# American Institute of Mathematical Sciences

September  2014, 3(3): 525-539. doi: 10.3934/eect.2014.3.525

## On the nonlinear stability of ternary porous media via only one necessary and sufficient algebraic condition

 1 University of Naples Federico II, Department of Mathematics and Applications, "Renato Caccioppoli", Via Cinzia 80126. Naples

Received  March 2013 Revised  December 2013 Published  August 2014

Research on convective-diffusive fluid motions in porous media has a notable relevance (increasing with the number of salts dissolved in the fluid) either for geophysical applications (engineering geology, subsurface and structural geology, subsurface contaminant transport, underground water flow, ...) or because porous materials occur very frequently in real life (fiber materials for insulating purposes, metallic foams in heat transfer devices ([1,13]). In the present paper porous horizontal layers, heated from below and salted from above and below, in the Darcy-Boussinesq scheme, are investigated. By virtue of the absence of subcritical instabilities ([20]), the bifurcating competition of Rayleigh and Prandtl numbers for promoting or inhibiting the onset of convection, investigated through the linearized equations, allows to show ([20]) that this competition can be restricted to the inequalities (2) which are necessary and sufficient for inhibiting the onset of convection. But while the onset of convection requires that only one of (3) holds, the stability requires that, at least, all the reverse of (3) hold. Therefore, either for theoretical reasons or for practical use of stability conditions, the problem of overcoming this gap arises. We call one stability condition (OSC) problem the looking for inequalities of type (4) able to inhibit the onset of convection for large sets of values of the bifurcating parameters with special attention to the case {$\alpha=3,g=0$} since in this case (4) is necessary and sufficient for inhibiting the onset of convection. To this goal new fields for the salts densities are introduced. These transformations allow to: i) discover skew-symmetries hidden in the Darcy-Boussinesq equations; ii) obtain meaningful contributions to the OSC problem.
Citation: Salvatore Rionero. On the nonlinear stability of ternary porous media via only one necessary and sufficient algebraic condition. Evolution Equations & Control Theory, 2014, 3 (3) : 525-539. doi: 10.3934/eect.2014.3.525
##### References:
 [1] V. V. Calmidi and R. L. Muhajan, Forced convection in high porosity metal foams,, J. Heat Trans. TASME, 122 (2000), 557. doi: 10.1115/1.1287793. Google Scholar [2] F. Capone and R. De Luca, Onset of convection for ternary fluid mixtures saturating horizontal porous layers with large pores,, Rend. Lincei Mat. Appl., 23 (2012), 405. doi: 10.4171/RLM/636. Google Scholar [3] F. Capone, R. De Luca and I. Torcicollo, Longtime behaviour of vertical throughflows for binary mixtures in porous layers,, Int. J. Nonlinear Mech., 5 (2013), 113. Google Scholar [4] F. Capone, M. Gentile and S. Rionero, Influence of linear concentration heat source and parabolic density on penetrative convection onset,, in 13th Conference on Waves and Stability in Continuum Media, (2003), 77. doi: 10.1142/9789812773616_0012. Google Scholar [5] F. Capone, M. Gentile and A. Hill, Anisotropy and symmetry in porous media convection,, Acta. Mech., 208 (2009), 205. doi: 10.1007/s00707-008-0135-2. Google Scholar [6] F. Capone and S. Rionero, Nonlinear stability of a convective motion in a porous layer driven by horizontally periodic temperature gradient,, Contin. Mech. Thermodyn., 15 (2003), 529. doi: 10.1007/s00161-003-0131-7. Google Scholar [7] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,, Dover. 1981., (1981). Google Scholar [8] J. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations. An Introduction,, Library of Engineering Mathematics. CRC Press, (1996). Google Scholar [9] A. A. Hill, S. Rionero and B. Straughan, Global stability for penetrative convection with throughflow in a porous material,, IMA J. Appl. Math., 72 (2007), 635. doi: 10.1093/imamat/hxm036. Google Scholar [10] A. V. Kuznetsov and D. A. Nield, The onset of double-diffusive convection in a nanofluid layer,, Int. J. of Heat and Mass Transfer, 32 (2011), 771. Google Scholar [11] A. V. Kuznetsov and D. A. Nield, The onset of double-diffusive nanofluid convection in a layer of a saturated porous medium,, Transp. Porous Media, 85 (2010), 941. doi: 10.1007/s11242-010-9600-1. Google Scholar [12] A. V. Kuznetsov and D. A. Nield, The Cheng-Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid,, Int. J. of Heat and Mass Transfer, 54 (2011), 374. doi: 10.1016/j.ijheatmasstransfer.2010.09.034. Google Scholar [13] L. P. Lefebvre, J. Bahnart and D. C. Dunand, Porous metals and metallic foams: Current status and recent developments,, Adv. Eng. Mater., 10 (2008), 775. doi: 10.1002/adem.200800241. Google Scholar [14] S. Lombardo, G. Mulone and S. Rionero, Global nonlinear exponential stability of the conduction-diffusion solution for Schmidt numbers greater than Prandtl numbers,, J. Math. Anal. Appl., 262 (2001), 191. doi: 10.1006/jmaa.2001.7556. Google Scholar [15] G. Mulone and S. Rionero, Unconditional nonlinear exponential stability in the Bénard problem for a mixture: Necessary and sufficient conditions,, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9, 9 (1998), 221. Google Scholar [16] D. A. Nield and A. Bejan, Convection in Porous Media,, Springer-Verlag. 1999., (1999). doi: 10.1007/978-1-4757-3033-3. Google Scholar [17] R. A. Noutly and D. G. Leaist, Quaternary diffusion in acqueous $KCl-KH_2PO_4-H_3PO_4$ mixtures,, J. Phys. Chem., 91 (1987), 1665. Google Scholar [18] A. J. Pearlstein, R. M. Harris and G. Terrones, The onset of convective instability in a triple diffusive fluid layer,, J.Fluid Mech., 202 (1989), 443. doi: 10.1017/S0022112089001242. Google Scholar [19] K. R. Rajagopal, G. Saccomandi and L. Vergori, Stability analysis of the Rayleigh-Bénard convection for a fluid with temperature and pressure dependent viscosity,, Zeitschrift fur angewandte Mathematik und Physik, 60 (2009), 739. doi: 10.1007/s00033-008-8062-6. Google Scholar [20] S. Rionero, Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures,, Phys. Fluids, 24 (2012), 405. doi: 10.1063/1.4757858. Google Scholar [21] S. Rionero, A new approach to nonlinear $L^2$ -stability of double diffusive convection in porous media: Necessary and sufficient conditions for global stability via a linearization principle,, J. Math. Anal, 333 (2007), 1036. doi: 10.1016/j.jmaa.2006.12.025. Google Scholar [22] S. Rionero, Global non-linear stability in double diffusive convection via hidden symmetries,, Int. J. Nonlinear Mech., 47 (2012), 61. Google Scholar [23] S. Rionero, Symmetries and skew-symmetries against the onset of convection in porous layers salted from above and below,, Int. J. Nonlinear Mech., 47 (2012), 61. Google Scholar [24] S. Rionero, Global non-linear stability for triple diffusive convection in porous layer,, Cont. Mech. and Thermodyn., 24 (2012), 629. doi: 10.1007/s00161-011-0219-4. Google Scholar [25] S. Rionero, Onset of convection in porous materials with vertically stratified porosity,, Acta. Mech., 227 (2012), 261. Google Scholar [26] S. Rionero, Triple diffusive convection in porous media,, Acta. Mech., 224 (2013), 447. doi: 10.1007/s00707-012-0749-2. Google Scholar [27] B. Straughan, The Energy Method, Stability, and Nonlinear Convection,, Springer-Verlag, (2004). doi: 10.1007/978-0-387-21740-6. Google Scholar [28] B. Straughan, Oscillatory convection and the Cattaneo law of heat conduction,, Ric. Mat., 58 (2009), 157. doi: 10.1007/s11587-009-0055-z. Google Scholar [29] B. Straughan, Stability and Wave Motion in Porous Media,, Media, (2008). Google Scholar [30] B. Straughan and D. W. Walker, Multi-component convection-diffusion and penetrative convection,, Fluid Dyn. Res., 19 (1997), 77. doi: 10.1016/S0169-5983(96)00031-7. Google Scholar [31] B. Straughan and J. Tracey, Multi-component convection-diffusion with internal heating or cooling,, Acta. Mech., 133 (1999), 219. doi: 10.1007/BF01179019. Google Scholar [32] J. Tracey, Multi-component convection-diffusion in a porous medium,, Continuum Mech. Thermodyn., 8 (1996), 361. doi: 10.1007/s001610050050. Google Scholar [33] L. Vergori and S. Rionero, Long time behaviour of fluid motions in porous media in the presence of Brinkman law,, Acta. Mech., 210 (2010), 221. doi: 10.1007/s00707-009-0205-0. Google Scholar

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##### References:
 [1] V. V. Calmidi and R. L. Muhajan, Forced convection in high porosity metal foams,, J. Heat Trans. TASME, 122 (2000), 557. doi: 10.1115/1.1287793. Google Scholar [2] F. Capone and R. De Luca, Onset of convection for ternary fluid mixtures saturating horizontal porous layers with large pores,, Rend. Lincei Mat. Appl., 23 (2012), 405. doi: 10.4171/RLM/636. Google Scholar [3] F. Capone, R. De Luca and I. Torcicollo, Longtime behaviour of vertical throughflows for binary mixtures in porous layers,, Int. J. Nonlinear Mech., 5 (2013), 113. Google Scholar [4] F. Capone, M. Gentile and S. Rionero, Influence of linear concentration heat source and parabolic density on penetrative convection onset,, in 13th Conference on Waves and Stability in Continuum Media, (2003), 77. doi: 10.1142/9789812773616_0012. Google Scholar [5] F. Capone, M. Gentile and A. Hill, Anisotropy and symmetry in porous media convection,, Acta. Mech., 208 (2009), 205. doi: 10.1007/s00707-008-0135-2. Google Scholar [6] F. Capone and S. Rionero, Nonlinear stability of a convective motion in a porous layer driven by horizontally periodic temperature gradient,, Contin. Mech. Thermodyn., 15 (2003), 529. doi: 10.1007/s00161-003-0131-7. Google Scholar [7] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability,, Dover. 1981., (1981). Google Scholar [8] J. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations. An Introduction,, Library of Engineering Mathematics. CRC Press, (1996). Google Scholar [9] A. A. Hill, S. Rionero and B. Straughan, Global stability for penetrative convection with throughflow in a porous material,, IMA J. Appl. Math., 72 (2007), 635. doi: 10.1093/imamat/hxm036. Google Scholar [10] A. V. Kuznetsov and D. A. Nield, The onset of double-diffusive convection in a nanofluid layer,, Int. J. of Heat and Mass Transfer, 32 (2011), 771. Google Scholar [11] A. V. Kuznetsov and D. A. Nield, The onset of double-diffusive nanofluid convection in a layer of a saturated porous medium,, Transp. Porous Media, 85 (2010), 941. doi: 10.1007/s11242-010-9600-1. Google Scholar [12] A. V. Kuznetsov and D. A. Nield, The Cheng-Minkowycz problem for the double-diffusive natural convective boundary layer flow in a porous medium saturated by a nanofluid,, Int. J. of Heat and Mass Transfer, 54 (2011), 374. doi: 10.1016/j.ijheatmasstransfer.2010.09.034. Google Scholar [13] L. P. Lefebvre, J. Bahnart and D. C. Dunand, Porous metals and metallic foams: Current status and recent developments,, Adv. Eng. Mater., 10 (2008), 775. doi: 10.1002/adem.200800241. Google Scholar [14] S. Lombardo, G. Mulone and S. Rionero, Global nonlinear exponential stability of the conduction-diffusion solution for Schmidt numbers greater than Prandtl numbers,, J. Math. Anal. Appl., 262 (2001), 191. doi: 10.1006/jmaa.2001.7556. Google Scholar [15] G. Mulone and S. Rionero, Unconditional nonlinear exponential stability in the Bénard problem for a mixture: Necessary and sufficient conditions,, Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9, 9 (1998), 221. Google Scholar [16] D. A. Nield and A. Bejan, Convection in Porous Media,, Springer-Verlag. 1999., (1999). doi: 10.1007/978-1-4757-3033-3. Google Scholar [17] R. A. Noutly and D. G. Leaist, Quaternary diffusion in acqueous $KCl-KH_2PO_4-H_3PO_4$ mixtures,, J. Phys. Chem., 91 (1987), 1665. Google Scholar [18] A. J. Pearlstein, R. M. Harris and G. Terrones, The onset of convective instability in a triple diffusive fluid layer,, J.Fluid Mech., 202 (1989), 443. doi: 10.1017/S0022112089001242. Google Scholar [19] K. R. Rajagopal, G. Saccomandi and L. Vergori, Stability analysis of the Rayleigh-Bénard convection for a fluid with temperature and pressure dependent viscosity,, Zeitschrift fur angewandte Mathematik und Physik, 60 (2009), 739. doi: 10.1007/s00033-008-8062-6. Google Scholar [20] S. Rionero, Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures,, Phys. Fluids, 24 (2012), 405. doi: 10.1063/1.4757858. Google Scholar [21] S. Rionero, A new approach to nonlinear $L^2$ -stability of double diffusive convection in porous media: Necessary and sufficient conditions for global stability via a linearization principle,, J. Math. Anal, 333 (2007), 1036. doi: 10.1016/j.jmaa.2006.12.025. Google Scholar [22] S. Rionero, Global non-linear stability in double diffusive convection via hidden symmetries,, Int. J. Nonlinear Mech., 47 (2012), 61. Google Scholar [23] S. Rionero, Symmetries and skew-symmetries against the onset of convection in porous layers salted from above and below,, Int. J. Nonlinear Mech., 47 (2012), 61. Google Scholar [24] S. Rionero, Global non-linear stability for triple diffusive convection in porous layer,, Cont. Mech. and Thermodyn., 24 (2012), 629. doi: 10.1007/s00161-011-0219-4. Google Scholar [25] S. Rionero, Onset of convection in porous materials with vertically stratified porosity,, Acta. Mech., 227 (2012), 261. Google Scholar [26] S. Rionero, Triple diffusive convection in porous media,, Acta. Mech., 224 (2013), 447. doi: 10.1007/s00707-012-0749-2. Google Scholar [27] B. Straughan, The Energy Method, Stability, and Nonlinear Convection,, Springer-Verlag, (2004). doi: 10.1007/978-0-387-21740-6. Google Scholar [28] B. Straughan, Oscillatory convection and the Cattaneo law of heat conduction,, Ric. Mat., 58 (2009), 157. doi: 10.1007/s11587-009-0055-z. Google Scholar [29] B. Straughan, Stability and Wave Motion in Porous Media,, Media, (2008). Google Scholar [30] B. Straughan and D. W. Walker, Multi-component convection-diffusion and penetrative convection,, Fluid Dyn. Res., 19 (1997), 77. doi: 10.1016/S0169-5983(96)00031-7. Google Scholar [31] B. Straughan and J. Tracey, Multi-component convection-diffusion with internal heating or cooling,, Acta. Mech., 133 (1999), 219. doi: 10.1007/BF01179019. Google Scholar [32] J. Tracey, Multi-component convection-diffusion in a porous medium,, Continuum Mech. Thermodyn., 8 (1996), 361. doi: 10.1007/s001610050050. Google Scholar [33] L. Vergori and S. Rionero, Long time behaviour of fluid motions in porous media in the presence of Brinkman law,, Acta. Mech., 210 (2010), 221. doi: 10.1007/s00707-009-0205-0. Google Scholar
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