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

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  • 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.
    Mathematics Subject Classification: 76S05, 76R10, 76U05, 35B35.


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