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For the spreading of a droplet covered with insoluble surfactant over an inclined random heterogeneous substrate, the base state and disturbance evolution equations for the film thickness and surfactant concentration are derived using the lubrication theory. Stability of the droplet spreading on the heterogeneous substrates, and effects of dimensionless parameters as well as the internal mechanism are investigated based on the non-modal stability theory. Results show that the disturbances of film thickness and surfactant concentration exihibit a double-hump shape and the crests lie in thinning regions at the bottom of the droplet. With increasing disturbance wave number, the stability of the droplet spreading is enhanced, but the maximal disturbance transfers from the right-side to the left-side of the droplet bottom. Increasing inclined angle leads to severe instability distinctly. Improving topographical height or increasing wave number of the substrate leads the spreading stability to firstly enhance and then to weaken. The droplet evolution displays an unstable spreading under smaller capillary number, with the maximum disturbance to be increasing before decreasing.
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Keywords:
- surfactant-laden droplet /
- topographical surface /
- spreading /
- non-modal stability
[1] Hamraoui A, Cachile M, Poulard C, Cazabat A M 2004 Colloids Surf. A 250 215
[2] Matar O K 2002 Phys. Fluids 14 4216
[3] Warner M R E, Craster R V, Matar O K 2004 Phys. Fluids 16 2933
[4] Edmonstone B D, Matar O K, Craster R V 2005 Physica D 209 62
[5] Mavromoustaki A, Matar O K, Craster R V 2012 J. Colloid Interface Sci. 371 121
[6] Ye X M, Jiang K, Shen L, Li C X 2013 Journal of Theoretical and Applied Mechanics 45 681 (in Chinese) [叶学民, 姜凯, 沈雷, 李春曦 2013 力学学报 45 681]
[7] Hu G H 2005 Phys. Fluids A 17 088105
[8] Wang W, Li Z X, Guo Z Y 2003 J. Eng. Thermophys. 24 85 (in Chinese) [王玮, 李志信, 过增元 2003 工程热物理学报 24 85]
[9] Gerbig Y B, Phani A R, Haefke H 2005 Appl. Surf. Sci. 242 251
[10] Craster R V, Matar O K 2009 Rev. Mod. Phys. 81 1131
[11] Pang H Y, Zhang X F, Zhang H Y, Du F P 2006 Chin. J. Pestic. Sci. 8 157 (in Chinese) [庞红宇, 张现峰, 张红艳, 杜凤沛 2006 农药学学报 8 157]
[12] Lee K S, Ivanova N, Starov V M Hilal N, Dutschk V 2008 Adv. Colloid Interfac. 244 54
[13] Kalliadasis S, Bielarz C, Homsy G M 2000 Phys. Fluids 12 1889
[14] Nonomura Y, Morita Y, Hikima T, Seino E, Chida S, Mayama H 2010 Langmuir 26 16150
[15] Liu M, Wang S L, Wu Z R 2014 Acta Phys. Sin. 63 154702 (in Chinese) [刘梅, 王松岭, 吴正人 2014 物理学报 63 154702]
[16] Tseluiko D, Blyth M G, Papageorgiou D T 2011 J. Eng. Math. 69 169
[17] Fischer B J, Troian S M 2003 Phys. Rev. B 67 016309
[18] Nouar C, Kabouya N, Duesk J 2007 Journal of Fluid Mechanics 577 211.
[19] Li C X, Pei J J, Ye X M 2013 Acta Phys. Sin. 62 174702 (in Chinese) [李春曦, 裴建军, 叶学民 2013 物理学报 62 174702]
[20] 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]
[21] Zhao Y P 2012 Physical Mechanics of Surface and Interface (Beijing: Science Press)pp185-186m (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]
[22] Edmonstone B D, Matar O K, Craster R V 2004 J. Eng. Math. 50 141
[23] Zhou H, Zhao G F 2003 Hydrodynamic Stability (Beijing: National Defense Press)pp2-4m (in Chinese) [周恒, 赵耕夫 2003 流动稳定性(北京: 国防工业出版社)第2–4页]
[24] Warner M R E, Craster R V, Matar O K 2004 J. Fluid Mech. 510 169
[25] Sadiq M R, Tatiana G R, Stephan P 2012 Phys. Fluids 24 014104
[26] Li C X, Pei J J, Ye X M 2013 CIESC J. 64 3308 (in Chinese) [李春曦, 裴建军, 叶学民 2013 化工学报 64 3308]
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[1] Hamraoui A, Cachile M, Poulard C, Cazabat A M 2004 Colloids Surf. A 250 215
[2] Matar O K 2002 Phys. Fluids 14 4216
[3] Warner M R E, Craster R V, Matar O K 2004 Phys. Fluids 16 2933
[4] Edmonstone B D, Matar O K, Craster R V 2005 Physica D 209 62
[5] Mavromoustaki A, Matar O K, Craster R V 2012 J. Colloid Interface Sci. 371 121
[6] Ye X M, Jiang K, Shen L, Li C X 2013 Journal of Theoretical and Applied Mechanics 45 681 (in Chinese) [叶学民, 姜凯, 沈雷, 李春曦 2013 力学学报 45 681]
[7] Hu G H 2005 Phys. Fluids A 17 088105
[8] Wang W, Li Z X, Guo Z Y 2003 J. Eng. Thermophys. 24 85 (in Chinese) [王玮, 李志信, 过增元 2003 工程热物理学报 24 85]
[9] Gerbig Y B, Phani A R, Haefke H 2005 Appl. Surf. Sci. 242 251
[10] Craster R V, Matar O K 2009 Rev. Mod. Phys. 81 1131
[11] Pang H Y, Zhang X F, Zhang H Y, Du F P 2006 Chin. J. Pestic. Sci. 8 157 (in Chinese) [庞红宇, 张现峰, 张红艳, 杜凤沛 2006 农药学学报 8 157]
[12] Lee K S, Ivanova N, Starov V M Hilal N, Dutschk V 2008 Adv. Colloid Interfac. 244 54
[13] Kalliadasis S, Bielarz C, Homsy G M 2000 Phys. Fluids 12 1889
[14] Nonomura Y, Morita Y, Hikima T, Seino E, Chida S, Mayama H 2010 Langmuir 26 16150
[15] Liu M, Wang S L, Wu Z R 2014 Acta Phys. Sin. 63 154702 (in Chinese) [刘梅, 王松岭, 吴正人 2014 物理学报 63 154702]
[16] Tseluiko D, Blyth M G, Papageorgiou D T 2011 J. Eng. Math. 69 169
[17] Fischer B J, Troian S M 2003 Phys. Rev. B 67 016309
[18] Nouar C, Kabouya N, Duesk J 2007 Journal of Fluid Mechanics 577 211.
[19] Li C X, Pei J J, Ye X M 2013 Acta Phys. Sin. 62 174702 (in Chinese) [李春曦, 裴建军, 叶学民 2013 物理学报 62 174702]
[20] 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]
[21] Zhao Y P 2012 Physical Mechanics of Surface and Interface (Beijing: Science Press)pp185-186m (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]
[22] Edmonstone B D, Matar O K, Craster R V 2004 J. Eng. Math. 50 141
[23] Zhou H, Zhao G F 2003 Hydrodynamic Stability (Beijing: National Defense Press)pp2-4m (in Chinese) [周恒, 赵耕夫 2003 流动稳定性(北京: 国防工业出版社)第2–4页]
[24] Warner M R E, Craster R V, Matar O K 2004 J. Fluid Mech. 510 169
[25] Sadiq M R, Tatiana G R, Stephan P 2012 Phys. Fluids 24 014104
[26] Li C X, Pei J J, Ye X M 2013 CIESC J. 64 3308 (in Chinese) [李春曦, 裴建军, 叶学民 2013 化工学报 64 3308]
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