# 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 and 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), 557-565. doi: 10.1115/1.1287793. [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), 405-428. doi: 10.4171/RLM/636. [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), 113-133. [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, (eds. R. Monaco, G. Mulone, S. Rionero and T. Ruggeri), World Scientific, (2003), 77-82. doi: 10.1142/9789812773616_0012. [5] F. Capone, M. Gentile and A. Hill, Anisotropy and symmetry in porous media convection, Acta. Mech., 208 (2009), 205-214. doi: 10.1007/s00707-008-0135-2. [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), 529-538. doi: 10.1007/s00161-003-0131-7. [7] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover. 1981. [8] J. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations. An Introduction, Library of Engineering Mathematics. CRC Press, Boca Raton, FL, 1996. [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), 635-643. doi: 10.1093/imamat/hxm036. [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), 771-776. [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), 941-951. doi: 10.1007/s11242-010-9600-1. [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), 374-378. doi: 10.1016/j.ijheatmasstransfer.2010.09.034. [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), 775-787. doi: 10.1002/adem.200800241. [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), 191-207. doi: 10.1006/jmaa.2001.7556. [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, Mat. Appl., 9 (1998), 221-236. [16] D. A. Nield and A. Bejan, Convection in Porous Media, Springer-Verlag. 1999. doi: 10.1007/978-1-4757-3033-3. [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), 1665-1658. [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), 443-465. doi: 10.1017/S0022112089001242. [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), 739-755. doi: 10.1007/s00033-008-8062-6. [20] S. Rionero, Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures, Phys. Fluids, 24 (2012), 405-428. doi: 10.1063/1.4757858. [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, Appl., 333 (2007), 1036-1057. doi: 10.1016/j.jmaa.2006.12.025. [22] S. Rionero, Global non-linear stability in double diffusive convection via hidden symmetries, Int. J. Nonlinear Mech., 47 (2012), 61-66. [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), 61-67. [24] S. Rionero, Global non-linear stability for triple diffusive convection in porous layer, Cont. Mech. and Thermodyn., 24, (2012), 629-641. doi: 10.1007/s00161-011-0219-4. [25] S. Rionero, Onset of convection in porous materials with vertically stratified porosity, Acta. Mech., 227 (2012), 261-272. [26] S. Rionero, Triple diffusive convection in porous media, Acta. Mech., 224 (2013), 447-458. doi: 10.1007/s00707-012-0749-2. [27] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer-Verlag, 2004 doi: 10.1007/978-0-387-21740-6. [28] B. Straughan, Oscillatory convection and the Cattaneo law of heat conduction, Ric. Mat., 58 (2009), 157-162. doi: 10.1007/s11587-009-0055-z. [29] B. Straughan, Stability and Wave Motion in Porous Media, Media, Appl. Math. Sci., 165, 2008. [30] B. Straughan and D. W. Walker, Multi-component convection-diffusion and penetrative convection, Fluid Dyn. Res., 19 (1997), 77-89. doi: 10.1016/S0169-5983(96)00031-7. [31] B. Straughan and J. Tracey, Multi-component convection-diffusion with internal heating or cooling, Acta. Mech., 133 (1999), 219-239. doi: 10.1007/BF01179019. [32] J. Tracey, Multi-component convection-diffusion in a porous medium, Continuum Mech. Thermodyn., 8 (1996), 361-381. doi: 10.1007/s001610050050. [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), 221-240. doi: 10.1007/s00707-009-0205-0.

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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), 557-565. doi: 10.1115/1.1287793. [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), 405-428. doi: 10.4171/RLM/636. [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), 113-133. [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, (eds. R. Monaco, G. Mulone, S. Rionero and T. Ruggeri), World Scientific, (2003), 77-82. doi: 10.1142/9789812773616_0012. [5] F. Capone, M. Gentile and A. Hill, Anisotropy and symmetry in porous media convection, Acta. Mech., 208 (2009), 205-214. doi: 10.1007/s00707-008-0135-2. [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), 529-538. doi: 10.1007/s00161-003-0131-7. [7] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover. 1981. [8] J. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations. An Introduction, Library of Engineering Mathematics. CRC Press, Boca Raton, FL, 1996. [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), 635-643. doi: 10.1093/imamat/hxm036. [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), 771-776. [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), 941-951. doi: 10.1007/s11242-010-9600-1. [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), 374-378. doi: 10.1016/j.ijheatmasstransfer.2010.09.034. [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), 775-787. doi: 10.1002/adem.200800241. [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), 191-207. doi: 10.1006/jmaa.2001.7556. [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, Mat. Appl., 9 (1998), 221-236. [16] D. A. Nield and A. Bejan, Convection in Porous Media, Springer-Verlag. 1999. doi: 10.1007/978-1-4757-3033-3. [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), 1665-1658. [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), 443-465. doi: 10.1017/S0022112089001242. [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), 739-755. doi: 10.1007/s00033-008-8062-6. [20] S. Rionero, Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures, Phys. Fluids, 24 (2012), 405-428. doi: 10.1063/1.4757858. [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, Appl., 333 (2007), 1036-1057. doi: 10.1016/j.jmaa.2006.12.025. [22] S. Rionero, Global non-linear stability in double diffusive convection via hidden symmetries, Int. J. Nonlinear Mech., 47 (2012), 61-66. [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), 61-67. [24] S. Rionero, Global non-linear stability for triple diffusive convection in porous layer, Cont. Mech. and Thermodyn., 24, (2012), 629-641. doi: 10.1007/s00161-011-0219-4. [25] S. Rionero, Onset of convection in porous materials with vertically stratified porosity, Acta. Mech., 227 (2012), 261-272. [26] S. Rionero, Triple diffusive convection in porous media, Acta. Mech., 224 (2013), 447-458. doi: 10.1007/s00707-012-0749-2. [27] B. Straughan, The Energy Method, Stability, and Nonlinear Convection, Springer-Verlag, 2004 doi: 10.1007/978-0-387-21740-6. [28] B. Straughan, Oscillatory convection and the Cattaneo law of heat conduction, Ric. Mat., 58 (2009), 157-162. doi: 10.1007/s11587-009-0055-z. [29] B. Straughan, Stability and Wave Motion in Porous Media, Media, Appl. Math. Sci., 165, 2008. [30] B. Straughan and D. W. Walker, Multi-component convection-diffusion and penetrative convection, Fluid Dyn. Res., 19 (1997), 77-89. doi: 10.1016/S0169-5983(96)00031-7. [31] B. Straughan and J. Tracey, Multi-component convection-diffusion with internal heating or cooling, Acta. Mech., 133 (1999), 219-239. doi: 10.1007/BF01179019. [32] J. Tracey, Multi-component convection-diffusion in a porous medium, Continuum Mech. Thermodyn., 8 (1996), 361-381. doi: 10.1007/s001610050050. [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), 221-240. doi: 10.1007/s00707-009-0205-0.
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