Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-23T15:05:49.488Z Has data issue: false hasContentIssue false
  • English
  • Français

Spreading and bistability of droplets on differentially heated substrates

Published online by Cambridge University Press:  17 May 2013

J. B. Bostwick
J. B. Bostwick*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

An axisymmetric drop spreads on a radially heated, partially wetting solid substrate in a rotating geometry. The lubrication approximation is applied to the field equations for this thin viscous drop to yield an evolution equation that captures the dependence of viscosity, surface tension, gravity, centrifugal forces and thermocapillarity. We study the quasi-static spreading regime, whereby droplet motion is controlled by a constitutive law that relates the contact angle to the contact-line speed. Non-uniform heating of the substrate can generate both vertical and radial temperature gradients along the drop interface, which produce distinct thermocapillary forces and equivalently flows that affect the spreading process. For the non-rotating system, competition between surface chemistry (wetting) and thermocapillary flows induced by the thermal gradients gives rise to bistability in certain regions of parameter space in which the droplets converge to an equilibrium shape. The centrifugal forces that develop in a rotating geometry enlarge the bistability regions. Parameter regimes in which the droplet spreads indefinitely are identified and spreading laws are computed to compare with experimental results from the literature.

JFM classification

Drops and Bubbles: Drops and Bubbles Interfacial Flows (free surface): Interfacial Flows (free surface) Drops and Bubbles: Thermocapillarity
Type
Papers
Information
Journal of Fluid Mechanics , Volume 725 , 25 June 2013 , pp. 566 - 587
Copyright
©2013 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

Benintendi, S. W. & Smith, M. K. 1999 The spreading of a non-isothermal liquid droplet. Phys. Fluids 11, 982–989.Google Scholar
Bonn, D., Eggers, J., Indekeu, J., Meunier, J. & Rolley, E. 2009 Wetting: statics and dynamics. Rev. Mod. Phys. 81, 739–805.CrossRefGoogle Scholar
Chen, C. D. 1988 Experiments on a spreading drop and its contact angle on a solid. J. Colloid Interface Sci. 122, 60–72.CrossRefGoogle 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, 014906.Google Scholar
Daniel, S., Chadhury, M. K. & Chen, J. C. 2001 Fast drop movements resulting from the phase change on a gradient surface. Science 291, 633–636.CrossRefGoogle ScholarPubMed
Daniels, K. E., Brausch, O., Pesch, W. & Bodenschatz, E. 2008 Competition and bistability of ordered undulations and undulation chaos in inclined layer convection. J. Fluid Mech. 597, 261–282.CrossRefGoogle Scholar
Darhuber, A. A. & Troian, S. M. 2005 Principles of microfluidic actuation by modulation of surface stresses. Annu. Rev. Fluid Mech. 37, 425–455.CrossRefGoogle Scholar
Darhuber, A. A., Troian, S. M. & Wagner, S. 2002 Generation of high-resolution surface temperature distributions. J. Appl. Phys. 91, 5686–5693.CrossRefGoogle Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19, 403–435.CrossRefGoogle Scholar
Dunn, G. J., Duffy, B. R., Wilson, S. K. & Holland, D. 2009 Quasi-steady spreading of a thin ridge of fluid with temperature-dependent surface tension on a heated or cooled substrate. Q. J. Mech. Appl. Maths 62, 365–402.Google Scholar
Dussan V., E. B. 1976 The moving contact line: the slip boundary condition. J. Fluid Mech. 77, 665–684.Google Scholar
Dussan V., E. B. 1979 On the spreading of liquid on solid surfaces: static and dynamic contact lines. Annu. Rev. Fluid Mech. 11, 371–400.Google Scholar
Dussan V., E. B. & Davis, S. H. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65, 71–95.CrossRefGoogle Scholar
Ehrhard, P. 1993 Experiments on isothermal and non-isothermal spreading. J. Fluid Mech. 257, 463–483.CrossRefGoogle Scholar
Ehrhard, P. 1994 The spreading of hanging drops. J. Colloid Interface Sci. 168, 242–246.CrossRefGoogle Scholar
Ehrhard, P. & Davis, S. H. 1991 Non-isothermal spreading of liquid drops on horizontal plates. J. Fluid Mech. 229, 365–388.Google Scholar
Ford, M. L. & Nadim, A. 1994 Thermocapillary migration of an attached drop on a solid surface. Phys. Fluids 6, 3183–3185.CrossRefGoogle Scholar
de Gennes, P. G. 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.CrossRefGoogle Scholar
Greenspan, H. P. 1978 On the motion of a small viscous droplet that wets a surface. J. Fluid Mech. 84, 125–143.Google Scholar
Hocking, L. M. 1983 The spreading of a thin drop by gravity and capillarity. Q. J. Mech. Appl. Maths 36, 55–69.Google Scholar
Hocking, L. M. 1992 Rival contact-angle models and the spreading of drops. J. Fluid Mech. 239, 671–681.CrossRefGoogle Scholar
Huh, C. & Scriven, L. E. 1971 Hydrodynamic model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85–101.Google Scholar
Lopez, C. A. & Hirsa, A. H. 2008 Fast focusing using a pinned-contact liquid lens. Nature Photonics 2 9, 610–613.Google Scholar
Matar, O. K. & Craster, R. V. 2009 Dynamics of surfactant-assisted spreading. Soft Matt. 5, 3801–3809.Google Scholar
Mukhopadhyay, S. & Behringer, R. P. 2009 Wetting dynamics of thin liquid films and drops under marangoni and centrifugal forces. J. Phys.: Condens. Matter 21, 464123.Google ScholarPubMed
Mukhopadhyay, S., Murisic, N., Behringer, R. P. & Kondic, L. 2011 Evolution of droplets of perfectly wetting liquid under the influence of thermocapillary forces. Phys. Rev. E 83, 046302.Google Scholar
Nguyen, H. B. & Chen, J. C. 2010a Numerical study of a droplet migration induced by combined thermocapillary-buoyancy convection. Phys. Fluids 22, 122101.Google Scholar
Nguyen, H. B. & Chen, J. C. 2010b A numerical study of thermocapillary migration of a small liquid droplet on a horizontal solid surface. Phys. Fluids 22, 062102.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.CrossRefGoogle Scholar
Rosenblat, S. & Davis, S. H. 1985 How do liquid drops spread on solids?. In Frontiers in Fluid Mechanics (ed. Davis, S. H. & Lumley, J. L.), pp. 171–183. Springer.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
Smith, M. K. & Davis, S. H. 1983a Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. J. Fluid Mech. 132, 119–144.Google Scholar
Smith, M. K. & Davis, S. H. 1983b Instabilities of dynamic thermocapillary liquid layers. Part 2. Surface-wave instabilities. J. Fluid Mech. 132, 145–162.Google Scholar
Spaid, M. A. & Homsy, G. M. 1996 Stability of viscoelastic dynamic contact lines: an experimental study. Phys. Fluids 9, 823–833.Google Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.CrossRefGoogle Scholar
Tanner, L. H. 1979 The spreading of silicone oil on horizontal surfaces. J. Phys. D: Appl. Phys. 12, 1473.Google Scholar
Vogel, M., Ehrhard, P. & Steen, P. 2005 The electroosmotic droplet switch: countering capillarity with electrokinetics. Proc. Natl Acad. Sci. 102, 11974–11979.Google Scholar
Vogel, M. J. & Steen, P. H. 2010 Capillarity-based switchable adhesion. Proc. Natl Acad. Sci. 107, 3377–3381.Google Scholar
Zimmerman, D. S., Triana, S. A. & Lathrop, D. P. 2011 Bi-stability in turbulent, rotating spherical couette flow. Phys. Fluids 23, 065104.Google Scholar