Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T05:12:07.458Z Has data issue: false hasContentIssue false
  • English
  • Français

On the breakup of air bubbles in a Hele-Shaw cell

Published online by Cambridge University Press:  21 December 2010

VLADIMIR ENTOV  and
PAVEL ETINGOF
PAVEL ETINGOF
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA e-mail: etingof@math.mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the problem of breakup of an air bubble in a Hele-Shaw cell. In particular, we propose some sufficient conditions of breakup of the bubble, and ways to find the contraction points of its parts. We also study regulated contraction of a pair of bubbles (in which the rates of air extraction from the bubbles are controlled) and study various asymptotic questions (such as the asymptotics of contraction of a bubble to a degenerate critical point, and asymptotics of contraction of a small bubble in the presence of a big bubble)

Keywords

bubbleHele-Shaw cellcontractionmoving boundary
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 2 , April 2011 , pp. 125 - 149
Copyright
Copyright © Cambridge University Press 2010

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

[1]Entov, V. M. & Etingof, P. I. (1991) Bubble contraction in Hele-Shaw cells. Q. J. Mech. Appl. Math. 44 (4), 507–535.CrossRefGoogle Scholar
[2]Entov, V. M. & Etingof, P. I. (1997) Viscous flows with time-dependent free boundaries in a non-planar Hele-Shaw cell. Eur. J. Appl. Math. 8 (1), 23–35.CrossRefGoogle Scholar
[3]Gustafsson, B. (1983) Quadrature identities and the Schottky double. Acta Appl. Math. 1 (3), 209–240.CrossRefGoogle Scholar
[4]Gustafsson, B. & Vasiliev, A. (2006) Conformal and Potential Analysis in Hele-Shaw Cells. Advances in Mathematical Fluid Mechanics, Birkhäuser, Basel.Google Scholar
[5]Howison, S. D. (1986) Cusp development in Hele-Shaw flow with a free surface. SIAM J. Appl. Math. 46 (1), 20–26.CrossRefGoogle Scholar
[6]Kufarev, P. P. (1950) The problem of the contour of the oil-bearing region for a circle with an arbitrary number of gaps (in Russian). Dokl. Akad. Nauk SSSR (NS) 75, 507–510.Google Scholar
[7]Lee, S.-Y., Bettelheim, E. & Wiegmann, P. (2006) Bubble break-off in Hele-Shaw flows: singularities and integrable structures. Physica D 219 (1), 22–34.Google Scholar
[8]Polubarinova-Kochina, P. Y. (1945) On the displacement of the oil-bearing contour. C. R. (Dokl.) Acad. Sci. URSS (NS) 47, 250–254.Google Scholar
[9]Richardson, S. (1981) Some Hele-Shaw flows with time-dependent free boundaries. J. Fluid Mech. 102, 263–278.CrossRefGoogle Scholar
[10]Varchenko, A. N. & Etingof, P. I. (1992) Why the Boundary of a Round Drop Becomes a Curve of Order Four. American Mathematical Society, Providence, RI.Google Scholar