Skip to main content Accessibility help
×
Home

Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 1. Linear stability analysis

  • L. Talon (a1) (a2), N. Goyal (a2) and E. Meiburg (a2)

Abstract

A computational investigation of variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells is presented. As a first step, two-dimensional base states are obtained by means of simulations of the Stokes equations, which are nonlinear due to the dependence of the viscosity on the local concentration. Here, the vertical position of the displacement front is seen to reach a quasisteady equilibrium value, reflecting a balance between viscous and gravitational forces. These base states allow for two instability modes: first, there is the familiar tip instability driven by the unfavourable viscosity contrast of the displacement, which is modulated by the presence of density variations in the gravitational field; second, a gravitational instability occurs at the unstably stratified horizontal interface along the side of the finger. Both of these instability modes are investigated by means of a linear stability analysis. The gravitational mode along the side of the finger is characterized by a wavelength of about one half to one full gap width. It becomes more unstable as the gravity parameter increases, even though the interface is shifted closer to the wall. The growth rate is largest far behind the finger tip, where the interface is both thicker, and located closer to the wall, than near the finger tip. The competing influences of interface thickness and wall proximity are clarified by means of a parametric stability analysis. The tip instability mode represents a gravity-modulated version of the neutrally buoyant mode. The analysis shows that in the presence of density stratification its growth rate increases, while the dominant wavelength decreases. This overall destabilizing effect of gravity is due to the additional terms appearing in the stability equations, which outweigh the stabilizing effects of gravity onto the base state.

Copyright

Corresponding author

Email address for correspondence: meiburg@engineering.ucsb.edu

References

Hide All
Aubertin, A., Gauthier, G., Martin, J., Salin, D. & Talon, L. 2009 Miscible viscous fingering in microgravity. Phys. Fluids 21, 054107.
Bacri, J.-C, Rakotomalala, N. & Salin, D. 1991 Three-dimensional miscible viscous fingering in porous media. Phys. Rev. Lett. 67, 2005.
Chen, C.-Y. & Meiburg, E. 1996 Miscible Displacement in capillary tubes. Part 2. Numerical simulations. J. Fluid Mech. 326, 5767.
d’Olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2008 Pearl and mushroom instability patterns in two miscible fluids core annular flow. Phys. Fluids 20, 24104.
d’Olce, M., Martin, J., Rakotomalala, N., Salin, D. & Talon, L. 2009 Convective/absolute instability in miscible core-annular flow. Part 1: Experiments. J. Fluid Mech. 618, 305311.
Fernandez, J., Kurowski, P., Petitjeans, P. & Meiburg, E. 2002 Density-driven unstable flows of miscible fluids in a Hele-Shaw cell. J. Fluid Mech. 451, 239260.
Govindarajan, R. 2004 Effect of miscibility on the linear instability of two-fluid channel flow. Intl J. Multiphase Flow 30, 1177.
Goyal, N. & Meiburg, E. 2004 Unstable density stratification of miscible fluids in a vertical Hele-Shaw cell: influence of variable viscosity on the linear stability. J. Fluid Mech. 516, 211238.
Goyal, N. & Meiburg, E. 2006 Miscible displacements in Hele-Shaw cells: two-dimensional base states and their linear stability. J. Fluid Mech. 558, 329355.
Goyal, N., Pichler, H. & Meiburg, E. 2007 Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability. J. Fluid Mech. 584, 357372.
Graf, F., Meiburg, E. & Härtel, C. 2002 Density-driven instabilities of miscible fluids in a Hele-Shaw cell: linear stability analysis of the three-dimensional Stokes equations. J. Fluid Mech. 451, 261282.
Härtel, C., Meiburg, E. & Necker, F. 2000 Analysis and direct numerical simulation of the flow at a gravity current head. Part 1: flow topology and front speed for slip and boundaries. J. Fluid Mech. 418, 189212.
Hinch, E. J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463465.
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19.
John, M. O., Oliveira, R. M., Heussler, F. H. C. & Meiburg, E. 2013 Variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells. Part 2. Nonlinear simulations. J. Fluid Mech. 721, 295323.
Lajeunesse, E., Martin, J., Rakotomalala, N. & Salin, D. 1997 3D instability of miscible displacements in a Hele-Shaw cell. Phys. Rev. Lett. 79, 52545257.
Lajeunesse, E., Martin, J., Rakotomalala, N., Salin, D. & Yortsos, Y. C. 1999 Miscible displacement in a Hell–Shaw cell at high rates. J. Fluid Mech. 398, 299319.
Manickam, F. J. & Homsy, G. M. 1993 Stability of miscible displacements in porous media with non-monotonic viscosity profiles. Phys. Fluids A.5, 13561367.
Martin, J., Rakotomalala, N. & Salin, D. 2002 Gravitational instability of miscible fluids in a Hele-Shaw cell. Phys. Fluids 14, 902905.
Martin, J., Rakotomalala, N., Talon, L. & Salin, D. 2011 Viscous lock-exchange in different geometries. J. Fluid Mech. 673, 132146.
Oliveira, R. M. & Meiburg, E. 2011 Miscible displacements in Hele-Shaw cells: three-dimensional Navier–Stokes simulations. J. Fluid Mech. 687, 431460.
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 3756.
Rakotomalala, N., Salin, D. & Watzky, P. 1997a Fingering in 2D parallel viscous flow. J. Phys. II France 7, 967972.
Rakotomalala, N., Salin, D. & Watzky, P. 1997b Miscible displacement between two parallel plates: BGK lattice gas simulations. J. Fluid Mech. 338, 277297.
Sahu, K. C., Ding, H., Valluri, P. & Matar, O. K. 2009a Linear stability analysis and numerical simulation of miscible two-layer channel flow. Phys. Fluids 21, 042104.
Sahu, K. C., Ding, H., Valluri, P. & Matar, O. K. 2009b Pressure-driven miscible two-fluid channel flow with density gradients. Phys. Fluids 21 (4).
Selvam, B., Merk, S., Govindarajan, R. & Meiburg, E. 2007 Stability of miscible core-annular flow with viscosity stratification. J. Fluid Mech. 592, 2349.
Selvam, B., Talon, L., Lesshaft, L. & Meiburg, E. 2009 Convective/absolute instability in miscible core-annular flow. Part 2: numerical simulation and nonlinear global modes. J. Fluid Mech. 618, 323348.
Séon, T., Hulin, J. P., Salin, D., Perrin, B. & Hinch, E. J. 2004 Buoyant mixing of miscible fluids in tilted tube. Phys. Fluids 16, 103.
Séon, T., Hulin, J. P., Salin, D., Perrin, B. & Hinch, E. J. 2005 Buoyancy driven miscible front dynamics in tilted tubes. Phys. Fluids 17 (3).
Séon, T., Hulin, J. P., Salin, D., Perrin, B. & Hinch, E. J. 2006 Laser-induced fluorescence measurements of buoyancy driven mixing in tilted tubes. Phys. Fluids 18 (4).
Séon, T., Znaien, J., Perrin, B., Hinch, E. J., Salin, D. & Hulin, J. P. 2007a Front dynamics and macroscopic diffusion in buoyant mixing in a tilted tube. Phys. Fluids 19 (12).
Séon, T., Znaien, J., Salin, D., Hulin, J. P., Hinch, E. J. & Perrin, B. 2007b Transient buoyancy-driven front dynamics in nearly horizontal tubes. Phys. Fluids 19, 123603.
Taghavi, S. M., Alba, K., Seon, T., Wielage-Burchard, K., Martinez, D. M. & Frigaard, I. A. 2012 Miscible displacement flows in near-horizontal ducts at low Atwood number. J. Fluid Mech. 696, 175214.
Taghavi, S. M., Seon, T., Martinez, D. M. & Frigaard, I. A. 2009 Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 135.
Taghavi, S. M., Seon, T., Martinez, D. M. & Frigaard, I. A. 2010 Influence of an imposed flow on the stability of a gravity current in a near horizontal duct. Phys. Fluids 22 (3), 031702.
Taghavi, S. M., Seon, T., Wielage-Burchard, K., Martinez, D. M. & Frigaard, I. A. 2011 Stationary residual layers in buoyant Newtonian displacement flows. Phys. Fluids 23 (4), 044105.
Talon, L. & Meiburg, E. 2011 Plane Poiseuille flow of miscible layers with different viscosities: instabilities in the Stokes flow regime. J. Fluid Mech. 686, 484506.
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186203.
Vanaparthy, S. H. & Meiburg, E. 2008 Variable density and viscosity, miscible displacements in capillary tubes. Eur. J. Mech. (B/Fluids 27 (3), 268289.
Vedernikov, A., Scheid, B., Istasse, E. & Legros, J. C. 2001 Viscous fingering in miscible liquids under microgravity conditions. Phys. Fluids 13 (9), S12.
Wooding, R. 1969 Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid Mech. 39, 477495.
Yih, C. S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337.
Yortsos, Y. C. & Zeybek, M. 1988 Dispersion driven instability in miscible displacement in porous media. Phys. Fluids 31, 35113518.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed