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On the instability of buoyancy-driven flows in porous media

Published online by Cambridge University Press:  03 April 2020

Shyam Sunder Gopalakrishnan Open the ORCID record for Shyam Sunder Gopalakrishnan [Opens in a new window]
Shyam Sunder Gopalakrishnan*
Affiliation:
Nonlinear Physical Chemistry Unit, Université Libre de Bruxelles (ULB), CP231,1050Brussels, Belgium Physique des Systèmes Dynamique, Université Libre de Bruxelles (ULB), CP231,1050Brussels, Belgium
*
†Email address for correspondence: shyam7sunder@gmail.com
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Abstract

The interface between two miscible solutions in porous media and Hele-Shaw cells (two glass plates separated by a thin gap) in a gravity field can destabilise due to buoyancy-driven and double-diffusive effects. In this paper the conditions for instability to arise are presented within an analytical framework by considering the eigenvalue problem based on the tools used extensively by Chandrasekhar. The model considered here is Darcy’s law coupled to evolution equations for the concentrations of different solutes. We have shown that, when there is an interval in the spatial domain where the first derivative of the base-state density profile is negative, the flows are unstable to stationary or oscillatory modes. Whereas for base-state density profiles that are strictly monotonically increasing downwards such that the first derivative of the base-state density profile is positive throughout the domain (for instance, when a lighter solution containing a species A overlies a denser solution containing another species B), a necessary and sufficient condition for instability is the presence of a point on either side of the initial interface where the second derivative of the base-state density profile is zero such that it changes sign. In such regimes the instability arises as non-oscillatory modes (real eigenvalues). The neutral stability curve, which delimits the stable from the unstable regime, that follows from the discussion presented here along with the other results are in agreement with earlier observations made using numerical computations. The analytical approach adopted in this work could be extended to other instabilities arising in porous media.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection in porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 892 , 10 June 2020 , A13
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Anderson, D. M. & Worster, M. G. 1996 A new oscillatory instabilitiy in a mushy layer during the solidification of binary alloys. J. Fluid Mech. 307, 245–267.CrossRefGoogle Scholar
Carballido-Landeira, J., Trevelyan, P. M. J., Almarcha, C. & De Wit, A. 2013 Mixed-mode instability of a miscible interface due to coupling between Rayleigh–Taylor and double-diffusive convective modes. Phys. Fluids 25, 024107.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.Google Scholar
Chen, F., Lu, J. W. & Yang, T. L. 1994 Convective instability in ammonium chloride solution directtionally solidified from below. J. Fluid Mech. 276, 163–187.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
Davis, S. H. 1969 On the principle of exchange of stabilities. Proc. R. Soc. Lond. A 310, 341–358.Google Scholar
De Paoli, M., Giurgiu, V., Zonta, F. & Soldati, A. 2019a Universal behavior of scalar dissipation rate in confined porous media. Phys. Rev. F 4, 101501.Google Scholar
De Paoli, M., Zonta, F. & Soldati, A. 2019b Rayleigh–Taylor convective dissolution in confined porous media. Phys. Rev. F 4, 023502.Google Scholar
De Wit, A. 2020 Chemo-hydrodynamic patterns and instabilities. Annu. Rev. Fluid Mech. 52, 531–555.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part II. Non-autonomous operators. Oper. J. Atmos. Sci. 53, 2041–2053.2.0.CO;2>CrossRefGoogle Scholar
Fernandez, J., Kurowski, P., Limat, L. & Petitjeans, P. 2001 Wavelength selection of fingering instability inside Hele-Shaw cells. Phys. Fluids 13, 3120–3125.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
Gandhi, J. & Trevelyan, P. M. J. 2014 Onset conditions for a Rayleigh–Taylor instability with step function density profiles. J. Engng Maths 86, 31–48.CrossRefGoogle Scholar
Gopalakrishnan, S. S., Carballido-Landeira, J., De Wit, A. & Knaepen, B. 2017 Relative role of convective and diffusive mixing in the miscible Rayleigh–Taylor instability in porous media. Phys. Rev. F 2, 012501.Google Scholar
Gopalakrishnan, S. S., Carballido-Landeira, J., Knaepen, B. & De Wit, A. 2018 Control of Rayleigh–Taylor instability onset time and convective velocity by differential diffusion effects. Phys. Rev. E 98, 011101(R).Google ScholarPubMed
Green, T. 1984 Scales for double-diffusive fingering in porous media. Water Resour. Res. 20, 1225–1229.CrossRefGoogle Scholar
Griffiths, R. W. 1981 Layered double-diffusive convection in porous media. J. Fluid Mech. 102, 221–248.CrossRefGoogle Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Hota, T. K. & Mishra, M. 2018 Non-modal stability analysis of miscible viscous fingering with non-monotonic viscosity profiles. J. Fluid Mech. 856, 552–579.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. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255–272.CrossRefGoogle Scholar
Huppert, H. E. & Sparks, R. S. J. 1984 Double-diffusive convection due to crystallization in magmas. Annu. Rev. Fluid Mech. 12, 11–37.Google Scholar
Huppert, H. E. & Turner, J. S. 1981 Double-diffusive convection. J. Fluid Mech. 106, 299–329.CrossRefGoogle Scholar
Kim, M. C. & Choi, C. K. 2011 The stability of miscible displacement in porous media: nonmonotonic viscosity profiles. Phys. Fluids 23, 084105.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., Rakotomala, N. & Salin, D. 2002 Gravitational instability of miscible fluids in a Hele-Shaw cell. Phys. Fluids 14, 902–905.CrossRefGoogle Scholar
Philip, J. R. 1970 Flow in porous media. Annu. Rev. Fluid Mech. 2, 177–204.CrossRefGoogle Scholar
Pramanik, S. & Mishra, M. 2013 Linear stability analysis of Korteweg stresses effect on miscible viscous fingering in porous media. Phys. Fluids 25, 074104.CrossRefGoogle 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
Radko, T. 2013 Double Diffusive Convection. Cambridge University Press.CrossRefGoogle Scholar
Rayleigh, Lord 1880 On the stability or instability of certain fluid motions. Proc. Lond. Math. Soc. 11, 57–70.Google Scholar
Roberts, P. H. 1960 Characteristic value problems posed by differential equations arising in hydrodynamics and hydromagnetics. J. Math. Anal. Appl. 1, 195–214.CrossRefGoogle Scholar
Saffman, P. & Taylor, G. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell contaning a more viscous liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Sardina, G., Brandt, L., Boffetta, G. & Mazzino, A. 2018 Buoyancy-driven flow through a bed of solid particles produces a new form of Rayleigh–Taylor turbulence. Phys. Rev. Lett. 121 (22), 224501.CrossRefGoogle Scholar
Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26, 255–285.CrossRefGoogle Scholar
Schmitt, R. W., Ledwell, J. R., Montgomery, E. T., Polzin, K. L. & Toole, J. M. 2005 Enhanced diapycnal mixing by salt fingers in the thermocline of the tropical Atlantic. Science 308, 685–688.CrossRefGoogle ScholarPubMed
Slim, A. C. 2014 Solutal-convection regimes in a two-dimensional porous medium. J. Fluid Mech. 741, 461–491.CrossRefGoogle Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements in porous media: rectilinear flow. Phys. Fluids 29, 3549.CrossRefGoogle Scholar
Taylor, G. I. 1915 Eddy motion in the atmosphere. Phil. Trans. R. Soc. 215, 1–26.Google Scholar
Trevelyan, P. M. J., Almarcha, C. & De Wit, A. 2011 Buoyancy-driven instabilities of miscible two-layer stratifications in porous media and Hele-Shaw cells. J. Fluid Mech. 670, 38–65.CrossRefGoogle Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Turner, J. S. 1985 Multicomponent convection. Annu. Rev. Fluid Mech. 17, 11–44.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