Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-23T07:17:11.606Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension

Published online by Cambridge University Press:  16 July 2009

S. Richardson
S. Richardson
Affiliation:
Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ, Scotland
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the two-dimensional quasi-steady Stokes flow of an incompressible Newtonian fluid occupying a time-dependent simply-connected region bounded by a free surface. The motion is driven by a constant surface tension acting at the free boundary so that, with the effects of gravity ignored, one expects the boundary to approach a circular form as time evolves. It is shown that, if at some initial instant the region occupied by the fluid is given by a rational conformal map of the unit disc, then it must retain this property as long as the region remains simply-connected. Moreover, its evolution may be described analytically; in simple cases this description is explicit, but in more complicated problems the numerical integration of a system of first order differential equations may be required.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 3 , Issue 3 , September 1992 , pp. 193 - 207
Copyright
Copyright © Cambridge University Press 1992

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. & Ryzhik, I. M. 1980 Tables of Integrals, Series, and Products. New York: Academic Press.Google Scholar
Hopper, R. W. 1984 Coalescence of two equal cylinders: Exact results for creeping viscous plane flow driven by capillarity. J. Am. Ceram. Soc. (Comm.) 67, C262–264. Errata, J. Fluid Mech. 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.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Richardson, S. 1968 Two-dimensional bubbles in slow viscous flows. J. Fluid Mech. 33, 475–493.CrossRefGoogle Scholar
Richardson, S. 1973 Two-dimensional bubbles in slow viscous flows. Part 2. J. Fluid Mech. 58, 115–127.CrossRefGoogle Scholar
Richardson, S. 1981 Some Hele Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263–278.CrossRefGoogle Scholar