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Parallel flow in Hele-Shaw cells

  • M. Zeybek (a1) and Y. C. Yortsos (a1)


We consider the parallel flow of two immiscible fluids in a Hele-Shaw cell. The evolution of disturbances on the fluid interfaces is studied both theoretically and experimentally in the large-capillary-number limit. It is shown that such interfaces support wave motion, the amplitude of which for long waves is governed by a set of KdV and Airy equations. The waves are dispersive provided that the fluids have unequal viscosities and that the space occupied by the inner fluid does not pertain to the Saffman-Taylor conditions (symmetric interfaces with half-width spacing). Experiments conducted in a long and narrow Hele-Shaw cell appear to validate the theory in both the symmetric and the non-symmetric cases.



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Ablowitz, M. J. & Segur H. 1981 Solitons and the Inverse Scattering Transforms. SIAM.
Aul, R. W. & Olbricht W. L. 1990 Stability of a thin annular film in pressure driven low Reynolds number flow through capillary. J. Fluid Mech. 215, 585599.
Bensimon D., Kadanoff L. P., Liang S., Shraiman, B. I. & Tang C. 1986 Viscous flows in two dimensions. Rev. Mod. Phys. 58, 977.
Burgess, D. & Foster M. R. 1990 Analysis of the boundary conditions for a Hele-Shaw bubble Phys. Fluids, 2, 11051117.
Coats K. R., Dempsey, J. R. & Anderson J. H. 1971 The use of vertical equilibrium in two dimensional simulation of three dimensional reservoir performance. Soc. Petrol. Engrs J. 11, 6371.
Collins R. E. 1967 Flow of Fluids Through Porous Media. Tulsa: Petroleum Publishing Co.
Combescot R., Dombre T., Hakim V., Pomeau, Y. & Pumir A. 1986 Shape selection of Saffman-Taylor finger. Phys. Rev. Lett. 56, 20362039.
Drazin, P. G. & Johnson R. J. 1989 Solitons: An Introduction. Cambridge University Press.
Emanuel A. S., Alameda G. K., Behrens, R. A. & Hewett T. A. 1989 Reservoir performance prediction methods based on fractal geostatistics. Soc. Petrol. Engrs Reservoir Engng 4, 311318.
Fayers F. J. 1988 An approximate model with physically interpretable parameters for representing miscible viscous fingering. Soc. Petrol. Engrs Reservoir Engng 3, 551558.
Fayers, F. J. & Newley T. M. J. 1988 Detailed validation of an empirical model for viscous fingering with gravity effects. Soc. Petrol. Engrs Reservoir Engng 3, 542551.
Fornberg, B. & Whitham G. B. 1978 A numerical and theoretical study of certain nonlinear wave phenomena Proc. R. Soc. Lond. A 289, 373403.
Hammack J., Scheffner, N. & Segur H. 1989 Two-dimensional periodic waves in shallow water. J. Fluid Mech. 209, 567589.
Homsy G. M. 1987 Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271311.
Kadanoff L. P. 1990 Exact solutions for the Saffman-Taylor problem with surface tension. Phys. Rev. Lett. 65, 29862988.
Kalaydjian, F. & Legait B. 1987 éAcoulement lent aG contre-courant en imbibition spontaneAe de deux fluides non miscibles dans capillaire preAsentant un reAtreAcissement. C.R. Acad. Sci. Paris 304, 869875.
Kevorkian, J. & Cole J. D. 1980 Peturbation Methods in Applied Mathematics. Springer.
King, M. J. & Scher H. 1990 Geometric dispersion and unstable flow Phys. Rev. A 41, 874883.
King P. R. 1987 The fractal nature of viscous fingering in porous media. J. Phys., Paris 20, L529L534.
Lee J., Coniglio, A. & Stanley H. E. 1990 Fractal-to-nonfractal crossover for viscous fingers Phys. Rev. A 41, 45894592.
Lenormand R. 1989 Flow through porous media: limits of fractal patterns Proc. R. Soc. Lond. A 423, 159168.
Lenormand R., Kalaydjian F., Bieber, M. T. & Lombard J. M. 1990 Use of a multifractal approach for multiphase flow in heterogeneous porous media: comparison with CT-scanning experiment. Paper SPE20475 presented at the 65th SPE Annual Conference, New Orleans, LA, 22–26 Sept.
Lenormand R., Touboul, E. & Zarcone C. 1988 Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189, 165187.
Lighthill J. 1978 Waves in Fluids. Cambridge University Press.
Maxworthy T. 1987 The nonlinear growth of a gravitationally unstable interface in a Hele-Shaw cell. J. Fluid Mech. 177, 207232.
Maxworthy T. 1989 Experimental study of interface stability in a Hele-Shaw cell Phys. Rev. A 39, 58635866.
Meiburg E. 1991 Stability of rising air-bubbles in a Hele-Shaw cell. Maxworthy, T. 1991 Appendix to: The stability of inclined interfaces in a Hele-Shaw cell. Phys. Fluids A (submitted).
Park C. W., Gorell, S. & Homsy G. M. 1984 Two phase displacement in Hele-Shaw cells: experiments on viscously driven instabilities. J. Fluid Mech. 141, 257287.
Park, C. W. & Homsy G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291308.
PelceA P. 1988 Dynamics of Curved Fronts. Academic.
Rothman D. H. 1990 Macroscopic laws for immiscible two phase flow in porous media: results from numerical experiments J. Geophys. Res. B 95, 86638674.
Saffman, P. G. & Taylor G. I. 1958 The penetration of a fluid into porous medium or Hele-Shaw cell containing a more viscous liquid Proc. R. Soc. Lond. A 245, 312329.
Tabeling P., Zocchi, G. & Libchaber A. 1987 An experimental study of Saffman-Taylor instability. J. Fluid Mech. 177, 6782.
Whitham G. B. 1974 Linear and Nonlinear Waves. John Wiley.
Witten, T. A. & Sanders L. M. 1981 Diffusion-Limited-Aggregation, a kinetic critical phenomenon. Phys. Rev. Lett. 47, 14001403.
Xu J. J. 1991 Global instability of viscous fingering in a Hele-Shaw cell: formation of oscillatory fingers. Paper presented at the 44th Annual APS Meeting, Tampa, AZ, 24–26 Nov.
Yokoyama, Y. & Lake L. W. 1981 The effects of capillary pressure on immiscible displacements in stratified porous media. Paper SPE 10109 presented at the 56th Annual SPE Conference, San Antonio, TX, 5–7 Oct.
Yortsos Y. C. 1990 Instabilities in displacement processes in porous media. J. Phys.: Condens. Matter 2, SA443SA448.
Yortsos, Y. C. & Zeybek M. 1989 Long waves on Hele-Shaw fingers. Paper presented at the 1989 AIChE Fall Meeting, San Francisco, CA, 5–10 Nov.
Zabusky, N. J. & Kruskal M. D. 1965 Interaction of solitons in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett. 15, 240243.
Zapata, V. J. & Lake L. W. 1981 A theoretical analysis of viscous crossflow. Paper SPE 10111 presented at the 56th Annual SPE Conference, San Antonio, TX, 5–7 Oct.
Zel'dovitch Y. B., Istratov A. G., Kidin, N. I. & Librovich V. B. 1980 Flame propagation in tubes: Hydrodynamics and stability. Combust. Sci. Tech. 24, 113.
Zeybek M. 1991 Two studies in porous medial flows: long waves in parallel flow and dispersion effects on viscous instabilities. PhD thesis, University of Southern California.
Zeybek, M. & Yortsos Y. C. 1991 Long waves in parallel flow in Hele-Shaw cells. Phys. Rev. Lett. 67, 14301433.
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Parallel flow in Hele-Shaw cells

  • M. Zeybek (a1) and Y. C. Yortsos (a1)


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