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The aim of the present paper is to investigate the gravity-driven draining process containing soluble surfactant when considering the coupling effects of surface elasticity and surfactant solubility. A nonlinear coupling evolution equation including liquid film thickness, surface velocity and surfactant concentration (both on the surface and in the bulk) is established based on the lubrication theory. Assuming that the top of liquid film is attached to the wireframe and the bottom is connected to a reservoir, the drainage evolution is simulated with the software called FreeFem. The effects of surface elasticity and solubility on liquid film draining are discussed under their coupling. The simulation results show that the surface elasticity is an indispensable factor in the process of liquid film drainage with soluble surfactant, and the surfactant solubility also has an important influence on the process. At the initial stage of liquid draining, the initial thickness of liquid film increases with increasing surface elasticity, and the surface tends to be more rigid; with the drainage proceeding, the liquid film with high and low elasticity illustrate different notable draining features:in the case of low surface elasticity, the distribution of surfactant forms a surface tension gradient from top to bottom on the film surface, leading to positive Marangoni effect that counteracts gravity. However, in the case of high elasticity, the film surface presents a surface tension gradient from bottom to top, resulting in a reverse Marangoni effect, which accelerates the draining and makes the film more susceptible to instability. The solubility of surfactant dominates the number of adsorbent molecules on the film surface, which affects the surface elasticity. When the solubility of the surfactant is great (β → 0), the film is extremely unstable, and it breaks down quickly. As the solubility decreases (namely, β increases), the stability of the film increases, and the initial surface elasticity also rises. The surface elasticity gradually approaches to the limiting dilational elasticity modulus due to the film being thinner.
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Keywords:
- vertical film /
- soluble surfactant /
- surface elasticity /
- solubility /
- drainage process
[1] Warner M R E, Craster R V, Matar O K 2004 Phys. Fluids 16 2933
[2] Lee K S, Ivanova N, Starov V M, Hilal N, Dutschk V 2008 Adv. Colloid Interface Sci. 144 54
[3] Afsar-Siddiqui A B, Luckham P F, Matar O K 2003 Adv. Colloid Interface Sci. 106 183
[4] Xe J N 2007 Int. Med. Health Guid. News 13 44 (in Chinese)[谢绛凝 2007 国际医药卫生导报 13 44]
[5] Couder Y, Chomaz J M, Rabaud M 1989 Physica D 37 384
[6] Langevin D 2014 Annu. Rev. Fluid Mech. 46 47
[7] Lucassen-Reynders E H, Cagna A, Lucassen J 2001 Colloids Surf. A 186 63
[8] Mysels K J, Shinoda K, Frankel S 1959 Soap Films: Studies of Their Thinning and a Bibilography (New York: Pergammon) p116
[9] Naire S, Braun R J, Snow S A 2001 Phys. Fluids 13 2492
[10] 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 物理学报 66 184702]
[11] Schwartz L W, Roy R V 1999 J. Colloid Interface Sci. 218 309
[12] Seiwert J, Benjamin D, Isabelle C 2014 J. Fluid Mech. 739 124
[13] 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 物理学报 66 194701]
[14] Ye X M, Li M L, Zhang X S, Li C X 2018 Acta Phys. Sin. 67 164701 (in Chinese)[叶学民, 李明兰, 张湘珊, 李春曦 2018 物理学报 67 164701]
[15] Yiantsios S G, Higgins B G 2010 Phys. Fluids 22 022102
[16] Lin C K, Hwang C C, Uen W Y 2000 J. Colloid Interface Sci. 231 379
[17] Shi D, Gu H X, Liu X Y, Fan Q L 2004 China Surf. Deterg. Cosmet. 34 229 (in Chinese) [史东, 谷惠先, 刘晓英, 樊全莲 2004 日用化学工业 34 229]
[18] Luo J, Gao B J, Wang J F, Cao Y, Yuan H 2000 Acta Polym. Sin. 1 262 (in Chinese) [罗娟, 高保娇, 王久芬, 曹远, 袁宏 2000 高分子学报 1 262]
[19] Bergeron V 1997 Langmuir 13 3474
[20] Monroy F, Kahn J G, Langevin D 1998 Colloids Surf. A 143 251
[21] Lucassen J, Tempel M V D 1972 Chem. Eng. Sci. 27 1283
[22] Santini E, Ravera F, Ferrari M, Stubenrauch C, Makievski A, Krägel J 2007 Colloids Surf. A 298 12
[23] Beneventi D, Pugh R J, Carré B, Gandini A 2003 J. Colloid Interface Sci. 268 221
[24] Georgieva D, Cagna A, Langevin D 2009 Soft Matter 5 2063
[25] Wang L, Yoon R H 2008 Int. J. Miner. Process. 85 101
[26] Wang L, Yoon R H 2006 Colloids Surf. A 282 84
[27] Karakashev S I, Ivanova D S 2010 J. Colloid Interface Sci. 343 584
[28] Champougny L, Scheid B, Restagno F, Vermant J, Rio E 2015 Soft Matter 11 2758
[29] Seiwert J, Cantat I 2015 Colloids Surf. A 473 2
[30] Jensen O E, Grotberg J B 1993 Phys. Fluids A 5 58
[31] 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]
[32] Afsarsiddiqui A B, And P F L, Matar O K 2003 Langmuir 19 703
[33] Ruschak K J 2010 Aiche J. 33 801
[34] Saulnier L, Restagno F, Delacotte J, Langevin D, Rio E 2011 Langmuir 27 13406
[35] Angarska J K, Ivanova D S, Manev E D 2015 Colloids Surf. A 481 87
[36] Xiong Z 2012 Farm Prod. Process. 3 67 (in Chinese)[熊拯 2012 农产品加工 3 67]
[37] Xiong Y, Chen D J, Wang J, Zhang Q, Wu W G, Yao Y 2008 J. Oil Gas Techn. 30 136 (in Chinese)[熊颖, 陈大钧, 王君, 张谦, 吴文刚, 尧艳 2008 石油天然气学报 30 136]
[38] Kumar N, Couzis A, Maldarelli C 2003 J. Colloid Interface Sci. 267 272
[39] Hsu C, Chang C, Lin S 1999 Langmuir 15 1952
[40] Berg S, Adelizzi E A, Troian S M 2005 Langmuir 21 3867
[41] Karakashev S I, Nguyen A V 2007 Colloids Surf. A 293 229
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[1] Warner M R E, Craster R V, Matar O K 2004 Phys. Fluids 16 2933
[2] Lee K S, Ivanova N, Starov V M, Hilal N, Dutschk V 2008 Adv. Colloid Interface Sci. 144 54
[3] Afsar-Siddiqui A B, Luckham P F, Matar O K 2003 Adv. Colloid Interface Sci. 106 183
[4] Xe J N 2007 Int. Med. Health Guid. News 13 44 (in Chinese)[谢绛凝 2007 国际医药卫生导报 13 44]
[5] Couder Y, Chomaz J M, Rabaud M 1989 Physica D 37 384
[6] Langevin D 2014 Annu. Rev. Fluid Mech. 46 47
[7] Lucassen-Reynders E H, Cagna A, Lucassen J 2001 Colloids Surf. A 186 63
[8] Mysels K J, Shinoda K, Frankel S 1959 Soap Films: Studies of Their Thinning and a Bibilography (New York: Pergammon) p116
[9] Naire S, Braun R J, Snow S A 2001 Phys. Fluids 13 2492
[10] 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 物理学报 66 184702]
[11] Schwartz L W, Roy R V 1999 J. Colloid Interface Sci. 218 309
[12] Seiwert J, Benjamin D, Isabelle C 2014 J. Fluid Mech. 739 124
[13] 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 物理学报 66 194701]
[14] Ye X M, Li M L, Zhang X S, Li C X 2018 Acta Phys. Sin. 67 164701 (in Chinese)[叶学民, 李明兰, 张湘珊, 李春曦 2018 物理学报 67 164701]
[15] Yiantsios S G, Higgins B G 2010 Phys. Fluids 22 022102
[16] Lin C K, Hwang C C, Uen W Y 2000 J. Colloid Interface Sci. 231 379
[17] Shi D, Gu H X, Liu X Y, Fan Q L 2004 China Surf. Deterg. Cosmet. 34 229 (in Chinese) [史东, 谷惠先, 刘晓英, 樊全莲 2004 日用化学工业 34 229]
[18] Luo J, Gao B J, Wang J F, Cao Y, Yuan H 2000 Acta Polym. Sin. 1 262 (in Chinese) [罗娟, 高保娇, 王久芬, 曹远, 袁宏 2000 高分子学报 1 262]
[19] Bergeron V 1997 Langmuir 13 3474
[20] Monroy F, Kahn J G, Langevin D 1998 Colloids Surf. A 143 251
[21] Lucassen J, Tempel M V D 1972 Chem. Eng. Sci. 27 1283
[22] Santini E, Ravera F, Ferrari M, Stubenrauch C, Makievski A, Krägel J 2007 Colloids Surf. A 298 12
[23] Beneventi D, Pugh R J, Carré B, Gandini A 2003 J. Colloid Interface Sci. 268 221
[24] Georgieva D, Cagna A, Langevin D 2009 Soft Matter 5 2063
[25] Wang L, Yoon R H 2008 Int. J. Miner. Process. 85 101
[26] Wang L, Yoon R H 2006 Colloids Surf. A 282 84
[27] Karakashev S I, Ivanova D S 2010 J. Colloid Interface Sci. 343 584
[28] Champougny L, Scheid B, Restagno F, Vermant J, Rio E 2015 Soft Matter 11 2758
[29] Seiwert J, Cantat I 2015 Colloids Surf. A 473 2
[30] Jensen O E, Grotberg J B 1993 Phys. Fluids A 5 58
[31] 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]
[32] Afsarsiddiqui A B, And P F L, Matar O K 2003 Langmuir 19 703
[33] Ruschak K J 2010 Aiche J. 33 801
[34] Saulnier L, Restagno F, Delacotte J, Langevin D, Rio E 2011 Langmuir 27 13406
[35] Angarska J K, Ivanova D S, Manev E D 2015 Colloids Surf. A 481 87
[36] Xiong Z 2012 Farm Prod. Process. 3 67 (in Chinese)[熊拯 2012 农产品加工 3 67]
[37] Xiong Y, Chen D J, Wang J, Zhang Q, Wu W G, Yao Y 2008 J. Oil Gas Techn. 30 136 (in Chinese)[熊颖, 陈大钧, 王君, 张谦, 吴文刚, 尧艳 2008 石油天然气学报 30 136]
[38] Kumar N, Couzis A, Maldarelli C 2003 J. Colloid Interface Sci. 267 272
[39] Hsu C, Chang C, Lin S 1999 Langmuir 15 1952
[40] Berg S, Adelizzi E A, Troian S M 2005 Langmuir 21 3867
[41] Karakashev S I, Nguyen A V 2007 Colloids Surf. A 293 229
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