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The primary and inverse instabilities of directionalviscous fingering

Published online by Cambridge University Press:  26 April 2006

D. A. Reinelt
D. A. Reinelt
Affiliation:
Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA
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Abstract

Consider two infinitely long cylinders of different radii with oneinside the other but off-centred. The gap between the two cylindersis partially filled with a viscous fluid. As the cylinders rotatewith independent velocities U1 andU2, a thin liquid film coats each oftheir surfaces all the way around except in the region where theviscous fluid completely fills the gap. Interface conditions thatconnect solutions of averaged equations in the viscous fluid regionwith solutions in the thin film region are derived. For thetwo-interface problem analysed here, two types of instabilitiesoccur depending on the amount of viscous fluid between thecylinders. For large fluid volume, the primary supercriticalinstability occurs when the front interface becomes unstable as thecylinder velocities are increased. For small fluid volume, the backinterface passes through the region where the gap width is a minimumto the same side as the front interface. Steady state solutions withstraight interface edges exhibit a turning point with respect to thecylinder velocities. The back interface becomes unstable at theturning point; this inverse instability is subcritical.

Information

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 285 , 25 February 1995 , pp. 303 - 327
Copyright
© 1995 Cambridge University Press

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References

Bretherton, F.P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Coyle, D. J., Macosko, C. W. & Scriven, L. E. 1986 Film-splitting flows in forward roll coating. J. Fluid Mech. 171, 183–207.Google Scholar
Coyle, D. J., Macosko, C. W. & Scriven, L. E. 1990 Stability of symmetric film-splitting between counter-rotating cylinders. J. Fluid Mech. 216, 437–458.Google Scholar
Hakim, V., Rabaud, M., Thomé, H. & Couder, Y. 1990 Directional growth in viscous fingering. In New Trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium (ed. P. Coullet & P. Huerre), pp. 327–337. Plenum.CrossRefGoogle Scholar
Landau, L. & Levich, B. 1942 Dragging of a liquid by a moving plate. Acta Physicochim. URSS 17, 42–54.Google Scholar
Michalland, S. 1992 Etude des differents regimes dynamiques de l'instabilite de l'imprimeur. Thesis, L'Ecole Normale Supérieure, Paris, France.Google Scholar
Mysels, K. J., Shinoda, K. & Frankel, S. 1959 Soap Films: Studies of Their Thinning. Pergamon.Google Scholar
Pan, L. & Bruyn, J. R. DE 1993 Spatially uniform traveling cellular patterns at a driven interface. Phys. Rev. E 49, 483–493.CrossRefGoogle Scholar
Park, C.-W. & Homsy, G. M. 1984 Two-phase displacement in Hele-Shaw cells: theory. J. Fluid Mech. 139, 291–308.Google Scholar
Pearson, J. R. A. 1960 The instability of uniform viscous flow under rollers and spreaders. J. Fluid Mech. 7, 481–500.Google Scholar
Pitts, E. & Greiller, J. 1961 The flow of thin liquid films between rollers. J. Fluid Mech. 11, 33–50.Google Scholar
Rabaud, M. & Hakim, V. 1991 Shape of stationary and travelling cells in the printer's instability. In Instabilities and Nonequilibrium Structures III (ed. E. Tirapegui & W. Zeller), pp. 217–223. Kluwer Academic.CrossRefGoogle Scholar
Rabaud, M., Michalland, S. & Couder, Y. 1990 Dynamical regimes of directional viscous fingering: spatiotemporal chaos and wave propagation. Phys. Rev. Lett. 64, 184–187.Google Scholar
Reinelt, D. A. 1987 Interface conditions for two-phase displacement in Hele-Shaw cells. J. Fluid Mech. 183, 219–234.Google Scholar
Ruschak, K. J. 1982 Boundary conditions at a liquid/air interface in lubrication flows. J. Fluid Mech. 119, 107–120.Google Scholar
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. A 245, 312–329.Google Scholar
Savage, M. D. 1977 Cavitation in lubrication. Part 1. On boundary conditions and cavity-fluid interfaces. J. Fluid Mech. 80, 743–755.Google Scholar