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Buoyancy-driven instabilities of miscible two-layer stratifications in porous media and Hele-Shaw cells

Published online by Cambridge University Press:  31 January 2011

P. M. J. TREVELYAN ,
C. ALMARCHA  and
A. DE WIT
P. M. J. TREVELYAN
Affiliation:
Nonlinear Physical Chemistry Unit, Center for Nonlinear Phenomena and Complex Systems, Faculté des Sciences, Université Libre de Bruxelles (ULB), CP 231, 1050 Brussels, Belgium
C. ALMARCHA
Affiliation:
Nonlinear Physical Chemistry Unit, Center for Nonlinear Phenomena and Complex Systems, Faculté des Sciences, Université Libre de Bruxelles (ULB), CP 231, 1050 Brussels, Belgium
A. DE WIT*
Affiliation:
Nonlinear Physical Chemistry Unit, Center for Nonlinear Phenomena and Complex Systems, Faculté des Sciences, Université Libre de Bruxelles (ULB), CP 231, 1050 Brussels, Belgium
*
†Email address for correspondence: adewit@ulb.ac.be
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Abstract

Buoyancy-driven instabilities of a horizontal interface between two miscible solutions in the gravity field are theoretically studied in porous media and Hele-Shaw cells (two glass plates separated by a thin gap). Beyond the classical Rayleigh–Taylor (RT) and double diffusive (DD) instabilities that can affect such two-layer stratifications right at the initial time of contact, diffusive-layer convection (DLC) as well as delayed-double diffusive (DDD) instabilities can set in at a later time when differential diffusion effects act upon the evolving density profile starting from an initial step-function profile between the two miscible solutions. The conditions for these instabilities to occur can therefore be obtained only by considering time evolving base-state profiles. To do so, we perform a linear stability analysis based on a quasi-steady-state approximation (QSSA) as well as nonlinear simulations of a diffusion–convection model to classify and analyse all possible buoyancy-driven instabilities of a stratification of a solution of a given solute A on top of another miscible solution of a species B. Our theoretical model couples Darcy's law to evolution equations for the concentration of species A and B ruling the density of the miscible solutions. The parameters of the problem are a buoyancy ratio R quantifying the ratio of the relative contribution of B and A to the density as well as δ, the ratio of diffusion coefficients of these two species. We classify the region of RT, DD, DDD and DLC instabilities in the (R, δ) plane as a function of the elapsed time and show that, asymptotically, the unstable domain is much larger than the one captured on the basis of linear base-state profiles which can only obtain stability thresholds for the RT and DD instabilities. In addition the QSSA allows one to determine the critical time at which an initially stable stratification of A above B can become unstable with regard to a DDD or DLC mechanism when starting from initial step function profiles. Nonlinear dynamics are also analysed by a numerical integration of the full nonlinear model in order to understand the influence of R and δ on the dynamics.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection in porous media Convection: Double diffusive convection
Type
Papers
Information
Journal of Fluid Mechanics , Volume 670 , 10 March 2011 , pp. 38 - 65
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Almarcha, C., Trevelyan, P. M. J., Grosfils, P. & De Wit, A. 2010 a Chemically driven hydrodynamic instabilities. Phys. Rev. Lett. 104, 044501.CrossRefGoogle ScholarPubMed
Almarcha, C., Trevelyan, P. M. J., Riolfo, L. A., Zalts, A., El Hasi, C., D'Onofrio, A. & De Wit, A. 2010 b Active role of a color indicator in buoyancy-driven instabilities of chemical fronts. J. Phys. Chem. Lett. 1, 752–757.CrossRefGoogle Scholar
Cooper, C. A., Glass, R. J. & Tyler, S. W. 1997 Experimental investigation of the stability boundary for double-diffusive finger convection in a Hele-Shaw cell. Water Resour. Res. 33, 517–526.CrossRefGoogle Scholar
De Wit, A. 2001 Fingering of chemical fronts in porous media. Phys. Rev. Lett. 87, 054502.CrossRefGoogle ScholarPubMed
De Wit, A., Bertho, Y. & Martin, M. 2005 Viscous fingering of miscible slices. Phys. Fluids 17, 054114.CrossRefGoogle Scholar
D'Hernoncourt, J., De Wit, A. & Zebib, A. 2007 Double-diffusive instabilities of autocatalytic chemical fronts. J. Fluid Mech. 576, 445–456.CrossRefGoogle Scholar
D'Hernoncourt, J., Zebib, A. & De Wit, A. 2006 Reaction driven convection around a stably stratified chemical front. Phys. Rev. Lett. 96, 154501.CrossRefGoogle ScholarPubMed
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
Griffiths, R. W. 1981 Layered double-diffusive convection in porous media. J. Fluid Mech. 102, 221–248.CrossRefGoogle Scholar
Huppert, H. E. & Manins, P. C. 1973 Limiting conditions for salt-fingering at an interface. Deep Sea Res. 20 315–323.Google Scholar
Huppert, H. E. & Turner, J. S. 1981 Double diffusive convection. J. Fluid Mech. 106, 299–329.CrossRefGoogle Scholar
Kalliadasis, S., Yang, J. & De Wit, A. 2004 Fingering instabilities of exothermic reaction–diffusion fronts in porous media. Phys. Fluids 16, 1395–1409.CrossRefGoogle Scholar
Kunze, E. 2003 A review of oceanic salt-fingering theory. Prog. Oceanogr. 56, 399–417.CrossRefGoogle Scholar
Linden, P. F. & Shirtcliffe, T. G. L. 1978 The diffusive interface in double diffusive convection. J. Fluid Mech. 87, 417–432.CrossRefGoogle Scholar
Manickam, O. & Homsy, G. M. 1995 Fingering instabilities in vertical miscible displacement flows in porous media. J. Fluid Mech. 288, 75–102.CrossRefGoogle Scholar
Martin, J., Rakotomalala, N. & Salin, D. 2002 Gravitational instability of miscible fluids in a Hele-Shaw cell. Phys. Fluids 14, 902–905.CrossRefGoogle Scholar
Nield, D. A. 1968 Onset of thermohaline convection in a porous medium. Water Resour. Res. 4, 553–560.CrossRefGoogle Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media. Springer.Google Scholar
Pringle, S. E. & Glass, R. J. 2002 Double-diffusive finger convection: influence of concentration at fixed buoyancy ratio. J. Fluid Mech. 462, 161–183.CrossRefGoogle Scholar
Pritchard, D. 2009 The linear stability of double-diffusive miscible rectilinear displacements in a Hele-Shaw cell. Eur. J. Mech. B/Fluids 28, 564–577.CrossRefGoogle Scholar
Sasaki, S. 1996 Development of coupled molecular diffusion to convective fingering. J. Phys. Chem. 100, 20164–20171.CrossRefGoogle Scholar
Schmitt, R. W. 2003 Observational and laboratory insights into salt finger convection. Prog. Oceanogr. 56, 419–433.CrossRefGoogle Scholar
Shirtcliffe, T. G. L. 1973 Transport and profile measurements of the diffusive interface in double diffusive convection with similar diffusivities. J. Fluid Mech. 57, 27–43.CrossRefGoogle Scholar
Stamp, A. P., Hughes, G. O., Nokes, R. I. & Griffiths, R. W. 1998 The coupling of waves and convection. J. Fluid Mech. 372, 231–271.CrossRefGoogle Scholar
Stern, M. E. 1960 The salt-fountain and thermohaline convection. Tellus 12, 172–175.CrossRefGoogle Scholar
Stern, M. E. & Turner, J. S. 1969 Salt fingers and convective layers. Deep Sea Res. 16, 497–511.Google Scholar
Stommel, H., Arons, A. B. & Blanchard, D. 1956 An oceanographic curiosity: the perpetual salt fountain. Deep-Sea Res. 3, 152–153.Google Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements in porous media: rectilinear flow. Phys. Fluids 29, 3549–3556.CrossRefGoogle Scholar
Tan, C. T. & Homsy, G. M. 1988 Simulation of nonlinear viscous fingering in miscible displacement. Phys. Fluids 31, 1330–1338.CrossRefGoogle Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Turner, J. S. & Chen, C. F. 1974 Two-dimensional effects in double diffusive convection. J. Fluid Mech. 63, 577–592.CrossRefGoogle Scholar
Turner, J. S. & Stommel, H. 1964 A new case of convection in the presence of combined vertical salinity and temperature gradients. Proc. Natl Acad. Sci. USA 52, 49–53.CrossRefGoogle ScholarPubMed
Wooding, R. A. 1969 Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid Mech. 39, 477–495.CrossRefGoogle Scholar
Zalts, A., El Hasi, C., Rubio, D., Urena, A. & D'Onofrio, A. 2008 Pattern formation driven by an acid–base neutralization reaction in aqueous media in a gravitational field. Phys. Rev. E 77, 015304.CrossRefGoogle Scholar