Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T04:19:01.462Z Has data issue: false hasContentIssue false
  • English
  • Français

Variable-density miscible displacements in a vertical Hele-Shaw cell: linear stability

Published online by Cambridge University Press:  25 July 2007

N. GOYAL ,
H. PICHLER  and
E. MEIBURG
N. GOYAL
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
H. PICHLER
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
E. MEIBURG*
Affiliation:
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93106, USA
*
† Author to whom correspondence should be addressed: meiburg@engineering.ucsb.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

A computational study based on the Stokes equations is conducted to investigate the effects of gravitational forces on miscible displacements in vertical Hele-Shaw cells. Nonlinear simulations provide the quasi-steady displacement fronts in the gap of the cell, whose stability to spanwise perturbations is subsequently examined by means of a linear stability analysis. The two-dimensional simulations indicate a marked thickening (thinning) and slowing down (speeding up) of the displacement front for flows stabilized (destabilized) by gravity. For the range investigated, the tip velocity is found to vary linearly with the gravity parameter. Strongly stable density stratifications lead to the emergence of flow patterns with spreading fronts, and to the emergence of a secondary needle-shaped finger, similar to earlier observations for capillary tube flows. In order to investigate the transition between viscously driven and purely gravitational instabilities, a comparison is presented between displacement flows and gravity-driven flows without net displacements.

The linear stability analysis shows that both the growth rate and the dominant wavenumber depend only weakly on the Péclet number. The growth rate varies strongly with the gravity parameter, so that even a moderately stable density stratification can stabilize the displacement. Both the growth rate and the dominant wavelength increase with the viscosity ratio. For unstable density stratifications, the dominant wavelength is nearly independent of the gravity parameter, while it increases strongly for stable density stratifications. Finally, the kinematic wave theory of Lajeunesse et al. (J. Fluid Mech. vol. 398, 1999, p. 299) is seen to capture the stability limit quite accurately, while the Darcy analysis misses important aspects of the instability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 584 , 10 August 2007 , pp. 357 - 372
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bacri, J., Salin, D. & Yortsos, Y. 1992 Analyse linéaire de la stabilité de l'écoulement de fluides miscibles en milieus poreux. C. R. Acad. Sci. Paris 314, 139–144.Google Scholar
Balasubramaniam, R., Rashidnia, N., Maxworthy, T. & Kuang, J. 2005 Instability of miscible interfaces in a cylindrical tube. Phys. Fluids 17 (5), 052103-1–052103-11.CrossRefGoogle Scholar
Chen, C. & Meiburg, E. 1996 Miscible displacements in capillary tubes. Part 2. Numerical simulations. J. Fluid Mech 326, 57–90.CrossRefGoogle Scholar
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, 239–260.CrossRefGoogle Scholar
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, 211–238.CrossRefGoogle Scholar
Goyal, N. & Meiburg, E. 2006 Miscible displacements in Hele-Shaw cells: two-dimensional base states and their linear stability. J. Fluid Mech. 558, 329–355.CrossRefGoogle Scholar
Graf, F. & Meiburg, E. 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, 261–282.CrossRefGoogle Scholar
Kuang, J., Maxworthy, T. & Petitjeans, P. 2003 Miscible displacements between silicone oils in capillary tubes. Eur. J. Mech. B/Fluids 22, 271–277.CrossRefGoogle Scholar
Kuang, J., Maxworthy, T. & Petitjeans, P. 2004 Velocity fields and streamline patterns of miscible displacements in cylindrical tubes. Exps. Fluids 37, 301–308.CrossRefGoogle Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N. & Salin, D. 1997 3D instability of miscible displacements in a Hele-Shaw cell. Phys. Rev. Lett. 79, 5254–5257.CrossRefGoogle Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N., Salin, D. & Yortsos, Y. 1999 Miscible displacement in a Hele-Shaw cell at high rates. J. Fluid Mech. 398, 299–319.CrossRefGoogle Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N., Salin, D. & Yortsos, Y. 2001 The threshold of the instability in miscible displacements in a Hele-Shaw cell. Phys. Fluids 13, 799–801.CrossRefGoogle Scholar
Manickam, O. & Homsy, G. 1995 Fingering instabilities in vertical miscisble displacement flows in porous media. J. Fluid Mech. 288, 75–102.CrossRefGoogle Scholar
Petitjeans, P. & Maxworthy, T. 1996 Miscible displacements in capillary tubes. Part 1. Experiments. J. Fluid Mech. 326, 37–56.CrossRefGoogle Scholar
Tan, C. & Homsy, G. 1986 Stability of miscible displacements in porous media: rectilinear flow. Phys. Fluids 29 (11), 3549–3556.CrossRefGoogle Scholar
Yang, Z. & Yortsos, Y. 1997 Asymptotic solutions of miscible displcements in geometries of large aspect ratio. Phys. Fluids 9 (2), 286–298.CrossRefGoogle Scholar