Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-06T01:09:47.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Cusp development in free boundaries, and two-dimensional slow viscous flows

Published online by Cambridge University Press:  26 September 2008

S. D. Howison  and
S. Richardson
S. D. Howison
Affiliation:
Mathematical Institute, 24–29 St Giles, Oxford 0X1 3LB, UK
S. Richardson
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, Edinburgh EH9 3JZ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider a family of problems involving two-dimensional Stokes flows with a time dependent free boundary for which exact analytic solutions can be found; the fluid initially occupies some bounded, simply-connected domain and is withdrawn from a fixed point within that domain. If we suppose there to be no surface tension acting, we find that cusps develop in the free surface before all the fluid has been extracted, and the mathematical solution ceases to be physically relevant after these have appeared. However, if we include a non-zero surface tension in the theory, no matter how small this may be, the cusp development is inhibited and the solution allows all the fluid to be removed.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 6 , Issue 5 , October 1995 , pp. 441 - 454
Copyright
Copyright © Cambridge University Press 1995

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

Bowman, F. 1961 Introduction to Elliptic Functions with Applications. Dover.Google Scholar
Constantin, P. & Pugh, M. 1993 Global solutions for small data to the Hele-Shaw problem. Nonlinearity 6, 393–415.CrossRefGoogle Scholar
Duchon, J. & Robert, R. 1984 Évolution d'une interface par capillarité et diffusion de volume: 1. Existence locale en temps. Ann. Inst. H. Poincaré 1, 361–378.CrossRefGoogle Scholar
Hohlov, Y. E. & Howison, S. D. 1993 On the classification of solutions to the zero-surface-tension model for Hele-Shaw free boundary flows. Quart. Appl. Math. 51, 777–789.CrossRefGoogle Scholar
Hohlov, Y. E., Howison, S. D., Huntingford, C. & Ockendon, J. R. 1994 A model for non-smooth free boundaries in Hele-Shaw flow. Quart. J. Mech. Appl. Math. 47, 107–128.CrossRefGoogle Scholar
Hopper, R. W. 1990 Plane Stokes flow driven by capillarity on a free surface. J. Fluid Mech. 213, 349–375.CrossRefGoogle 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.CrossRefGoogle Scholar
Hopper, R. W. 1992 Stokes flow of a cylinder and half-space driven by capillarity. J. Fluid Mech. 243, 171–181.CrossRefGoogle Scholar
Howison, S. D. 1986 Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Math. 46, 20–26.CrossRefGoogle Scholar
Jeong, J.-T. & Moffatt, H. K. 1992 Free-surface cusps associated with flow at low Reynolds number. J. Fluid Mech. 241, 1–22.CrossRefGoogle Scholar
Joseph, D. D. & Renardy, Y. Y. 1993 Fundamentals of Two-Fluid Dynamics. Part I: Mathematical Theory and Applications. Springer-Verlag.Google Scholar
Lacey, A. A., Howison, S. D., Ockendon, J. R. & Wilmott, P. 1990 Irregular morphologies in unstable Hele-Shaw free-boundary problems. Quart. J. Mech. Appl. Math. 43, 387–405.CrossRefGoogle Scholar
Lawrence, J. D. 1972 A Catalog of Special Plane Curves. Dover.Google Scholar
Richardson, S. 1972 Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618.CrossRefGoogle Scholar
Richardson, S. 1992a Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension. Euro. J. Appl. Math. 3, 193–207.CrossRefGoogle Scholar
Richardson, S. 1992b Hele-Shaw flows with time-dependent free boundaries involving injection through slits. Stud. Appl. Math. 87, 175–194.CrossRefGoogle Scholar
Saffman, P. G. 1991 Selection mechanisms and stability of fingers and bubbles in Hele-Shaw cells. IMA J. Appl. Math. 46, 137–145.CrossRefGoogle Scholar
Vasconcelos, G. L. & Kadanoff, L. P. 1991 Stationary solutions for the Saffman-Taylor problem with surface tension. Phys. Rev. A. 44, 6490–6495.CrossRefGoogle ScholarPubMed