Skip to main content Accessibility help

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

  • T. Maxworthy (a1)


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.



Hide All
Bretherton, F. P. 1961 The motion of long bubbles in a tube. J. Fluid Mech. 10, 166188.
Couder, Y., Cardoso, A., Dupuy, D., Tavernier, P. & Thom, W. 1986 Dendritic growth in the Saffman—Taylor experiment. Europhys. Lett. (to appear).
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, 383400.
Fairbrother, F. & Stubbs, A. E. 1935 The bubble tube method of measurement. J. Chem. Soc. 1, 527529.
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, 303327.
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, 455469.
Maxworthy, T. 1987 The non-linear growth of a gravitationally unstable interface in a Hele-Shaw cell. J. Fluid Mech. (in press).
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in a Hele-Shaw cell. J. Fluid Mech. 139, 291308.
Park, C.-W. & Homsy, G. M. 1985 The instability of long fingers in a Hele-Shaw cell. Phys. Fluids 28, 15831585.
Pitts, E. 1980 Penetration of fluid into a Hele—Shaw cell: the Saffman—Taylor experiment. J. Fluid Mech. 97, 5364.
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, 542561.
Riegels, F. 1938 Zur Kritik des Hele-Shaw-Versuchs, Z. angew. Math. Mech. 18, 95106.
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, 312329.
Tabeling, P. & Libchaber, A. 1986 Film draining and the Saffman—Taylor problem. Phys. Rev. A 33, 794796.
Tanveer, S. 1986 The effect of surface tension on the shape of a Hele-Shaw bubble. Phys. Fluids (submitted).
Taylor, G. I. 1961 Deposition of a viscous fluid on the wall of a tube. J. Fluid Mech. 10, 161165.
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, 265279.
Tryggvason, G. & Aref, H. 1983 Numerical experiments on Hele-Shaw flow with a sharp interface. J. Fluid Mech. 136, 130.
MathJax is a JavaScript display engine for mathematics. For more information see

Related content

Powered by UNSILO

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

  • T. Maxworthy (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.