Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-25T01:43:56.854Z Has data issue: false hasContentIssue false
  • English
  • Français

Stokes flow of a cylinder and half-space driven by capillarity

Published online by Cambridge University Press:  26 April 2006

Robert W. Hopper
Robert W. Hopper
Affiliation:
Chemistry and Materials Science Department, Lawrence Livermore National Laboratory. Livermore, CA 94550, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The coalescence of a cylinder with half-space by creeping viscous flow driven solely by surface tension is analysed using methods developed previously. The evolution of the shape with time is described, exactly, in terms of a time-dependent mapping function z = ω(ζ,t) of the upper half-plane, conformal on Im ζ [Gt ] 0. The results are in closed analytic form except for the time, which requires a quadrature. The height of the figure decays as t−1 as t → ∞, which is consistent with Kuiken's analysis of an isolated disturbance. (Previously, the author reported an erroneous solution which behaved otherwise.) The results are compared with the coalescence of equal cylinders obtained previously. For a modest degree of coalescence, the shapes are rather alike. In the limit as t → 0. the time dependence of the minimum widths (necks) are the same. At the times when the minimum widths disappear, the heights of the two shapes are equal.

Appended is a note providing a counter-example to earlier conjecture. A simply connected region undergoing this type of flow need not remain so.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 243 , October 1992 , pp. 171 - 181
Copyright
© 1992 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

Gradshteyn, I. S. & Rhyzhik, I. M. 1980 Table of Integrals, Series and Products, (transl. A. Jeffrey, corrected and enlarged edn). Academic.
Hopper, R. W. 1984 Coalescence of two equal cylinders — exact results for creeping viscous plane flow driven by capillarity. J. Am. Ceram. Soc. (Commun.) 67, C262–264. See also errata ibid. 68, C138 (1985).Google Scholar
Hopper, R. W. 1990 Plane stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349–375.[dagger]Google Scholar
Hopper, R. W. 1991 Plane stokes flow driven by capillarity on a free surface. Part 2. Further developments. J. Fluid Mech. 230, 355–364.[dagger]Google Scholar
Hopper, R. W. 1992 Coalescence of two viscous cylinders by capillarity. J. Am. Ceram. Soc. (to be submitted).Google Scholar
Jagota, A. & Dawson, P. R. 1988 Micromechanical modelling of powder compact — I. Unit problems for sintering and traction induced deformation. Acta Metall. 36. 2551–2561.Google Scholar
Korwin, D. M., Lange, S. R., Eaton, W. C., Joseph, I. & Pye, L. D. 1992 A study of the sintering behavior of glass in two geometric configurations. Proc. 16th Intl. Congress on Glass. In Boletin de la Sociedad Española de Cerámica y Vidrio (to appear).
Kuiken, H. K. 1990 Viscous sintering: the surface-tension-driven flow of a liquid form under the influence of curvature gradients at its surface, J. Fluid Mech. 214. 503–515.Google Scholar
Richardson, S. 1992 Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Eur. J. App. Maths. (to appear).Google Scholar