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The aim of the present paper is to investigate the gravity-driven draining process containing insoluble surfactants, with the coupling effects of surface elasticity and disjoining pressure taken into consideration. A set of evolution equations including liquid film thickness, surface velocity and surfactant concentration, is established based on the lubrication theory. Assuming that the top of the liquid film is attached to the wireframe and the bottom is connected to the reservoir, the drainage stability is simulated with the FreeFem software. The characteristics of film evolution under the coupled effects of surface elasticity and disjoining pressure are examined, respectively. The simulated results show that the surface elasticity and the disjoining pressure have significant influences on the vertical thin film draining process. Under the effect of the surface elasticity alone, the initial film thickness increases with the elasticity increasing and the black film only forms on the top of the liquid film, but cannot stably exist and breaks quickly. The addition of the surface elasticity can increase the liquid film thickness and the drainage time, reduce the surface velocity, and rigidify the interface. When the disjoining pressure is applied merely, the surfactant flows into the reservoir continuously; hardly can the liquid film form a surface tension gradient and thus cannot form a countercurrent phenomenon. Under the coupling effect of the surface elasticity and disjoining pressure, a more stable liquid film forms. In the early stage of drainage, surface elasticity increases the film thickness, reduces the surface speed and generates the liquid countercurrent to slow the drainage process. When the black film appears, the electrostatic repulsion of the disjoining pressure is notable and makes the black film stable. The results obtained in the paper are in agreement with some of the experimental results in the literature. However, the elasticity-related surface tension and surfactant concentration model used is a simplified model. The nonlinear relationship between surface tension and surfactant concentration should be further considered in future theoretical models.
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Keywords:
- vertical film /
- film drainage /
- surface elasticity /
- disjoining pressure
[1] Wang J, Nguyen A V, Farrokhpay S 2015 Adv. Colloid Interface 228 55
[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] Wang L, Yoon R H 2008 Int. J. Miner. Process. 85 101
[5] Wang J, Nguyen A V, Farrokhpay S 2016 Colloid Surface A 488 70
[6] Sett S, Sinharay S, Yarin A L 2013 Langmuir 29 4934
[7] Wang L, Yoon R H 2006 Colloid Surface A 282 84
[8] Mysels K J, Cox M C, Skewis J D 1961 J. Phys. Chem. 65 1107
[9] Champougny L, Scheid B, Restagno F, Vermant J, Rio E 2015 Soft Matter 11 2758
[10] Saulnier L, Champougny L, Bastien G, Restagno F, Langevin D, Rio E 2014 Soft Matter 10 2899
[11] Liggieri L, Attolini V, Ferrari M, Ravera F 2002 J. Colloid Interface Sci. 255 225
[12] Persson C M, Claesson P M, Lunkenheimer K 2002 J. Colloid Interface Sci. 251 182
[13] Karakashev S I, Ivanova D S, Angarska Z K, Manev E D, Tsekov R, Radoev B, Slavchov R, Nguyen A V 2010 Colloid Surface A 365 122
[14] Seiwert J, Dollet B, Cantat I 2014 J. Fluid Mech. 739 124
[15] Exerowa D, Zacharieva M, Cohen R, Platikanov D 1979 Colloid Polym. Sci. 257 1089
[16] Churaev N V 2003 Colloid J. 103 197
[17] Manev E D, Pugh R J 1991 Langmuir 7 2253
[18] Bhakta A, Ruckenstein E 1997 J. Colloid Interface Sci. 191 184
[19] Carey E, Stubenrauch C 2010 J. Colloid Interface Sci. 343 314
[20] Buchavzov N, Stubenrauch C 2007 Langmuir 23 5315
[21] Stubenrauch C, Schlarmann J, Strey R 2002 Phys. Chem. Chem. Phys. 4 4504
[22] Karakashev S I, Ivanova D S 2010 J. Colloid Interface Sci. 343 584
[23] Teletzke G F, Davis H T, Scriven L E 1988 Rev. Phys. Appl. 23 989
[24] Mitlin V S, Petviashvili N V 1994 Phys. Lett. A 192 323
[25] Frastia L, Archer A J, Thiele U 2012 Soft Matter 8 11363
[26] Ye X M, Yang S D, Li C X 2017 Acta Phys. Sin. 66 184702 (in Chinese) [叶学民, 杨少东, 李春曦 2017 66 184702]
[27] Ye X M, Yang S D, Li C X 2017 Acta Phys. Sin. 66 194701 (in Chinese) [叶学民, 杨少东, 李春曦 2017 66 194701]
[28] Tabakova S S, Danov K D 2009 J. Colloid Interface Sci. 336 273
[29] Georgieva D, Cagna A, Langevin D 2009 Soft Matter 5 2063
[30] Bergeron V 1997 Langmuir 13 3474
[31] Panaiotov I, Dimitrov D S, Ter-Minassian-Saraga L 1979 J. Colloid Interface Sci. 72 49
[32] Park C W 1991 J. Colloid Interface Sci. 146 382
[33] Zhao Y P 2012 Physical Mechanics of Surface and Interface (Beijing: Science Press) p185, 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]
[34] Naire S, Braun R J, Snow S A 2000 J. Colloid Interface Sci. 230 91
[35] Heidari A H, Braun R J, Hirsa A H, Snow S A, Naire S 2002 J. Colloid Interface Sci. 253 295
[36] Ruschak K J 2010 Aiche J. 24 705
[37] Vitasari D, Grassia P, Martin P 2016 Appl. Math. Model. 40 1941
[38] Berg S, Adelizzi E A, Troian S M 2005 Langmuir 21 3867
[39] Schwartz L W, Roy R V 1999 J. Colloid. Interface Sci. 218 309
[40] Saulnier L, Boos J, Stubenrauch C, Rio E 2014 Soft Matter 10 7117
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[1] Wang J, Nguyen A V, Farrokhpay S 2015 Adv. Colloid Interface 228 55
[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] Wang L, Yoon R H 2008 Int. J. Miner. Process. 85 101
[5] Wang J, Nguyen A V, Farrokhpay S 2016 Colloid Surface A 488 70
[6] Sett S, Sinharay S, Yarin A L 2013 Langmuir 29 4934
[7] Wang L, Yoon R H 2006 Colloid Surface A 282 84
[8] Mysels K J, Cox M C, Skewis J D 1961 J. Phys. Chem. 65 1107
[9] Champougny L, Scheid B, Restagno F, Vermant J, Rio E 2015 Soft Matter 11 2758
[10] Saulnier L, Champougny L, Bastien G, Restagno F, Langevin D, Rio E 2014 Soft Matter 10 2899
[11] Liggieri L, Attolini V, Ferrari M, Ravera F 2002 J. Colloid Interface Sci. 255 225
[12] Persson C M, Claesson P M, Lunkenheimer K 2002 J. Colloid Interface Sci. 251 182
[13] Karakashev S I, Ivanova D S, Angarska Z K, Manev E D, Tsekov R, Radoev B, Slavchov R, Nguyen A V 2010 Colloid Surface A 365 122
[14] Seiwert J, Dollet B, Cantat I 2014 J. Fluid Mech. 739 124
[15] Exerowa D, Zacharieva M, Cohen R, Platikanov D 1979 Colloid Polym. Sci. 257 1089
[16] Churaev N V 2003 Colloid J. 103 197
[17] Manev E D, Pugh R J 1991 Langmuir 7 2253
[18] Bhakta A, Ruckenstein E 1997 J. Colloid Interface Sci. 191 184
[19] Carey E, Stubenrauch C 2010 J. Colloid Interface Sci. 343 314
[20] Buchavzov N, Stubenrauch C 2007 Langmuir 23 5315
[21] Stubenrauch C, Schlarmann J, Strey R 2002 Phys. Chem. Chem. Phys. 4 4504
[22] Karakashev S I, Ivanova D S 2010 J. Colloid Interface Sci. 343 584
[23] Teletzke G F, Davis H T, Scriven L E 1988 Rev. Phys. Appl. 23 989
[24] Mitlin V S, Petviashvili N V 1994 Phys. Lett. A 192 323
[25] Frastia L, Archer A J, Thiele U 2012 Soft Matter 8 11363
[26] Ye X M, Yang S D, Li C X 2017 Acta Phys. Sin. 66 184702 (in Chinese) [叶学民, 杨少东, 李春曦 2017 66 184702]
[27] Ye X M, Yang S D, Li C X 2017 Acta Phys. Sin. 66 194701 (in Chinese) [叶学民, 杨少东, 李春曦 2017 66 194701]
[28] Tabakova S S, Danov K D 2009 J. Colloid Interface Sci. 336 273
[29] Georgieva D, Cagna A, Langevin D 2009 Soft Matter 5 2063
[30] Bergeron V 1997 Langmuir 13 3474
[31] Panaiotov I, Dimitrov D S, Ter-Minassian-Saraga L 1979 J. Colloid Interface Sci. 72 49
[32] Park C W 1991 J. Colloid Interface Sci. 146 382
[33] Zhao Y P 2012 Physical Mechanics of Surface and Interface (Beijing: Science Press) p185, 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]
[34] Naire S, Braun R J, Snow S A 2000 J. Colloid Interface Sci. 230 91
[35] Heidari A H, Braun R J, Hirsa A H, Snow S A, Naire S 2002 J. Colloid Interface Sci. 253 295
[36] Ruschak K J 2010 Aiche J. 24 705
[37] Vitasari D, Grassia P, Martin P 2016 Appl. Math. Model. 40 1941
[38] Berg S, Adelizzi E A, Troian S M 2005 Langmuir 21 3867
[39] Schwartz L W, Roy R V 1999 J. Colloid. Interface Sci. 218 309
[40] Saulnier L, Boos J, Stubenrauch C, Rio E 2014 Soft Matter 10 7117
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