Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T21:42:18.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Thermocapillary migration of droplets under molecular and gravitational forces

Published online by Cambridge University Press:  17 May 2018

J. R. Mac Intyre Open the ORCID record for J. R. Mac Intyre [Opens in a new window] ,
J. M. Gomba ,
Carlos Alberto Perazzo ,
P. G. Correa  and
M. Sellier
J. R. Mac Intyre*
Affiliation:
Instituto de Física Arroyo Seco - IFAS (UNCPBA) and CIFICEN (UNCPBA-CICPBA-CONICET), Pinto 399, 7000, Tandil, Argentina
J. M. Gomba
Affiliation:
Instituto de Física Arroyo Seco - IFAS (UNCPBA) and CIFICEN (UNCPBA-CICPBA-CONICET), Pinto 399, 7000, Tandil, Argentina
Carlos Alberto Perazzo
Affiliation:
IMeTTyB, Universidad Favaloro-CONICET, Solís 453, C1078AAI Buenos Aires, Argentina Departamento de Física y Química, FICEN, Universidad Favaloro, Sarmiento 1853, C1044AAA Buenos Aires, Argentina
P. G. Correa
Affiliation:
Instituto de Física Arroyo Seco - IFAS (UNCPBA) and CIFICEN (UNCPBA-CICPBA-CONICET), Pinto 399, 7000, Tandil, Argentina
M. Sellier
Affiliation:
Department of Mechanical Engineering, University of Canterbury, Private Bag 4800, Christchurch 8140, New Zealand
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniformly heated surface. The effect of a non-zero contact angle is imposed through a disjoining–conjoining pressure term. The numerical results for two different molecular interactions are compared: on the one hand, London–van der Waals and ionic–electrostatics molecular interactions that account for polar liquids; on the other hand, long- and short-range molecular forces that model molecular interactions of non-polar fluids. In addition, the effect of gravity on the velocity of the drop is analysed. We find that for small contact angles, the long-time dynamics is independent of the molecular potential, and the footprint of the droplet increases with the square root of time. For intermediate contact angles we observe that polar droplets are more likely to break up into smaller volumes than non-polar ones. A linear stability analysis allows us to predict the number of droplets after breakup occurs. In this regime, the effect of gravity is stabilizing: it reduces the growth rates of the unstable modes and increases the shortest unstable wavelength. When breakup is not observed, the droplet moves steadily with a profile that consists in a capillary ridge followed by a film of constant thickness, for which we find power law dependencies with the cross-sectional area of the droplet, the contact angle and the temperature gradients. For large contact angles, non-polar liquids move faster than polar ones, and the velocity is proportional to the Marangoni stress. We find power law dependencies for the velocity for the different regimes of flow. The numerical results allow us to shed light on experimental facts such as the origin of the elongation of droplets and the existence of saturation velocity.

JFM classification

Drops and Bubbles: Drops Low-Reynolds-number flows: Lubrication theory Drops and Bubbles: Thermocapillarity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 847 , 25 July 2018 , pp. 1 - 27
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Brochard-Wyart, F. 1989 Motion of droplets on solid surfaces induced by chemical or thermal gradients. Langmuir 5 (3), 432–438.Google Scholar
Brzoska, J. B., Brochard-Wyart, F. & Rondelez, F. 1993 Motions of droplets on hydrophobic model surfaces induced by thermal gradients. Langmuir 9 (8), 2220–2224.Google Scholar
Cachile, M., Schneemilch, M., Hamraoui, A. & Cazabat, A. M. 2002 Films driven by surface tension gradients. Adv. Colloid Interface Sci. 96 (1), 59–74.Google Scholar
Campana, D. M., Ubal, S., Giavedoni, M. D., Saita, F. A. & Carvalho, M. S. 2016 Three dimensional flow of liquid transfer between a cavity and a moving roll. Chem. Engng Sci. 149, 169–180.CrossRefGoogle Scholar
Casadevall i Solvas, X. & DeMello, A. 2011 Droplet microfluidics: recent developments and future applications. Chem. Commun. 47 (7), 1936–1942.Google Scholar
Chaudhury, K. & Chakraborty, S. 2015 Spreading of a droplet over a nonisothermal substrate: multiple scaling regimes. Langmuir 31 (14), 4169–4175.Google Scholar
Chen, J. Z., Troian, S. M., Darhuber, A. A. & Wagner, S. 2005 Effect of contact angle hysteresis on thermocapillary droplet actuation. J. Appl. Phys. 97 (1), 014906.Google Scholar
Choi, K., Ng, A. H. C., Fobel, R. & Wheeler, A. R. 2012 Digital microfluidics. Annu. Rev. Anal. Chem. 5 (1), 413–440.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.Google Scholar
Dai, Q., Khonsari, M. M., Shen, C., Huang, W. & Wang, X. 2017 On the migration of a droplet on an incline. J. Colloid Interface Sci. 494, 8–14.Google Scholar
Derjaguin, B. V. & Churaev, N. V. 1974 Structural component of disjoining pressure. J. Colloid Interface Sci. 49 (2), 249–255.Google Scholar
Derjaguin, B. V., Rabinovich, Y. I. & Churaev, N. V. 1978 Direct measurement of molecular forces. Nature 272 (5651), 313–318.Google Scholar
Ehrhard, P. & Davis, S. H. 1991 Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365–388.Google Scholar
Eötvös, R. 1886 Ueber den Zusammenhang der Oberflächenspannung der Flüssigkeiten mit ihrem Molecularvolumen. Annalen der Physik 263 (3), 448–459.CrossRefGoogle Scholar
Ford, M. L. & Nadim, A. 1994 Thermocapillary migration of an attached drop on a solid surface. Phys. Fluids 6 (9), 3183–3185.Google Scholar
Fote, A. A., Slade, R. A. & Feuerstein, S. 1977 Thermally induced migration of hydrocarbon oil. Trans. ASME J. Lubr. Technol. 99 (2), 158–162.Google Scholar
Gaskell, P. H., Jimack, P. K., Sellier, M. & Thompson, H. M. 2004 Efficient and accurate time adaptive multigrid simulations of droplet spreading. Intl J. Numer. Meth. Fluids 45 (11), 1161–1186.Google Scholar
Glasner, K. B. & Witelski, T. P. 2003 Coarsening dynamics of dewetting films. Phys. Rev. E 67 (1), 016302.Google Scholar
Gomba, J. M., Diez, J. A., González, A. G. & Gratton, R. 2005 Spreading of a micrometric fluid strip down a plane under controlled initial conditions. Phys. Rev. E 71 (1), 016304.Google Scholar
Gomba, J. M., Diez, J. A., Gratton, R., González, A. G. & Kondic, L. 2007 Stability study of a constant-volume thin film flow. Phys. Rev. E 76 (4), 046308.Google Scholar
Gomba, J. M. & Homsy, G. M. 2009 Analytical solutions for partially wetting two-dimensional droplets. Langmuir 25 (10), 5684–5691.Google Scholar
Gomba, J. M. & Homsy, G. M. 2010 Regimes of thermocapillary migration of droplets under partial wetting conditions. J. Fluid Mech. 647, 125–142.Google Scholar
Goodwin, R. & Homsy, G. M. 1991 Viscous flow down a slope in the vicinity of a contact line. Phys. Fluids A 3 (1991), 515–528.Google Scholar
Gotkis, Y., Ivanov, I., Murisic, N. & Kondic, L. 2006 Dynamic structure formation at the fronts of volatile liquid drops. Phys. Rev. Lett. 97 (18), 186101.Google Scholar
Guggenheim, E. A. 1945 The principle of corresponding states. J. Chem. Phys. 13 (7), 253–261.Google Scholar
Huebner, A., Sharma, S., Srisa-Art, M., Hollfelder, F., Edel, J. B. & Demello, A. J. 2008 Microdroplets: a sea of applications? Lab on a Chip 8 (8), 1244–1254.CrossRefGoogle ScholarPubMed
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.Google Scholar
Karapetsas, G., Sahu, K. C. & Matar, O. K. 2013 Effect of contact line dynamics on the thermocapillary motion of a droplet on an inclined plate. Langmuir 29 (28), 8892–8906.Google Scholar
Karapetsas, G., Sahu, K. C., Sefiane, K. & Matar, O. K. 2014 Thermocapillary-driven motion of a sessile drop: effect of non-monotonic dependence of surface tension on temperature. Langmuir 30 (15), 4310–4321.Google Scholar
Karbalaei, A., Kumar, R. & Cho, H. 2016 Thermocapillarity in microfluidics – a review. Micromachines 7 (1), 13.Google Scholar
Mac Intyre, J. R., Gomba, J. M. & Perazzo, C. A. 2016 New analytical solutions for static two-dimensional droplets under the effects of long- and short-range molecular forces. J. Engng Math. 101 (1), 55–69.CrossRefGoogle Scholar
Nguyen, H. B. & Chen, J. C. 2010 A numerical study of thermocapillary migration of a small liquid droplet on a horizontal solid surface. Phys. Fluids 22 (6), 062102.Google Scholar
Oron, A. & Bankoff, S. G. 2001 Dynamics of a condensing liquid film under conjoining/disjoining pressures. Phys. Fluids 13 (5), 1107–1117.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931–980.Google Scholar
Perazzo, C. A. & Gratton, J. 2004 Navier–Stokes solutions for parallel flow in rivulets on an inclined plane. J. Fluid Mech. 507, 367–379.Google Scholar
Perazzo, C. A., Mac Intyre, J. R. & Gomba, J. M. 2014 Final state of a perturbed liquid film inside a container under the effect of solid–liquid molecular forces and gravity. Phys. Rev. E 89 (4), 043010.Google ScholarPubMed
Pratap, V., Moumen, N. & Subramanian, R. S. 2008 Thermocapillary motion of a liquid drop on a horizontal solid surface. Langmuir 24 (7), 5185–5193.Google Scholar
Schwartz, L. W. & Eley, R. R. 1998 Simulation of droplet motion on low-energy and heterogeneous surfaces. J. Colloid Interface Sci. 202 (1), 173–188.Google Scholar
Schwartz, L. W. & Roy, R. V. 2004 Theoretical and numerical results for spin coating of viscous liquids. Phys. Fluids 16 (3), 569–584.CrossRefGoogle Scholar
Smith, M. K. 1995 Thermocapillary migration of a two-dimensional liquid droplet on a solid surface. J. Fluid Mech. 294, 209–230.Google Scholar
Solomentsev, Y. & White, L. R. 1999 Microscopic drop profiles and the origins of line tension. J. Colloid Interface Sci. 218, 122–136.Google Scholar
Starov, V. M., Velarde, M. G. & Radke, C. J. 2007 Wetting and Spreading Dynamics, Surfactant Science Series, p. 11. CRC Press.Google Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices. Annu. Rev. Fluid Mech. 36 (1), 381–411.Google Scholar
Sur, J., Bertozzi, A. L. & Behringer, R. P. 2003 Reverse undercompressive shock structures in driven thin film flow. Phys. Rev. Lett. 90 (12), 126105.Google Scholar
Ubal, S., Campana, D. M., Giavedoni, M. D. & Saita, F. A. 2008 Stability of the steady-state displacement of a liquid plug driven by a constant pressure difference along a prewetted capillary tube. Ind. Engng Chem. Res. 47 (16), 6307–6315.Google Scholar