Abstract
The process of displacement of a viscous fluid from a Hele-Shaw cell consisting of two plates separated by a small gap is investigated. The front formed when the fluid is displaced from the cell by another, lower-viscosity fluid is unstable. The lower-viscosity fluid breaks through the layer of displaced fluid and forms channels called viscous fingers. As a result, a mixing zone occupied by both displaced and displacing fluids is formed. The structure of the unstable displacement front is investigated when the surface tension forces have no effect on the shape of the fingers. This situation is realized when a water-glycerin mixture is rapidly displaced from the cell by water. Equations taking the inertial and viscous forces acting in the plane of the plates into account are obtained by averaging the Navier-Stokes equations over the cell gap. Using the equations obtained the stability of a plane displacement front traveling in the direction of its normal and the stability of the lateral surfaces of the viscous fingers is investigated when the fluid velocities are parallel to the interface. From the solution for stability of the transverse displacement front it follows that the viscous forces acting in the plane of the plates determine the finger width (when there is no surface tension). Instability also develops in the flow on the longitudinal fluid interface. In this case the destabilizing factor is the inertial forces. Under the action of this instability the fingers, in their turn, lose stability and disintegrate into viscous bubbles.
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References
P.G. Saffman and G. 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, No. 1242, 312–329 (1958).
H. J. S. Hele-Shaw, “On the Motion of a Viscous Fluid between Two Parallel Plates,” Nature 58, No. 1489, 34–36 (1898).
C.-W. Park and G.M. Homsy, “The Instability of Long Fingers in Hele-Shaw Flows,” Phys. Fluid 28, No. 6, 1583–1585 (1985).
S. Tanveer, “Surprises in Viscous Fingering,” J. Fluid Mech. 409, ZZP 273–308 (2000).
E. Meiburg and G.M. Homsy, “Nonlinear Instable Viscous Fingers in Hele-Shaw Flows. II. Numerical Simulation,” Phys. Fluid 31, No. 3, 429–439 (1987).
N.N. Smirnov, V.F. Nikitin, O.E. Ivashnyov, A. Maximenko, M. Thiercelin, A. Vedernikov, B. Scheid, E. Istasse, and J.C. Legros, “Microgravity Investigations of Instability and Mixing Flux in Frontal Displacement of Fluids,” Microgravity Sci. Technol. 15, No. 2, 35–51 (2004).
H. Schlichting, Boundary Layer Theory (McGraw-Hill, New York, 1968; Mir, Moscow, 1974).
L.I. Sedov, Mechanics of Continuous Media, Vol. 1 (Nauka, Moscow, 1973; World Scientific, Singapore, New Jersey, London, Hong Kong, 1997).
H. Lamb, Hydrodynamics (Cambridge University Press, Cambridge, 1932; Gostekhizdat, Moscow, Leningrad, 1947).
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Original Russian Text © A.V. Zvyagin, O.E. Ivashnev, O.A. Logvinov, 2007, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2007, Vol. 42, No. 4, pp. 16–26.
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Zvyagin, A.V., Ivashnev, O.E. & Logvinov, O.A. Effect of small parameters on the structure of the front formed by unstable viscous-fluid displacement from a Hele-Shaw cell. Fluid Dyn 42, 518–527 (2007). https://doi.org/10.1134/S0015462807040023
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DOI: https://doi.org/10.1134/S0015462807040023