Abstract.
We study the thermocapillary creeping flow induced by a thermal gradient at the liquid-air interface in the presence of insoluble surfactants (impurities). Convective sweeping of the surfactants causes density inhomogeneities that confers in-plane elastic features to the interface. This mechanism is discussed for radially symmetric temperature fields, in both the deep and shallow water regimes. When mass transport is controlled by convection, it is found that surfactants are depleted from a region whose size is inversely proportional to the interfacial elasticity. Both the concentration and the velocity fields follow power laws at the border of the depleted region. Finally, it is shown that this singular behavior is smeared out when molecular diffusion is accounted for.
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References
L.E. Scriven, Nature 187, 186 (1987)
V.G. Levich, Physicochemical Hydrodynamics (Prentice Hall, 1962)
S.H. Davis, Annu. Rev. Fluid Mech. 19, 403 (1987)
G. Van der Mensbrugghe, Mem. Cour. Acad. R. Sci. Belg. 34, 1 (1869)
K. Sefiane, C.A. Ward, Adv. Colloid Interface Sci. 134-135, 201 (2007)
A. Karbalaei, R. Kumar, H.J. Cho, Micromachines 7, 13 (2016)
R. Muruganathan, Y. Zhang, T.M. Fischer, J. Am. Chem. Soc. 128, 3474 (2006)
C.N. Baroud, M.R. de Saint-Vincent, J.-P. Delville, Lab Chip 7, 1029 (2007)
B. Pottier, C. Frétigny, L. Talini, Phys. Rev. Lett. 114, 227801 (2015)
C. Maggi, F. Saglimbeni, M. Dipalo, F. De Angelis, R. Di Leonardo, Nat. Commun. 6, 7855 (2015)
A. Girot, N. Danné, A. Würger, T. Bickel, F. Ren, J.-C. Loudet, B. Pouligny, Langmuir 32, 2687 (2016)
M.-C. Zhong, Z.-Q. Wang, Y.-M. Li, Opt. Express 25, 2481 (2017)
I.K. Bratukhin, L.N. Maurin, J. Appl. Math. Mech. 31, 605 (1967)
R.V. Birikh, V.A. Briskman, M.G. Velarde, J.-C. Legros, Liquid Interfacial Systems (Marcel Dekker, 2003)
A. Ito, S.K. Choudhury, T. Fukano, JSME Int. J. -- Ser. II 33, 128 (1990)
T.-C. Wu, Y.-M. Yang, J.-R. Maa, Int. Commun. Heat Mass Transfer 27, 655 (2000)
T.-C. Wu, Y.-M. Yang, J.-R. Maa, Int. Commun. Heat Mass Transfer 28, 357 (2001)
E. Favre, L. Blumenfeld, F. Daviaud, Phys. Fluids 9, 1473 (1997)
A. Mizev, Phys. Fluids 17, 122107 (2005)
V. Shtern, F. Hussain, J. Fluid Mech. 256, 535 (1993)
J.C. Berg, A. Acrivos, Chem. Eng. Sci. 20, 737 (1965)
C. Ybert, J.-M. di Meglio, Eur. Phys. J. E 3, 143 (2000)
G.M. Homsy, E. Meiburg, J. Fluid Mech. 139, 443 (1984)
B. Carpenter, G.M. Homsy, J. Fluid Mech. 155, 429 (1985)
A. Shmyrov, A. Mizev, V. Demin, M. Petukhov, D. Bratsun, Adv. Colloid Interface Sci. 225, 10 (2018)
Z. Khattari, P. Steffen, T.M. Fischer, J. Phys.: Condens. Matter 14, 4823 (2002)
S. Das, S. mandal, S.K. Som, S. Chakraborty, Phys. Fluids 29, 012002 (2017)
A. Srivastava, N. Tiwari, Eur. Phys. J. E 41, 56 (2018)
P.A. Kralchevsky, K.D. Danov, N.D. Denkov, Chemical Physics of Colloid Systems and Interfaces, in Handbook of Surface and Colloid Chemistry (CRC Press, 2015)
T. Bickel, Soft Matter 15, 3644 (2019)
A. Würger, J. Fluid Mech. 752, 589 (2014)
R. Piessens, The Hankel Transform, in Transforms and Applications Handbook (CRC Press, 2010)
H. Chraibi, J.-P. Delville, Phys. Fluids 24, 032102 (2012)
T. Bickel, Phys. Rev. E 75, 041403 (2007)
I.N. Sneddon, Mixed Boundary Value Problems in Potential Theory (J. Wiley & Sons, 1966)
D.G. Duffy, Mixed Boundary Value Problems (Chapman & Hall/CRC, 2008).
M.L. Cordero, E. Verneuil, F. Gallaire, C.N. Baroud, Phys. Rev. E 79, 011201 (2009)
D. Rivière, B. Selva, H. Chraibi, U. Delabre, J.-P. Delville, Phys. Rev. E 93, 023112 (2016)
M. Robert de Saint Vincent, J.-P. Delville, Phys. Rev. Fluids 1, 043901 (2016)
N. Kavokine, M. Anyfantakis, M. Morel, S. Rudiuk, T. Bickel, D. Baigl, Angew. Chem. Int. Ed. 55, 11183 (2016)
R. Leite Pinto, S. Le Roux, I. Cantat, A. Saint-Jalmes, Phys. Rev. Fluids 3, 024003 (2018)
R.V. Craster, O.K. Matar, Rev. Mod. Phys. 81, 1131 (2009)
T. Bickel, J.-C. Loudet, G. Koleski, B. Pouligny, arXiv:1909.13540 (2019)
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Bickel, T. Effect of surface-active contaminants on radial thermocapillary flows. Eur. Phys. J. E 42, 131 (2019). https://doi.org/10.1140/epje/i2019-11896-5
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DOI: https://doi.org/10.1140/epje/i2019-11896-5