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Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On diffusive slowdown in three-layer Hele-Shaw flows


Authors: Prabir Daripa and Gelu Pasa
Journal: Quart. Appl. Math. 68 (2010), 591-606
MSC (2000): Primary 76E17, 76T30, 76R50, 65F99, 65Q05
DOI: https://doi.org/10.1090/S0033-569X-2010-01174-3
Published electronically: May 19, 2010
MathSciNet review: 2676978
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Abstract | References | Similar Articles | Additional Information

Abstract: In a recently published article of Daripa and Pasa [Transp. Porous Media (2007) 70:11-23], the stabilizing effect of diffusion in three-layer Hele-Shaw flows was proved using an exact analysis of normal modes. In particular, this was established from an upper bound on the growth rate of instabilities which was derived from analyzing stability equations. However, the method used there is not constructive in the sense that the upper bound derived from actual numerical discretization of the problem could be significantly different from the exact one reported depending on the scheme used. In this paper, a numerical approach to solve the stability equations using a finite difference scheme is presented and analyzed. An upper bound on the growth rate is derived from numerical analysis of the discrete system which also shows the diffusive slowdown of instabilities. Upper bounds obtained by this numerical approach and by the analytical approach are compared. The present approach is constructive and directly leads to the implementation of the numerical approach to obtain approximate solutions in the presence of diffusion. The contributions of the paper are the novelty of the approach and a bound on the growth rates that does not depend on the solution itself.

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References
  • Robert Almgren, Crystalline Saffman-Taylor fingers, SIAM J. Appl. Math. 55 (1995), no. 6, 1511–1535. MR 1358787, DOI https://doi.org/10.1137/S0036139993259908
  • R. L. Chouke, P. van Meurs, and C. Van der Poel, The stability of a slow, immiscible, viscous liquid-liquid displacement in a permeable media, Petrol. Trans. AIME. 216, 188-194 (1959)
  • Prabir Daripa and G. Paşa, New bounds for stabilizing Hele-Shaw flows, Appl. Math. Lett. 18 (2005), no. 11, 1293–1303. MR 2170886, DOI https://doi.org/10.1016/j.aml.2005.02.027
  • Prabir Daripa and G. Paşa, On the growth rate for three-layer Hele-Shaw flows: variable and constant viscosity cases, Internat. J. Engrg. Sci. 43 (2005), no. 11-12, 877–884. MR 2163169, DOI https://doi.org/10.1016/j.ijengsci.2005.03.006
  • P. Daripa and G. Pasa, A simple derivation of an upper bound in the presence of viscosity gradient in three-layer Hele-Shaw flows, J. Stat. Mech. 11 pages, P01014. (2006) doi:10.1088/1742-5468/2006/01/P01014
  • Prabir Daripa and G. Paşa, Stabilizing effect of diffusion in enhanced oil recovery and three-layer Hele-Shaw flows with viscosity gradient, Transp. Porous Media 70 (2007), no. 1, 11–23. MR 2336218, DOI https://doi.org/10.1007/s11242-007-9122-7
  • Joachim Escher and Gieri Simonett, On Hele-Shaw models with surface tension, Math. Res. Lett. 3 (1996), no. 4, 467–474. MR 1406012, DOI https://doi.org/10.4310/MRL.1996.v3.n4.a5
  • F. J. Hickernell and Y. C. Yortsos, Linear stability of miscible displacement processes in porous media in the absence of dispersion, Stud. Appl. Math. 74 (1986), no. 2, 93–115. MR 836292, DOI https://doi.org/10.1002/sapm198674293
  • G. M. Homsy, Viscous fingering in porous media, Ann. Rev. Fluid Mech. 19, 271-311 (1987)
  • S. D. Howison, Complex variable methods in Hele-Shaw moving boundary problems, European J. Appl. Math. 3 (1992), no. 3, 209–224. MR 1182213, DOI https://doi.org/10.1017/S0956792500000802
  • Sheldon B. Gorell and G. M. Homsy, A theory of the optimal policy of oil recovery by secondary displacement processes, SIAM J. Appl. Math. 43 (1983), no. 1, 79–98. MR 687791, DOI https://doi.org/10.1137/0143007
  • L. P. Kadanoff, Exact Solutions for the Saffman-Taylor Problem with Surface Tension, Phys. Rev. Letts. 65, 2986-2988 (1990)
  • D. A. Kessler, J. Koplik, and H. Levine, Pattern selection in fingered growth phenomena, Adv. in Phys. 37, 255-329 (1998)
  • W. Littman, Polymer Flooding: Developments in Petroleum Science. 24 Elsevier, Amsterdam (1988)
  • D. Loggia, N. Rakotomalala, D. Salin, and Y. C. Yortsos, The effect of mobility gradients on viscous instabilities in miscible flows in porous media, Phys. Fluids. 11(3), 740-742 (1999)
  • J. W. McLean and P. G. Saffman, The effect of surface tension on the shape of fingers in a Hele-Shaw cell, J. Fluid Mech. 102, 445-469 (1981)
  • R. B. Needham and P. H. Doe, Polymer flooding review, J. Pet. Technol. 12, 1503-1507 (1987)
  • Qing Nie and Fei Ran Tian, Singularities in Hele-Shaw flows, SIAM J. Appl. Math. 58 (1998), no. 1, 34–54. MR 1610037, DOI https://doi.org/10.1137/S0036139996297924
  • P. G. Saffman, Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium or Hele-Shaw cell, Quart. J. Mech. Appl. Math. 12 (1959), 146–150. MR 104422, DOI https://doi.org/10.1093/qjmam/12.2.146
  • P. G. Saffman, Viscous fingering in Hele-Shaw cells, J. Fluid Mech. 173 (1986), 73–94. MR 877015, DOI https://doi.org/10.1017/S0022112086001088
  • P. G. Saffman and Geoffrey Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. Roy. Soc. London Ser. A 245 (1958), 312–329. (2 plates). MR 97227, DOI https://doi.org/10.1098/rspa.1958.0085
  • Peter J. Schmid and Dan S. Henningson, Stability and transition in shear flows, Applied Mathematical Sciences, vol. 142, Springer-Verlag, New York, 2001. MR 1801992
  • D. Shah and R. Schecter, Improved Oil Recovery by Surfactants and Polymer Flooding, Academic Press, New York, 1977
  • M. Shariati and Y. C. Yortsos, Stability of miscible displacements across stratified porous media, Phys. Fluids 13(8) 2245-2257 (2001)
  • K. S. Sorbie, Polymer-Improved Oil Recovery. CRC Press, Boca Raton, Florida, 1991
  • S. Tanveer, Surprises in viscous fingering, J. Fluid Mech. 409 (2000), 273–308. MR 1756392, DOI https://doi.org/10.1017/S0022112099007788
  • Fei Ran Tian, A Cauchy integral approach to Hele-Shaw problems with a free boundary: the case of zero surface tension, Arch. Rational Mech. Anal. 135 (1996), no. 2, 175–196. MR 1418464, DOI https://doi.org/10.1007/BF02198454
  • A. C. Uzoigwe, F. C. Scanlon, and R. L. Jewett, Improvement in polymer flooding: The programmed slug and the polymer-conserving agent, J. Petrol. Tech. 26, 33-41 (1974)
  • G. L. Vasconceles and L. P. Kadanoff, Stationary solutions for the Saffman-Taylor problem with surface tension Phys. Rev. A. 44, 6490-6495 (1991)
  • Y. C. Yortsos and M. Zeybek, Dispersion driven instability in miscible displacement in porous-media, Phys. Fluids 31, 3511-3518 (1988)

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Additional Information

Prabir Daripa
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: daripa@math.tamu.edu

Gelu Pasa
Affiliation: Institute of Mathematics “Simion Stoillow” of Romanian Academy, Bucharest, Romania 70700
Email: Gelu.Pasa@imar.ro

Received by editor(s): February 1, 2009
Published electronically: May 19, 2010
Article copyright: © Copyright 2010 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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