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TECHNICAL PAPERS

Viscous Fingering in a Hele-Shaw Cell With Finite Viscosity Ratio and Interfacial Tension

[+] Author and Article Information
X. Guan, R. Pitchumani

Department of Mechanical Engineering, University of Connecticut, Storrs, CT 06269-3139

J. Fluids Eng 125(2), 354-364 (Mar 27, 2003) (11 pages) doi:10.1115/1.1524589 History: Received February 01, 2002; Revised August 15, 2002; Online March 27, 2003
Copyright © 2003 by ASME
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References

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H., Lamb, 1932, Hydrodynamics, Cambridge Univ. Press, Cambridge, UK.
Park,  C. W., and Homsy,  G. M., 1985, “The Instability of Long Fingers in Hele-Shaw Flows,” Phys. Fluids, 28(6), p. 1583.
Maxworthy,  T., 1987, “The Nonlinear Growth of a Gravitationally Unstable Interface in a Hele-Shaw Cell,” J. Fluid Mech., 177, p. 207.
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Nie,  Q., and Tian,  F. R., 1998, “Singularities in Hele-Shaw Flows,” SIAM (Soc. Ind. Appl. Math.) J. Appl. Math., 58, p. 34.
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Figures

Grahic Jump Location
(a) Calculation of boundary fractions and interface slope based on volume of fill fractions in a 3×3 stencil around each cell. Numerics at the upper-right corner of each cell denote the volume fractions of the displacing fluid (depicted by the shaded region). (b) Schematic of the “shooting” method to correct estimations on boundary fractions in situations where interface orientation has an abrupt change within a 3×3 stencil. (c) Choice of the appropriate neighboring interface cells for obtaining a good approximation of curvature of center cell in a 3×3 stencil.
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Validation of curvature calculation on y=1/2+(3/10)sin 2πx. Continuous line denotes the exact curvature as C(x)=±(1.2π2)sin 2πx/[1+(9π2/25)cos2 2πx]3/2, and the symbols represent the curvatures computed by the numerical method.
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Dimensionless width λ of a Saffman-Taylor finger as a function of 2π(3τ)1/2. Solid line is McLean and Saffman’s solution, and the symbols are simulation results using the present algorithm.
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Finger morphologies as a function of Ca and μr for the case of constant flow rate injection. The corresponding Ca and μr values for each frame are given in Table 1.
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Finger-tip location as a function of time for a constant flow rate injection. The labels correspond to the finger patterns in Fig. 4.
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Finger morphology diagram on a Ca−μr space for the case of constant flow rate injection. All the simulated cases are discerned as being one of multiscale (square), coalescence (triangle), transition (inverted-triangle), and stable (circle) fingers.
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Finger morphologies as a function of P* and μr for select cases of constant pressure injection
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Finger-tip location as a function of time for constant pressure injection, corresponding to the finger patterns in Fig. 7
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Finger morphology diagram on a P*−μr space for constant pressure injection. All the simulated cases are classified as being one of multiscale (square), coalescence (triangle), transition (inverted-triangle), and stable (circle) fingers.
Grahic Jump Location
Evolution of a multiscaled finger pattern in simulation and experiment for a constant pressure injection of air into glycerin. Frames with dark background and white fingers denote experiments, while frames with black fingers on a white background are the simulation results.
Grahic Jump Location
Evolution of a coalescing finger pattern in simulation and experiment for injection of mineral oil into glycerin. Frames with dark background and white fingers denote experiments, while frames with black fingers on a white background are the simulation results.

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