DOI: 10.1017/s0022112086002604. A boundary-integral method for two-phase displacement in Hele-Shaw cells

A boundary-integral method for two-phase displacement in Hele-Shaw cells

10.1017/s0022112086002604

Crossref journal-article

Cambridge University Press (CUP)

Journal of Fluid Mechanics (56)

Abstract

We develop a time-dependent numerical algorithm, using a boundary-integral approach, to investigate fingering in Hele-Shaw cells. Starting from a sinusoidal variation in the initial interface, stable fingers quickly form for a wide range of the dimensionless surface-tension parameter. For very low values of the parameter, the incipient finger bifurcates. The stable fingers are clearly the same as those obtained by McLean & Saffman (1981) using a steady-state algorithm. These steady-state solutions were found to be linearly unstable. We resolve this apparent discrepancy regarding stability by tracing the fate of small disturbances placed on and about the finger tip. We show that some small disturbances do, indeed, grow initially; however, they reach a maximum amplitude and decay as they convect backward from the tip of the finger to regions where stabilizing surface tension is the major physical force. Relatively large imposed disturbances, on the other hand, cause a finger to bifurcate; the critical disturbance amplitude decreases as the surface tension is reduced

Bibliography

Degregoria, A. J., & Schwartz, L. W. (1986). A boundary-integral method for two-phase displacement in Hele-Shaw cells. Journal of Fluid Mechanics, 164, 383â400.

Authors 2

A. J. Degregoria (first)
L. W. Schwartz (additional)

References 22 Referenced 132

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.
Botha, J. F. & Pinder, G. F. 1983 Fundamental Concepts in the Numerical Solution of Differential Equations. Wiley. (10.1115/1.3167711)
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.
Chouke, R. L. , van Meurs, P. & van Der Poel, C. 1959 The instability of slow, immiscible, viscous, liquid—liquid displacements in permeable media.Trans. AIME 216,188–194.
Nittmann, J. , Daccord, G. & Stanley, H. E. 1985 Fractal growth of viscous fingers: quantitative characterization of a fluid instability phenomenon.Nature 314,141–144. (10.1038/314141a0)
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. A245,312–329.
Jebatjd, J. R. , Davis, H. T. & Scriven, L. E. 1984 SPE/DOE Preprint No. 12691.
Vanden-Broeck, J. M. 1983 Fingers in a Hele-Shaw cell with surface tension.Phys. Fluids 26,2033–34.
Jaswon, M. A. & Symm, G. T. 1977 Integral Equation Methods in Potential Theory and Elastostatics. Academic. (10.1115/1.3424468)
Tryggvason, G. & Aref, H. 1983 Numerical experiments on Hele-Shaw flow with a sharp interface.J. Fluid Mech. 136,1–30.
Peters, E. J. & Flock, D. L. 1981 The onset of instability during two-phase immiscible displacement in porous media.SPE J. 21,249–258.
Romero, L. 1982 Ph.D. thesis, California Institute of Technology.
Park, C.-W. & Homsy, G. M. 1985 The instability of long fingers in Hele-Shaw flows.Phys. Fluids 28,1583–1585.
Hindmarsh, A. C. , Livermore Solver for Ordinary Differential Equations (L8ODE) Sept. 23,1980 version, Lawrence Livermore Laboratory, Livermore CA.
Paterson, L. 1981 Radial fingering in a Hele-Shaw cell.J. Fluid Mech. 113,513–529.
Meng, J. C. S. & Thomson, J. A. L. 1978 Numerical studies of some nonlinear hydrodynamic problems by discrete vortex element methods.J. Fluid Mech. 84,433–453.
Pitts, E. 1980 Penetration of fluid into a Hele-Shaw cell: the Saffman—Taylor experiment.J. Fluid Mech. 97,53–64.
Tryggvason, G. & Ahef, H. 1985 Finger interaction mechanisms in stratified Hele-Shaw flow.J. Fluid Mech. 154,287–301.
Wehausen, J. & Laitone, E. 1960 Surface waves. Handbuch der Physik, vol. 9 (ed. S. Flügge ), pp. 446–778.
DeGregoria, A. J. & Schwartz, L. W. 1985 Finger breakup in Hele-Shaw cells.Phys. Fluids 28,2313–2314.
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory.J. Fluid Mech. 139,291–308.
Reshotko, E. 1976 Boundary-layer stability and transition.Ann. Rev. Fluid Mech. 8,311–348.

Dates

Type	When
Created	19 years, 4 months ago (April 21, 2006, 3:59 a.m.)
Deposited	6 years, 2 months ago (June 7, 2019, 11:59 a.m.)
Indexed	1 week, 5 days ago (Aug. 21, 2025, 1:18 p.m.)
Issued	39 years, 6 months ago (March 1, 1986)
Published	39 years, 6 months ago (March 1, 1986)
Published Online	19 years, 4 months ago (April 21, 2006)
Published Print	39 years, 6 months ago (March 1, 1986)

Funders 0

None

BibTeX

@article{Degregoria_1986, title={A boundary-integral method for two-phase displacement in Hele-Shaw cells}, volume={164}, ISSN={1469-7645}, url={http://dx.doi.org/10.1017/s0022112086002604}, DOI={10.1017/s0022112086002604}, journal={Journal of Fluid Mechanics}, publisher={Cambridge University Press (CUP)}, author={Degregoria, A. J. and Schwartz, L. W.}, year={1986}, month=mar, pages={383–400} }

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