-field fluid models, J. Comput. Phys., 229 (2010), 6135-6155. doi: 10.1016/j.jcp.2010.04.045. [2] Qingming Chang and J. I. D. Alexander, Analysis of single droplet dynamics on striped surface domains using a lattice Boltzmann method, Microfluid Nanofluid, 2 (2006), 309-326. doi: 10.1007/s10404-005-0075-2. [3] Yana Di, Ruo Li and Tao Tang, A general moving mesh framework in 3D and its application for simulating the mixture of multi-phase flows, Commun. Comput. Phys., 3 (2008), 582-602. [4] Yana Di and Xiao-Ping Wang, Precursor simulations in spreading using a multi-mesh adaptive finite elment method, J. Comput. Phys., 228 (2009), 1380-1390. doi: 10.1016/j.jcp.2008.10.028. [5] Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322. doi: 10.1137/0728069. [6] Qiang Du and Jian Zhang, Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations, SIAM J. Sci. Comput., 30 (2008), 1634-1657. doi: 10.1137/060656449. [7] A. Dupuis and J. M. Yeomans, Droplet dynamics on patterned substrates, Pramana J. Phys., 64 (2005), 1019-1027. doi: 10.1007/BF02704164. [8] Xiaobing Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two phase flows, SIAM J. Numer. Anal., 44 (2006), 1049-1072. doi: 10.1137/050638333. [9] Xiaobing Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99 (2004), 47-84. doi: 10.1007/s00211-004-0546-5. [10] Min Gao and Xiao-Ping Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), 1372-1386. doi: 10.1016/j.jcp.2011.10.015. [11] V. Girault and P.-A. Raviart, "Finite Element Method for Navier-Stokes Equations. Theory and Algorithms," Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [12] Qiaolin He, R. Glowinski and Xiao-Ping Wang, A least square/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. doi: 10.1016/j.jcp.2011.03.022. [13] Yinnian He, Yunxian Liu and Tao Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616-628. doi: 10.1016/j.apnum.2006.07.026. [14] Xianliang Hu, Ruo Li and Tao Tang, A multi-mesh adaptive finite element approximation to phase field models, Commun. Comput. Phys., 5 (2009), 1012-1029. [15] J. Léopoldés, A. Dupuis, D. G. Bucknall and J. M. Yeomans, Jetting micron-scale droplets onto chemically heterogeneous surfaces, Langmuir, 19 (2003), 9818-9822. doi: 10.1021/la0353069. [16] Xiongping Luo, Xiao-Ping Wang, Tiezheng Qian and Ping Sheng, Moving contact line over undulating surfaces, Solid. Stat. Commun., 139 (2006), 623-629. doi: 10.1016/j.ssc.2006.04.040. [17] Nikolas Provatas, Nigel Goldenfeld and Jonathan Dantzig, Adaptive mesh refinement computation of solidification microstructures using dynamic data structures, J. Comput. Phys., 148 (1999), 265-290. doi: 10.1006/jcph.1998.6122. [18] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306, 15 pp. doi: 10.1103/PhysRevE.68.016306. [19] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, Molecular hydrodynamics of the moving contact line in two-phase immiscible flows, Commun. Comput. Phys., 1 (2006), 1-52. [20] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, A variational approach to the moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360. doi: 10.1017/S0022112006001935. [21] Jie Shen and Xiaofeng 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. doi: 10.1137/09075860X. [22] R. Verfurth, "A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques," Wiley-Teubner, 1996. [23] J. M. Yeomans and H.Kusumaatmaja, Modelling drop dynamics on patterned surfaces, Bull. Pol. Acad. Sci.: Tech. Sci., 55 (2007), 203-210. [24] Pengtao Yue, James J. Feng, Chun Liu and Jie Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293-317. doi: 10.1017/S0022112004000370. [25] Pengtao Yue, Chunfeng Zhou, James J. Feng, Carl F. Ollivier-Gooch and Howard H. Hu, Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing, J. Comput. Phys., 219 (2006), 47-67. doi: 10.1016/j.jcp.2006.03.016. [26] Linbo Zhang, Parallel hierarchical grid., Available from: , (). [27] Chunfeng Zhou, Pengtao Yue and James J. Feng, 3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids, J. Comput. Phys., 229 (2010), 498-511. doi: 10.1016/j.jcp.2009.09.039. [28] Xiao-Ping Wang, Tiezheng Qian and Ping Sheng, Moving contact line on chemically patterned surfaces, J. Fluid Mech., 605 (2008), 59-78. doi: 10.1017/S0022112008001456.

show all references

##### References:
 [1] H. D. Ceniceros, Rudimar L. Nos and Alexandre M. Roma, Theree-dimensional, fully adaptive simulations of phase-field fluid models, J. Comput. Phys., 229 (2010), 6135-6155. doi: 10.1016/j.jcp.2010.04.045. [2] Qingming Chang and J. I. D. Alexander, Analysis of single droplet dynamics on striped surface domains using a lattice Boltzmann method, Microfluid Nanofluid, 2 (2006), 309-326. doi: 10.1007/s10404-005-0075-2. [3] Yana Di, Ruo Li and Tao Tang, A general moving mesh framework in 3D and its application for simulating the mixture of multi-phase flows, Commun. Comput. Phys., 3 (2008), 582-602. [4] Yana Di and Xiao-Ping Wang, Precursor simulations in spreading using a multi-mesh adaptive finite elment method, J. Comput. Phys., 228 (2009), 1380-1390. doi: 10.1016/j.jcp.2008.10.028. [5] Q. Du and R. A. Nicolaides, Numerical analysis of a continuum model of phase transition, SIAM J. Numer. Anal., 28 (1991), 1310-1322. doi: 10.1137/0728069. [6] Qiang Du and Jian Zhang, Adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations, SIAM J. Sci. Comput., 30 (2008), 1634-1657. doi: 10.1137/060656449. [7] A. Dupuis and J. M. Yeomans, Droplet dynamics on patterned substrates, Pramana J. Phys., 64 (2005), 1019-1027. doi: 10.1007/BF02704164. [8] Xiaobing Feng, Fully discrete finite element approximations of the Navier-Stokes-Cahn-Hilliard diffuse interface model for two phase flows, SIAM J. Numer. Anal., 44 (2006), 1049-1072. doi: 10.1137/050638333. [9] Xiaobing Feng and A. Prohl, Error analysis of a mixed finite element method for the Cahn-Hilliard equation, Numer. Math., 99 (2004), 47-84. doi: 10.1007/s00211-004-0546-5. [10] Min Gao and Xiao-Ping Wang, A gradient stable scheme for a phase field model for the moving contact line problem, J. Comput. Phys., 231 (2012), 1372-1386. doi: 10.1016/j.jcp.2011.10.015. [11] V. Girault and P.-A. Raviart, "Finite Element Method for Navier-Stokes Equations. Theory and Algorithms," Springer Series in Computational Mathematics, 5, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61623-5. [12] Qiaolin He, R. Glowinski and Xiao-Ping Wang, A least square/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. doi: 10.1016/j.jcp.2011.03.022. [13] Yinnian He, Yunxian Liu and Tao Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl. Numer. Math., 57 (2007), 616-628. doi: 10.1016/j.apnum.2006.07.026. [14] Xianliang Hu, Ruo Li and Tao Tang, A multi-mesh adaptive finite element approximation to phase field models, Commun. Comput. Phys., 5 (2009), 1012-1029. [15] J. Léopoldés, A. Dupuis, D. G. Bucknall and J. M. Yeomans, Jetting micron-scale droplets onto chemically heterogeneous surfaces, Langmuir, 19 (2003), 9818-9822. doi: 10.1021/la0353069. [16] Xiongping Luo, Xiao-Ping Wang, Tiezheng Qian and Ping Sheng, Moving contact line over undulating surfaces, Solid. Stat. Commun., 139 (2006), 623-629. doi: 10.1016/j.ssc.2006.04.040. [17] Nikolas Provatas, Nigel Goldenfeld and Jonathan Dantzig, Adaptive mesh refinement computation of solidification microstructures using dynamic data structures, J. Comput. Phys., 148 (1999), 265-290. doi: 10.1006/jcph.1998.6122. [18] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306, 15 pp. doi: 10.1103/PhysRevE.68.016306. [19] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, Molecular hydrodynamics of the moving contact line in two-phase immiscible flows, Commun. Comput. Phys., 1 (2006), 1-52. [20] Tiezheng Qian, Xiao-Ping Wang and Ping Sheng, A variational approach to the moving contact line hydrodynamics, J. Fluid Mech., 564 (2006), 333-360. doi: 10.1017/S0022112006001935. [21] Jie Shen and Xiaofeng 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. doi: 10.1137/09075860X. [22] R. Verfurth, "A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques," Wiley-Teubner, 1996. [23] J. M. Yeomans and H.Kusumaatmaja, Modelling drop dynamics on patterned surfaces, Bull. Pol. Acad. Sci.: Tech. Sci., 55 (2007), 203-210. [24] Pengtao Yue, James J. Feng, Chun Liu and Jie Shen, A diffuse-interface method for simulating two-phase flows of complex fluids, J. Fluid Mech., 515 (2004), 293-317. doi: 10.1017/S0022112004000370. [25] Pengtao Yue, Chunfeng Zhou, James J. Feng, Carl F. Ollivier-Gooch and Howard H. Hu, Phase-field simulations of interfacial dynamics in viscoelastic fluids using finite elements with adaptive meshing, J. Comput. Phys., 219 (2006), 47-67. doi: 10.1016/j.jcp.2006.03.016. [26] Linbo Zhang, Parallel hierarchical grid., Available from: , (). [27] Chunfeng Zhou, Pengtao Yue and James J. Feng, 3D phase-field simulations of interfacial dynamics in Newtonian and viscoelastic fluids, J. Comput. Phys., 229 (2010), 498-511. doi: 10.1016/j.jcp.2009.09.039. [28] Xiao-Ping Wang, Tiezheng Qian and Ping Sheng, Moving contact line on chemically patterned surfaces, J. Fluid Mech., 605 (2008), 59-78. doi: 10.1017/S0022112008001456.
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