On the stability of vertical constant throughflows for binary mixtures in porous layers F. Capone a,n , V. De Cataldis a , R. De Luca a , I. Torcicollo b a Department of Mathematics and Applications ‘R. Caccioppoli’, University of Naples Federico II, Complesso Universitario Monte S. Angelo, Via Cinzia, 80126 Naples, Italy b Istituto per le Applicazioni del Calcolo “M. Picone”, C.N.R., Via P. Castellino 111, 80131 Naples, Italy article info Article history: Received 10 August 2013 Received in revised form 23 October 2013 Accepted 23 October 2013 Available online 30 October 2013 Keywords: Porous media Convection Vertical throughflows Stability Routh–Hurwitz conditions abstract A system modeling fluid motions in horizontal porous layers, uniformly heated from below and salted from above by one salt, is analyzed. The definitely boundedness of solutions (existence of absorbing sets) is proved. Necessary and sufficient conditions ensuring the linear stability of a vertical constant throughflow have been obtained via a new approach. Moreover, conditions guaranteeing the global non-linear asymptotic stability are determined. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction The research concerned with fluid motions in porous media has been widely studied, in the past as nowadays, due to the large applications in the real world phenomena like, for instance, in geophysics, in contaminant transport and underground water flow [1–20]. The models describing the fluid motion in porous media are, in general, reaction–diffusion dynamical systems of P.D.Es, which, as it is well known, play an important role in modeling and studying many phenomena (see, for instance [21–29], and refer- ences therein). In particular, convection in porous layers in the presence of vertical throughflows finds relevant applications in cloud physics, in hydrological studies, in subterranean pollution and in many industrial processes [30–46,49,50. The effect of vertical throughflow on the onset of convection has been con- sidered in many cases, like, for example in a rectangular box [37]; combined with a magnetic field [38]; in a cubic Forchheimer model [39]; when the density is a quadratic function of tempera- ture [40] and under the action of an inclined temperature gradient [41,42]. The present paper is devoted to study the effects of both temperature gradient and salt concentration on the stability of a vertical throughflow. Already in [30,31] the authors consider both the effects. Precisely, the effect of variable thermal and solute diffusivities on the onset of convection for non-constant through- flows has been analyzed in [30]; while in [31] the stability of a vertical constant throughflow in a porous layer, uniformly heated and salted from below, has been investigated for determining conditions ensuring the stability in the L 2 -norm. In the present paper, we will analyze the more destabilizing case of horizontal porous layers uniformly heated from below and salted from above by one salt and, by using a new approach proposed by Rionero [13–16], conditions ensuring the stability of a vertical constant throughflow have been obtained. The plan of the paper is as follows. Section 2 concerns with the introduction of the mathematical model and the determination of a vertical constant throughflow. The definitely boundedness of solutions (existence of absorbing sets) is proved in Section 3, while, in Section 4, the main boundary value problem is analyzed and the independent unknown fields are determined. Section 5 is devoted to the introduction of the linear auxiliary system, while the non-linear equation governing each Fourier component of the perturbations is introduced in Section 6 (according to the meth- odology introduced by Rionero [13]). In Section 7 the global non- linear stability of the vertical throughflow is investigated. The paper ends with a section devoted to the final remarks. 2. Preliminaries Let Oxyz be an orthogonal frame of reference with fundamental unit vectors i, j, k (k pointing vertically upwards). The model Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/nlm International Journal of Non-Linear Mechanics 0020-7462/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijnonlinmec.2013.10.010 n Corresponding author. Tel.: þ39 81675645; fax: þ39 817662106. E-mail addresses: fcapone@unina.it (F. Capone), valentina.decataldis@unina.it (V. De Cataldis), roberta.deluca@unina.it (R. De Luca), i.torcicollo@iac.cnr.it (I. Torcicollo). International Journal of Non-Linear Mechanics 59 (2014) 1–8