Synergistic effects of disjoining pressure and surface viscosity on film drainage process

Ye Xue-Min; Yang Shao-Dong; Li Chun-Xi

doi:10.7498/aps.66.194701

Abstract

A mathematical model is established to investigate the gravity-driven draining process of a vertical thin liquid film containing insoluble surfactants when considering the synergistic effect of surface viscosity and disjoining pressure. Lubrication theory is used to derive a coupled equation set describing the evolution of the film thickness, surfactant concentration and surface velocity. The equation set is solved numerically by the FreeFem program based on the finite element method. The film is assumed to be supported by the wire frame at both the top and bottom, thus the mass of liquid and the mass of total surfactants are conserved in the simulation. The characteristics of film evolution under the constant and variable surface viscosity are examined. Simulation results show that the surface viscosity is a crucial factor affecting the film drainage process. When neglecting the effect of surface viscosity, the film surface exhibits the mobile mode, while the film surface presents the rigid mode in the case of the surface viscosity considered. Increasing the surface viscosity, the rate of film drainage is slowed down significantly, leading to a reduction of the film thinning and enhancement of film stability, which is consistent with the results obtained by Naire et al. The disjoining pressure is a key factor in the formation of black film. When the disjoining pressure is only involved in the model, the length of the black film region is relatively short, nevertheless, if the effect of surface viscosity is only considered, a stable black film does not form. Under the synergistic effect of the disjoining pressure and surface viscosity, a very long and thin but stable black film is found in the middle segment of the film. Additionally, the thickness of black film increases and the appearance time postpones with the increase of surface viscosity. Considering the influence of concentration-dependent surface viscosity, the drainage rate is greatly affected. In the early stage, due to the smaller overall surface viscosity, the surface velocity is relatively large. With increasing surface viscosity at the bottom of film, the strength of the film surface tends to be enhanced, and then the anti-perturbation ability of the film is promoted and the film thinning is retarded. There is no significant difference in the length nor the appearance time of black film except that the thickness of black film with concentration-dependent surface viscosity is lower than that with the constant viscosity, thus the flow stability is weaker than that with the constant viscosity. In addition, the presence of the disjoining pressure slows down the thinning of blackest portion of the film and the surfactant concentration at this position. In the numerical results of the variable surface viscosity given by Braun et al. it is observed that the concentration of surfactant could almost be swept to clean in the top part of film. That is possibly because the effect of the disjoining pressure is neglected by Braun et al. It should be pointed out that the surface elasticity plays an important role in the stability of film. Therefore, it is necessary to consider the effect of surface elasticity in the future investigation.

Keywords:

Authors and contacts

1.
Key Laboratory of Condition Monitoring and Control for Power Plant Equipment, North China Electric Power University, Baoding 071003, China

Corresponding author: Li Chun-Xi, leechunxi@163.com

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11202079) and the Fundamental Research for the Central Universities, China (Grant No. 13MS97).

References

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[2]	Bournival G, Du Z, Ata S, Jameson G J 2014 Chem. Eng. Sci. 116 536
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[35]	Ivanov I B, Danov K D, Ananthapadmanabhan K P, Lips A 2005 Adv. Colloid Interfac. 114-115 61
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[38]	Li C X, Pei J J, Ye X M 2013 Acta Phys. Sin. 62 214704 (in Chinese)[李春曦, 裴建军, 叶学民 2013 物理学报 62 214704]
[39]	de Wit A, Gallez D, Christov C I 1994 Phys. Fluids 6 3256
[40]	Sakata E K, Berg J C 1972 J. Colloid Interface Sci. 40 99
[41]	Saulnier L, Boos J, Stubenrauch C, Rio E 2014 Soft Matter 10 7117
[42]	Lu K Q, Liu J X 2007 Introduction of Soft Matter Physics (Beijing:Peking University Press) pp278, 279 (in Chinese)[陆坤权, 刘寄星 2007 软物质物理学导论 (北京大学出版社) 第278, 279页]
[43]	Braun R J, Snow S A, Naire S 2002 J. Eng. Math. 43 281
[44]	Karakashev S I, Nguyen A V 2007 Colloid Surface A 293 229
[45]	Ivanov I B, Dimitrov D S 1974 Colloid Polym. Sci. 252 98

Cited By

[1]	Huang J, Sun Q C 2007 Acta Phys. Sin. 56 6124 (in Chinese)[黄晋, 孙其诚 2007 物理学报 56 6124]
[2]	Bournival G, Du Z, Ata S, Jameson G J 2014 Chem. Eng. Sci. 116 536
[3]	Firouzi M, Nguyen A V 2014 Adv. Powder Technol. 25 1212
[4]	Jun S, Pelot D D, Yarin A L 2012 Langmuir 28 5323
[5]	Anazadehsayed A, Naser J 2017 Chem. Eng. Sci. 166 11
[6]	Mysels K J, Shinoda K, Frankel S 1959 Soap Films:Studies of Their Thinning and a Bibilography (New York:Pergammon) p116
[7]	Wang J, Nguyen A V, Farrokhpay S 2016 Adv. Colloid Interfac. 228 55
[8]	Benjamin Dollet, Isabelle Cantat 2014 J. Fluid Mech. 739 124
[9]	Zang D Y, Rio E, Langevin D, Wei B, Binks B P 2010 Eur. Phys. J. E 31 125
[10]	Schwartz L W, Roy R V 1999 J. Colloid Interface Sci. 218 309
[11]	Carey E, Stubenrauch C 2010 J. Colloid Interface Sci. 343 314
[12]	Tabakova S S, Danov K D 2009 J. Colloid Interface Sci. 336 273
[13]	Matar O K 2002 Phys. Fluids 14 4216
[14]	Boussinesq J 1913 Ann. Chim. Phys. 29 349
[15]	Li T M, Jia S Y 1995 CIESC J. 5 532 (in Chinese)[李佟茗, 贾绍义 1995 化工学报 5 532]
[16]	Naire S, Braun R J, Snow S A 2000 J. Colloid Interface Sci. 230 91
[17]	Naire S, Braun R J, Snow S A 2004 J. Comput. Appl. Math. 166 385
[18]	Naire S, Braun R J, Snow S A 2001 Phys. Fluids 13 2492
[19]	Heidari A H, Braun R J, Hirsa A H, Snow S A, Naire S 2002 J. Colloid Interface Sci. 253 295
[20]	Snow S A, Pernisz U C, Nugent B M 1996 Dow Corning Corporation Research Report 1996-I0000-41395
[21]	Lopez J M, Hirsa A H 2000 J. Colloid Interface Sci. 229 575
[22]	Vitasari D, Grassia P, Martin P 2015 Appl. Math. Model. 40 1941
[23]	Elfring G J, Leal L G, Squires T M 2016 J. Fluid Mech. 792 712
[24]	Yekeen N, Idris A K, Manan M A, Samin A M, Risal A R, Tan X K 2017 Chin. J. Chem. Eng. 25 347
[25]	Han G B, Wu J T, Xu X M 2000 Acta Phys. Chim. Sin. 16 507 (in Chinese)[韩国彬, 吴金添, 徐晓明 2000 物理化学学报 16 507]
[26]	Gauchet S, Durand M, Langevin D 2014 J. Colloid Interface Sci. 449 373
[27]	Saulnier L, Champougny L, Bastien G, Restagno F, Langevin D, Rio E 2014 Soft Matter 10 2899
[28]	Sett S, Sinha-Ray S, Yarin A L 2013 Langmuir 29 4934
[29]	Zang D Y, Rio E, Delon G, Langevin D, Wei B, Binks B P 2011 Mol. Phys. 109 1057
[30]	Pu W, Wei P, Sun L, Jin F Y, Wang S 2016 J. Ind. Eng. Chem. 47 360
[31]	Zhang C R 2007 Ph. D. Dissertation (Beijing:Technical Institute of Physical and Chemistry of Chinese Academy of Sciences) (in Chinese)[张春荣 2007 博士学位论文 (北京:中国科学院理化技术研究所)]
[32]	Tian Y, Holt R G, Apfel R E 1997 J. Colloid Interface Sci. 87 1
[33]	Avramidis K S, Jiang T S 1991 J. Colloid Interface Sci. 147 262
[34]	Murai Y, Shiratori T, Kumagai I, Rhs P A, Fischer P 2015 Flow Meas. Instrum. 41 121
[35]	Ivanov I B, Danov K D, Ananthapadmanabhan K P, Lips A 2005 Adv. Colloid Interfac. 114-115 61
[36]	Manor O, Lavrenteva O, Nir A 2008 J. Colloid Interface Sci. 321 142
[37]	Zhao Y P 2012 Physical Mechanics of Surface and Interface (Beijing:Science Press) pp185, 186 (in Chinese)[赵亚溥 2012 表面与界面物理力学 (北京:科学出版社) 第185, 186页]
[38]	Li C X, Pei J J, Ye X M 2013 Acta Phys. Sin. 62 214704 (in Chinese)[李春曦, 裴建军, 叶学民 2013 物理学报 62 214704]
[39]	de Wit A, Gallez D, Christov C I 1994 Phys. Fluids 6 3256
[40]	Sakata E K, Berg J C 1972 J. Colloid Interface Sci. 40 99
[41]	Saulnier L, Boos J, Stubenrauch C, Rio E 2014 Soft Matter 10 7117
[42]	Lu K Q, Liu J X 2007 Introduction of Soft Matter Physics (Beijing:Peking University Press) pp278, 279 (in Chinese)[陆坤权, 刘寄星 2007 软物质物理学导论 (北京大学出版社) 第278, 279页]
[43]	Braun R J, Snow S A, Naire S 2002 J. Eng. Math. 43 281
[44]	Karakashev S I, Nguyen A V 2007 Colloid Surface A 293 229
[45]	Ivanov I B, Dimitrov D S 1974 Colloid Polym. Sci. 252 98

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Vol. 66, Issue 19 - 2017-10-05

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