Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T06:39:45.297Z Has data issue: false hasContentIssue false
  • English
  • Français

Bubble formation, motion and interaction in a Hele-Shaw cell

Published online by Cambridge University Press:  21 April 2006

T. Maxworthy
T. Maxworthy
Affiliation:
Departments of Mechanical and Aerospace Engineering, University of Southern California, Los Angeles, CA 90089-1453, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the motion of the flattened bubbles which form when air is injected into a viscous fluid contained in the narrow gap between two flat, parallel plates which make up a conventional Hele-Shaw cell, inclined at an angle x to the horizontal. We present a number of qualitative observations on the formation and interaction of the streams of bubbles that appear when air is injected continuously into the cell. The majority of this paper is then concerned with the shape and velocity of rise of single, isolated bubbles over a wide range of bubble size and cell inclination. We compare these results to theories by Taylor & Saffman (1959), and Tanveer (1986). It appears that the bubble characteristics found by an ad hoc speculation in Taylor & Saffman (1959) and by Tanveer (1986) only agree with the experimental results in the limit α → 0, and for large bubble widths (D). For finite values of α, it is necessary to use the measured bubble shape in order to calculate the rise velocity using the more general Taylor & Saffman (1959) formulation. Deviations from these theories for small D can be explained by considering the effects of the detailed flow close to the bubble surface.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 173 , December 1986 , pp. 95 - 114
Copyright
© 1986 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

Bretherton, F. P. 1961 The motion of long bubbles in a tube. J. Fluid Mech. 10, 166–188.Google Scholar
Couder, Y., Cardoso, A., Dupuy, D., Tavernier, P. & Thom, W. 1986 Dendritic growth in the Saffman—Taylor experiment. Europhys. Lett. (to appear).Google Scholar
Davidson, J. F. & Harrison, D. 1977 Fluidisation. Academic.
Degregoria, A. J. & Schwartz, L. W. 1986 A boundary-integral method for two phase displacements in Hele-Shaw cells. J. Fluid Mech. 164, 383–400.Google Scholar
Fairbrother, F. & Stubbs, A. E. 1935 The bubble tube method of measurement. J. Chem. Soc. 1, 527–529.Google Scholar
Happel, J. & Brenner, H. 1973 Low Reynolds number hydrodynamics. Leyden: Noordhof.
Hele-Shaw, H. S. & Hay, A. 1901 Lines of induction in a magnetic field. Phil. Trans. R. Soc. Lond. A 195, 303–327.Google Scholar
Levich, B. G. 1962 Physiochemical Hydrodynamics. Prentice-Hall.
Mclean, J. W. & Saffman, P. G. 1981 The effect of surface tension on the shape of fingers in a Hele-Shaw cell. J. Fluid Mech. 102, 455–469.Google Scholar
Maxworthy, T. 1987 The non-linear growth of a gravitationally unstable interface in a Hele-Shaw cell. J. Fluid Mech. (in press).Google Scholar
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in a Hele-Shaw cell. J. Fluid Mech. 139, 291–308.Google Scholar
Park, C.-W. & Homsy, G. M. 1985 The instability of long fingers in a Hele-Shaw cell. Phys. Fluids 28, 1583–1585.Google Scholar
Pitts, E. 1980 Penetration of fluid into a Hele—Shaw cell: the Saffman—Taylor experiment. J. Fluid Mech. 97, 53–64.Google Scholar
Reinelt, D. A. & Saffman, P. G. 1985 The penetration of a finger into a viscous fluid in a channel and a tube. SIAM. J. Sci. Stat. Comput. 6, 542–561.Google Scholar
Riegels, F. 1938 Zur Kritik des Hele-Shaw-Versuchs, Z. angew. Math. Mech. 18, 95–106.Google Scholar
Shaw, R. 1984 The Dripping Faucet as a Model Chaotic System. Santa Cruz: Aerial.
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Tabeling, P. & Libchaber, A. 1986 Film draining and the Saffman—Taylor problem. Phys. Rev. A 33, 794–796.Google Scholar
Tanveer, S. 1986 The effect of surface tension on the shape of a Hele-Shaw bubble. Phys. Fluids (submitted).Google Scholar
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161–165.Google Scholar
Taylor, G. I. & Saffman, P. G. 1959 A note on the motion of bubbles in a Hele-Shaw cell and porous medium. Q. J. Mech. Appl. Maths 12, 265–279.Google Scholar
Tryggvason, G. & Aref, H. 1983 Numerical experiments on Hele-Shaw flow with a sharp interface. J. Fluid Mech. 136, 1–30.Google Scholar