Hostname: page-component-68945f75b7-z8dg2 Total loading time: 0 Render date: 2024-08-05T23:08:04.044Z Has data issue: false hasContentIssue false
  • English
  • Français

A theoretical study of two-phase flow through a narrow gap with a moving contact line: viscous fingering in a Hele-Shaw cell

Published online by Cambridge University Press:  26 April 2006

Steven J. Weinstein ,
E. B. Dussan  and
Lyle H. Ungar
Steven J. Weinstein
Affiliation:
Emulsion Coating Technologies, Building 35, 2nd Floor, Eastman Kodak Company, Rochester, NY 14652–3701, USA
E. B. Dussan
Affiliation:
Schlumberger-Doll Research, Old Quarry Road, Ridgefield, CT 06877, USA
Lyle H. Ungar
Affiliation:
University of Pennsylvania. Department of Chemical Engineering. 220 S. 33rd Street, Towne Building, Philadelphia, PA 19104, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The problem of viscous fingering in a Hele-Shaw cell with moving contact lines is considered. In contrast to the usual situation where the displaced fluid coats the solid surface in the form of thin films, here, both the displacing and the displaced fluids make direct contact with the solid. The principal differences between these two situations are in the ranges of attainable values of the gapwise component of the interfacial curvature (the component due to the bending of the fluid interface across the small gap of the Hele-Shaw cell), and in the introduction of two additional parameters for the case with moving contact lines. These parameters are the receding contact angle, and the sensivity of the dynamic angle to the speed of the contact line. Our objective is the prediction of the shape and widths of the fingers in the limit of small capillary number, Uμ/σ. Here, U denotes the finger speed, μ denotes the dynamic viscosity of the more viscous displaced fluid, and σ denotes the surface tension of the fluid interface. As might be expected, there are similarities and differences between the two problems. Despite the fact that different equations arise, we find that they can be analysed using the techniques introduced by McLean & Saffman and Vanden-Broeck for the thin-film case. The nature of the multiplicity of solutions also appears to be similar for the two problems. Our results indicate that when contact lines are present, the finger shapes are sensitive to the value of the contact angle only in the vicinity of its nose, reminiscent of experiments where bubbles or wires are placed at the nose of viscous fingers when thin films are present. On the other hand, in the present problem at least two distinct velocity scales emerge with well-defined asymptotic limits, each of these two cases being distinguished by the relative importance played by the two components of the curvature of the fluid interface. It is found that the widths of fingers can be significantly smaller than half the width of the cell.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 221 , December 1990 , pp. 53 - 76
Copyright
© 1990 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

Bensimon, D.: 1986 Stability of viscous fingering. Phys. Rev. A 33, 1302.Google Scholar
Bretherton, F. P.: 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166.Google Scholar
Couder, Y., Gérard, N. & Rabaud, M. 1986 Narrow fingers in the Saffman-Taylor instability. Phys. Rev. A 34, 5175.Google Scholar
Dussan, V. E. B.: 1979 On the spreading of liquids on solid surfaces: static and dynamic contact lines. Ann. Rev. Fluid Mech. 11, 371.Google Scholar
Kessler, D. A. & Levine, H., 1985 Stability patterns in Hele-Shaw cells. Phys. Rer. A 32, 1930.Google Scholar
Kopf-Sill, A. R. & Homsy, G. M. 1987 Narrow fingers in a Hele-Shaw cell. Phys. Fluids 31, 2607.Google Scholar
Kopf-Sill, A. R. & Homsy, G. M. 1988 Bubble motion in a Hele-Shaw cell. Phys. Fluids 31, 18.Google Scholar
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.Google Scholar
Ngan, C. G. & Dussan, V. E. B. 1989 On the dynamics of liquid spreading on solid surfaces. J. Fluid Mech. 209, 191.Google Scholar
Park, C. W. & Homsy, G. M., 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291.Google Scholar
Reinelt, D. A.: 1987a Interface conditions for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 183, 219.Google Scholar
Reinelt, D. A.: 1987b The effect of thin film variations and transverse curvature on the shape of fingers in a Hele-Shaw cell. Phys. Fluids 30, 2617.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 (3), 542.Google Scholar
Romero, I. A.: 1981 The fingering problem in a Hele-Shaw cell. PhD thesis, California Institute of Technology.
Saffman, P. G. & Tanveer, S., 1989 Prediction of bubble velocity in a Hele-Shaw cell: Thin film and contact angle effects. Phys. Fluids A 1 (2), 219.Google Scholar
Saffman, P. G. & Taylor, G. I., 1958 The penetration of a fluid into a porous medium of Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245, 312.Google Scholar
Schwartz, L. W. & Degregoria, A. J., 1987 Simulation of Hele-Shaw fingering with finite-capillary-number effects included. Phys. Rev. A 35, 276.Google Scholar
Tabeling, P. & Libchaber, A., 1986 Film draining and the Saffman-Taylor Problem. Phys. Rev. A 33, 794.Google Scholar
Tabeling, P., Zocchi, G. & Libchaber, A., 1987 An experimental study of the Saffman-Taylor instability. J. Fluid Mech. 177, 67.Google Scholar
Tanveer, S. & Saffman, P. G., 1987 Stability of bubbles in a Hele-Shaw cell. Phys. Fluids 30, 2624.Google Scholar
Vanden-Broeck, J. M.: 1983 Fingers in a Hele-Shaw cell with surface tension. Phys. Fluids 26, 2033.Google Scholar
Weinstein, S. J.: 1988 A theoretical study of two-phase Hele-Shaw flow with a moving contact line. PhD thesis, University of Pennsylvania.