Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-19T09:20:08.272Z Has data issue: false hasContentIssue false
  • English
  • Français

Interface conditions for two-phase displacement in Hele-Shaw cells

Published online by Cambridge University Press:  21 April 2006

D. A. Reinelt
D. A. Reinelt
Affiliation:
Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In displacing a viscous fluid from the gap between two closely spaced parallel plates, a thin film of the original fluid remains on the surface of each plate. Boundary conditions which connect the approximate equations in the region in front of the interface with the approximate solutions in the thin-film region are determined from local solutions of the equations in the vicinity of the interface edge. These interface conditions depend on both b/R (gap half-width/radius of curvature) and μUn/T, where μ is the viscosity of the original fluid, Un is the normal velocity of the interface edge, and T is the interfacial tension. These conditions are determined using perturbation method when μUn/T [Lt ] 1 and numerical methods when μUn/T is O(1). Though previous theories have shown qualitative agreement with experiments, it is hoped that these new boundary conditions improve the quantitative agreement.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 183 , October 1987 , pp. 219 - 234
Copyright
© 1987 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

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Lamb, H. 1932 Hydrodynamics, sixth edn. Dover.
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
Nittman, J., Daccord, G. & Stanley, H. E. 1985 Fractal growth of viscous fingers: quantitative characterization of a fluid instability phenomenon. Nature 314, 141–144.Google Scholar
Park, C. W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Park, C. W. & Homsy, G. M. 1985 The instability of long fingers in Hele-Shaw flows. 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 tube. SIAM J. Sci. Stat. Comput. 6, 542–561.Google Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Tabeling, P., Zocchi, G. & Libchaber, A. 1987 An experimental study of the Saffman-Taylor instability. J. Fluid Mech. 117, 67–82.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
Vanden-Broeck, J.-M. 1983 Fingers in a Hele-Shaw cell with surface tension. Phys. Fluids 26, 2033–2034.Google Scholar