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September  2014, 19(7): 2279-2296. doi: 10.3934/dcdsb.2014.19.2279

## Onset of convection in rotating porous layers via a new approach

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

Received  April 2013 Revised  September 2013 Published  August 2014

Via a new approach, ternary fluid mixtures saturating rotating horizontal porous layers, heated from below and salted from above and below, are investigated. With or without the presence of Brinkman viscosity, the absence of subcritical instabilities is shown together with the coincidence of linear and non-linear global stability of the thermal conduction solution. The stability-instability conditions are found to be given by simple algebraic conditions in closed forms.
Citation: Salvatore Rionero. Onset of convection in rotating porous layers via a new approach. Discrete and Continuous Dynamical Systems - B, 2014, 19 (7) : 2279-2296. doi: 10.3934/dcdsb.2014.19.2279
##### References:
 [1] 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. [2] F. Capone and R. De Luca, Ultimately boundedness and stability of triply diffusive mixtures in rotating porous layers under the action of Brinkman law, Int. J. Nonlinear Mech., 47 (2012), 799-805. [3] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, 1981. [4] J. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations, An Introduction. CRC Press, 1996. [5] F. R. Gantmaker, The Theory Of Matrices, Vol 1-2 AMS (Chelsea Publishing), 2000. [6] M. Lappa, Rotating Thermal Flows in Natural and Industrial Processes, Wiley, 2012. doi: 10.1002/9781118342411. [7] A. Lopez, L. Romero and A. Pearlstein, Effect of rigid boundaries on the onset of convection in a triply diffusive fluid layer, Phys. Fluids A, 2 (1990), 897-902. [8] D. R. Merkin, Introduction to the Theory of Stability, Texts in Applied Mathematics, v. 24, Springer-Verlag, New York, 1997. xx+319 pp. [9] D. A. Nield and A. Bejan, Conduction in Porous Media, 2nd edition, Springer, 1999. doi: 10.1007/978-1-4757-3033-3. [10] R. A. Noutly and D. G. Leaist, Quaternary diffusion in acqueous $KCl-KH_2PO_4-H_3PO_4$ mixtures, J. Phys. Chem., 91 (1987), 1655-1658. [11] A. J. Pearlstein, R. M. Harris and G. Terrones, The onset of convective instability in a triply diffusive fluid layer, J.Fluid Mech., 202 (1989), 443-465. doi: 10.1017/S0022112089001242. [12] S. Rionero, Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures, Phys. Fluids, 24 (2012), 104101, 17pp. doi: 10.1063/1.4757858. [13] S. Rionero, Long time behaviour of multicomponent fluid mixture in porous media, Int. J. of Eng. Sci., 48 (2010), 1519-1533. doi: 10.1016/j.ijengsci.2010.07.007. [14] S. Rionero, Global nonlinear stability for a triply diffusive-convection in a porous layer, Contin. Mech. Thermodyn., 24 (2012), 629-641. doi: 10.1007/s00161-011-0219-4. [15] S. Rionero, On the nonlinear stability of ternary porous media via only one necessary and sufficient algebraic condition, (submitted to EECT, evolution equation and control theory). [16] S. Rionero, Symmetries and skew-symmetries against onset of convection in porous layers salted from above and below, Int. J. Non-linear Mech., 47 (2012), 61-67. [17] S. Rionero, Multicomponent diffusive-convective fluid motions in porous layers: ultimately boundedness, absence of subcritical instabilities and global nonlinear stability for any number of salts, Phys. Fluids, 25 (2013), 054104. doi: 10.1063/1.4802629. [18] B. Straughan, The Energy Method, Stability and Nonlinear Convection, Second edition. Applied Mathematical Sciences, 91. Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21740-6. [19] B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, 165. Springer, New York, 2008. [20] B. Straughan and J. Tracey, Multi-component convection-diffusion with internal heating or cooling, Acta Mechanica, 133 (1999), 219-239. [21] J. Tracey, Multi-component convection-diffusion in a porous medium, Continuum Mech. Thermodyn., 8 (1996), 361-381.

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##### References:
 [1] 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. [2] F. Capone and R. De Luca, Ultimately boundedness and stability of triply diffusive mixtures in rotating porous layers under the action of Brinkman law, Int. J. Nonlinear Mech., 47 (2012), 799-805. [3] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover, 1981. [4] J. Flavin and S. Rionero, Qualitative Estimates for Partial Differential Equations, An Introduction. CRC Press, 1996. [5] F. R. Gantmaker, The Theory Of Matrices, Vol 1-2 AMS (Chelsea Publishing), 2000. [6] M. Lappa, Rotating Thermal Flows in Natural and Industrial Processes, Wiley, 2012. doi: 10.1002/9781118342411. [7] A. Lopez, L. Romero and A. Pearlstein, Effect of rigid boundaries on the onset of convection in a triply diffusive fluid layer, Phys. Fluids A, 2 (1990), 897-902. [8] D. R. Merkin, Introduction to the Theory of Stability, Texts in Applied Mathematics, v. 24, Springer-Verlag, New York, 1997. xx+319 pp. [9] D. A. Nield and A. Bejan, Conduction in Porous Media, 2nd edition, Springer, 1999. doi: 10.1007/978-1-4757-3033-3. [10] R. A. Noutly and D. G. Leaist, Quaternary diffusion in acqueous $KCl-KH_2PO_4-H_3PO_4$ mixtures, J. Phys. Chem., 91 (1987), 1655-1658. [11] A. J. Pearlstein, R. M. Harris and G. Terrones, The onset of convective instability in a triply diffusive fluid layer, J.Fluid Mech., 202 (1989), 443-465. doi: 10.1017/S0022112089001242. [12] S. Rionero, Absence of subcritical instabilities and global nonlinear stability for porous ternary diffusive-convective fluid mixtures, Phys. Fluids, 24 (2012), 104101, 17pp. doi: 10.1063/1.4757858. [13] S. Rionero, Long time behaviour of multicomponent fluid mixture in porous media, Int. J. of Eng. Sci., 48 (2010), 1519-1533. doi: 10.1016/j.ijengsci.2010.07.007. [14] S. Rionero, Global nonlinear stability for a triply diffusive-convection in a porous layer, Contin. Mech. Thermodyn., 24 (2012), 629-641. doi: 10.1007/s00161-011-0219-4. [15] S. Rionero, On the nonlinear stability of ternary porous media via only one necessary and sufficient algebraic condition, (submitted to EECT, evolution equation and control theory). [16] S. Rionero, Symmetries and skew-symmetries against onset of convection in porous layers salted from above and below, Int. J. Non-linear Mech., 47 (2012), 61-67. [17] S. Rionero, Multicomponent diffusive-convective fluid motions in porous layers: ultimately boundedness, absence of subcritical instabilities and global nonlinear stability for any number of salts, Phys. Fluids, 25 (2013), 054104. doi: 10.1063/1.4802629. [18] B. Straughan, The Energy Method, Stability and Nonlinear Convection, Second edition. Applied Mathematical Sciences, 91. Springer-Verlag, New York, 2004. doi: 10.1007/978-0-387-21740-6. [19] B. Straughan, Stability and Wave Motion in Porous Media, Applied Mathematical Sciences, 165. Springer, New York, 2008. [20] B. Straughan and J. Tracey, Multi-component convection-diffusion with internal heating or cooling, Acta Mechanica, 133 (1999), 219-239. [21] J. Tracey, Multi-component convection-diffusion in a porous medium, Continuum Mech. Thermodyn., 8 (1996), 361-381.
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