Abstract
In the Darcy–Boussinesq–Brinkman scheme, the onset of convection in a porous horizontal layer \(\mathtt L \) with depth-dependent permeability and viscosities is investigated. The linear instability is studied and the global nonlinear stability is investigated via the auxiliary system method. By looking for symmetries and skew-symmetries hidden in the Darcy–Boussinesq–Brinkman model, a condition, in closed form, guaranteeing the global nonlinear stability is furnished. Applications to the earth’s mantle and to artificial porous materials are furnished.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In Rionero (2014b) the onset of convection in ternary porous layers with depth-dependent permeability and viscosity has been studied in the absence of Brinkman term.
References
Alloui, Z., Bennacer, R., Vasseur, P.: Variable permeability effect on convection in binary mixtures saturating a porous layer. Heat Mass Transf. 45, 1117–1127 (2009)
Capone, F., De Luca, R.: Ultimately boundedness and stability of triply diffusive mixtures in rotating porous layers under the action of Brinkman law. Int. J. of Non-Linear Mech. 47(7), 799–805 (2012a)
Capone, F., De Luca, R.: Onset of convection for ternary fluid mixtures saturating horizontal porous layers with large pores. Rendiconti Lincei Matematica e Applicazioni 23(4), 405–428 (2012b)
Capone, F., Rionero, S.: Inertia effect on the onset of convection in rotating porous layers via the “auxiliary system method”. Int. J. Non-Linear Mech. 57, 192–200 (2013)
Flavin, J.N., Rionero, S.: Qualitative Estimates for Partial Differential Equations: An Introduction. CRC Press, Boca Raton (1996)
Fontaine, FJh, Rabinowicz, M., Boulegue, J.: Permeability changes due to mineral diagenesis in fractured crust. Earth Planet. Sci. Lett. 184, 407–425 (2001)
Gantmacher, F.R.: The Theory of Matrices, vol. 2. AMS (Chelsea Plublishing), Providence (2000)
Hamdan, M.H., Kamel, M.T., Siyyam, H.I.: A permeability function for Brinkman’s equation. In: Proceedings of the 11th WSEAS International Conference on Mathematical Methods, Computational Techniques and intelligent systems. (2009)
Hamdan, M.H., Kamel, M.T.: Flow through variable permeability porous layers. Adv. Theor. Appl. Mech. 4(3), 135–145 (2011)
Kassoy, D.R., Zebib, A.: Variable viscosity effects on the onset of convection in porous media. Phys. Fluids 18, 1649–1651 (1975)
McKibbin, R.: Heat transfer in a vertically-layered porous medium heated from below. Transp. Porous Media. 1, 361–370 (1986)
Merkin, D.R.: Introduction to the Theory of Stability. Springer Texts in Applied Mathematics, vol. 24. Springer, Heidelberg (1997)
Nield, D.A., Bejan, A.: Convection in Porous Media, 4th edn. Springer, Heidelberg (2013)
Nield, D.A., Kuznetsov, A.V.: The effect of a transition layer between a fluid and a porous medium: shear flow in a channel. Transp. Porous Media 72, 477–487 (2009)
Nield, D.A., Kuznetsov, A.V.: Optimization of forced convection heat transfer in a composite porous medium channel. Transp. Porous Media 99, 349–357 (2013)
Rees, D.A.S., Pop, I.: Vertical free convection in a porous medium with variable permeability effects. Int. J. Heat Mass Transf. 43, 2565–2571 (2000)
Rionero, S.: Longtime behaviour of multicomponent fluid mixture in porous media. J. Eng. Sci. 48, 1519–1533 (2010)
Rionero, S.: Onset of convection in porous materials with vertically stratified porosity. Acta Mech. 222, 261–272 (2011)
Rionero, S.: Absence of subcritical instabilities and global non linear stability for porous ternary diffusive-convective fluid mixtures. Phys. Fluids 24(10), 104101, p. 17 (2012a)
Rionero, S.: Symmetries and skew-symmetries against onset of convection in porous layers salted from above and below. Int. J. Nonlinear Mech. 47, 61–67 (2012b)
Rionero, S.: Multicomponent diffusive-convective fluid motions in porous layers: ultimately boundedness, absence of subcritical instabilities and global non linear stability for any number of salts. Phys. Fluids 25, 054104, p. 23 (2013a)
Rionero, S.: Soret effects on the onset of convection in rotating porous layers via the “auxiliary system method”. Ricerche di Matematica 62(2), 183–208 (2013b)
Rionero, S.: Heat and mass transfer by convection in multicomponent Navier-stokes mixtures: absence of subcritical instabilities and global nonlinear stability via the Auxiliary System Method. Rend. Lincei Mat. Appl. 25(1), 1–44 (2014a)
Rionero, S.:Convection in ternary porous layers with depth-dependent permeability and viscosity. (2014b) (In Press).
Rosenberg, N.J., Spera, F.J.: Role of anisotropic and/or layered permeability in hydrothermal system. Geophys. Res. Lett. 17, 235–238 (1990)
Straughan, B.: Stability criteria for convection with large viscosity variations. Acta Mech. 61, 59–72 (1986)
Straughan, B.: Stability and wave motion in porous media. Appl. Math. Sc. 165, (2008)
Torrance, K.E., Turcotte, D.L.: Thermal convection with large viscosity variations. J. Fluid Mech. 47, 113–125 (1971)
Acknowledgments
This paper has been performed under the auspices of G.N.F.M. of I.N.d.A.M. and Leverhulm Trust, “Tipping points: mathematics, metaphors and meanings”.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Rionero, S. Influence of Depth-Dependent Brinkman Viscosity on the Onset of Convection in Ternary Porous Layers. Transp Porous Med 106, 221–236 (2015). https://doi.org/10.1007/s11242-014-0397-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-014-0397-1