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Analysis of an irregular boundary layer behavior for the steady state flow of a Boussinesq fluid
A dynamic theory for contact angle hysteresis on chemically rough boundary
1. | Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China |
2. | LSEC, Institute of Computational Mathematics and Scientific/Engineering Computing, NCMIS, Chinese Academy of Sciences, Beijing 100190, China |
We study the interface dynamics and contact angle hysteresis in a two dimensional, chemically patterned channel described by the Cahn-Hilliard equation with a relaxation boundary condition. A system for the dynamics of the contact angle and contact point is derived in the sharp interface limit. We then analyze the behavior of the solution using the phase plane analysis. We observe the stick-slip of the contact point and the contact angle hysteresis. As the size of the pattern decreases to zero, the stick-slip becomes weaker but the hysteresis becomes stronger in the sense that one observes either the advancing contact angle or the receding contact angle without any switching in between. Numerical examples are presented to verify our analysis.
References:
[1] |
X. Chen, X.-P. Wang and X. Xu,
Analysis of the cahn-hilliard equation with a relaxation boundary condition modeling the contact angle dynamics, Arch. Rational Mech. Anal., 213 (2014), 1-24.
doi: 10.1007/s00205-013-0713-x. |
[2] |
P. de Gennes,
Wetting: Statics and dynamics, Simple Views on Condensed Matter, 12 (2003), 357-394.
doi: 10.1142/9789812564849_0041. |
[3] |
A. DeSimone, N. Gruenewald and F. Otto,
A new model for contact angle hysteresis, Networks and Heterogeneous Media, 2 (2007), 211-225.
doi: 10.3934/nhm.2007.2.211. |
[4] |
R. Pego,
Front migration in the nonlinear cahn-hilliard equation, Proc. R. Soc. Lond., A, 422 (1989), 261-278.
doi: 10.1098/rspa.1989.0027. |
[5] |
T. Qian, X. Wang and P. Sheng,
Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.
doi: 10.1103/PhysRevE.68.016306. |
[6] |
T. Qian, X. Wang and P. Sheng,
Power-law slip profile of the moving contact line in two-phase immiscible flows, Phys. Rev. Lett., 93 (2004), 094501.
doi: 10.1103/PhysRevLett.93.094501. |
[7] |
A. Turco, F. Alouges and A. DeSimone, Wetting on rough surfaces and contact angle hysteresis: Numerical experiments based on a phase field model, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 1027-1044, http://www.esaim-m2an.org/article_S0764583X09000168. Google Scholar |
[8] |
X. Wang, T. Qian and P. Sheng,
Moving contact line on chemically patterned surfaces, Journal of Fluid Mechanics, 605 (2008), 59-78.
doi: 10.1017/S0022112008001456. |
[9] |
X. Xu and X. P. Wang,
Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779.
doi: 10.1137/110829593. |
[10] |
T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87. Google Scholar |
show all references
References:
[1] |
X. Chen, X.-P. Wang and X. Xu,
Analysis of the cahn-hilliard equation with a relaxation boundary condition modeling the contact angle dynamics, Arch. Rational Mech. Anal., 213 (2014), 1-24.
doi: 10.1007/s00205-013-0713-x. |
[2] |
P. de Gennes,
Wetting: Statics and dynamics, Simple Views on Condensed Matter, 12 (2003), 357-394.
doi: 10.1142/9789812564849_0041. |
[3] |
A. DeSimone, N. Gruenewald and F. Otto,
A new model for contact angle hysteresis, Networks and Heterogeneous Media, 2 (2007), 211-225.
doi: 10.3934/nhm.2007.2.211. |
[4] |
R. Pego,
Front migration in the nonlinear cahn-hilliard equation, Proc. R. Soc. Lond., A, 422 (1989), 261-278.
doi: 10.1098/rspa.1989.0027. |
[5] |
T. Qian, X. Wang and P. Sheng,
Molecular scale contact line hydrodynamics of immiscible flows, Phys. Rev. E, 68 (2003), 016306.
doi: 10.1103/PhysRevE.68.016306. |
[6] |
T. Qian, X. Wang and P. Sheng,
Power-law slip profile of the moving contact line in two-phase immiscible flows, Phys. Rev. Lett., 93 (2004), 094501.
doi: 10.1103/PhysRevLett.93.094501. |
[7] |
A. Turco, F. Alouges and A. DeSimone, Wetting on rough surfaces and contact angle hysteresis: Numerical experiments based on a phase field model, ESAIM: Mathematical Modelling and Numerical Analysis, 43 (2009), 1027-1044, http://www.esaim-m2an.org/article_S0764583X09000168. Google Scholar |
[8] |
X. Wang, T. Qian and P. Sheng,
Moving contact line on chemically patterned surfaces, Journal of Fluid Mechanics, 605 (2008), 59-78.
doi: 10.1017/S0022112008001456. |
[9] |
X. Xu and X. P. Wang,
Analysis of wetting and contact angle hysteresis on chemically patterned surfaces, SIAM J. Appl. Math., 71 (2011), 1753-1779.
doi: 10.1137/110829593. |
[10] |
T. Young, An essay on the cohesion of fluids, Philos. Trans. R. Soc. London, 95 (1805), 65-87. Google Scholar |







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