Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-20T11:14:27.651Z Has data issue: false hasContentIssue false
  • English
  • Français

Thin static drops with a free attachment boundary

Published online by Cambridge University Press:  26 April 2006

E. O. Tuck  and
L. W. Schwartz
E. O. Tuck
Affiliation:
University of Adelaide, S.A. 5001, Australia
L. W. Schwartz
Affiliation:
University of Delaware, Newark, DE 19716, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

A thin fluid drop is at rest on a plane vertical surface, supported against gravity by surface tension. The perimeter of the drop is required to lie on a given closed curve, upon which the contact angle is arbitrary. If the drop is of sufficient volume, it can wet the whole area interior to this curve. However, for any given curve, there is a certain critical volume below which this fully wetted configuration is not physically acceptable, the formal solution having negative thickness. It is suggested here as an alternative that the upper portion of the drop, above a free boundary to be determined, must drain completely. Some time-dependent computations in two dimensions are presented to illustrate this draining property. In three dimensions, the static free boundary has zero contact angle, and must be determined as part of the solution. An example solved here is that where the original boundary is a circle, and the free boundary is a non-trivial curve lying inside it, whose shape is found by numerical methods. This problem also has relevance to the shape of a raindrop on a windowpane where surface contamination prevents contact-line motion, and the drop may again be considered to be confined within a prescribed boundary.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 223 , February 1991 , pp. 313 - 324
Copyright
© 1991 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

Dussan, V. E. B.: 1979 On the spreading of liquids on solid surfaces: static and dynamic contact angles. Ann. Rev. Fluid Mech. 11, 371–400.Google Scholar
Finn, R.: 1988 Non uniqueness and uniqueness of capillary surfaces. Manuscripta Mathematica 61, 347–372.Google Scholar
de Gennes, P. G.: 1985 Wetting: statics and dynamics. Rev. Mod. Phys. 57, 827–863.Google Scholar
Levich, V. G.: 1962 Physicochemical Hydrodynamics. Prentice-Hall.
Moriarty, J. A., Schwartz, L. W. & Tuck, E. O., 1990 Unsteady spreading of liquid films with small surface tension. Rep. 296. University of Delaware, Department of Mechanical Engineering, June 1990.
Princen, H. M.: 1969 The equilibrium shape of interfaces, drops and bubbles. In Surface and Colloid Science (ed. E. Matijevic & F. Eirich), vol. 2, pp. 1–84. Interscience.
Schwartz, L. W.: 1989 Viscous flows down an inclined plane: instability and finger formation. Phys. Fluids A 1, 443–445.Google Scholar
Schwartz, L. W. & Garoff, S., 1985 Contact angle hysteresis and the shape of the three-phase line. J. Colloid and Interface Sci. 106, 422–437.Google Scholar
Tuck, E. O. & Schwartz, L. W., 1990 A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev. 32, 453–469.Google Scholar
Tuck, E. O. & Vanden–Broeck, J.–M. 1984 Influence of surface tension on jet-stripped continuous coating of sheet materials. Am. Inst. Chem. Engrs J. 30, 808–811.Google Scholar