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A binary mixture saturating a horizontal porous layer, with large pores and uniformly heated from below, is considered. The instability of a vertical fluid motion (throughflow) when the layer is salted by one salt (either from above or from below) is analyzed. Ultimately boundedness of solutions is proved, via the existence of positively invariant and attractive sets (i.e. absorbing sets). The critical Rayleigh numbers at which steady or oscillatory instability occurs are recovered. Sufficient conditions guaranteeing that a secondary steady motion or a secondary oscillatory motion can be observed after the loss of stability are found. When the layer is salted from above, a condition guaranteeing the occurrence of “cold” instability is determined. Finally, the influence of the velocity module on the increasing/decreasing of the instability thresholds is investigated.

Convection in fluid mixtures saturating porous media has attracted—in the past as nowadays—the attention of many scientists due to its practical applications like, for example, in geothermal energy exploitation, extraction of oil from underground reservoirs, ground-water pollution, underground flows movement, thermal engineering, crystal growth, polymer engineering, and ceramic processing. For this reason, several studies have been addressed to this topic [

The models describing the fluid motion in porous media are reaction-diffusion dynamical systems of P.D.Es, which, as it is well known, play an important role in the modeling and studying of many phenomena

The effect of vertical throughflow on convective instability, in either the fluid or the porous layer, has been extensively discussed by several investigators. In the fluid layer, the problem is of interest because of the possibility of controlling the convective instability by adjusting the throughflow [

In the present paper, we will focus on the instability analysis of a vertical constant throughflow in a horizontal porous layer, with large pores, uniformly heated from below and uniformly salted by one chemical either from above or below. In particular we determine the critical Rayleigh thermal numbers at which instability occurs and investigate for the kind of secondary motion arising.

Section

Let us consider a fluid moving mixture saturating a horizontal porous layer of depth

The subscripts

the perturbations

Let us denote by

We recall that the solutions of (

The set

The proof is obtained by following, step by step, the procedure given in [

In this section we study the instability of the throughflow solution (_{1}, one obtains that the linear system governing the evolution of

From (

Now, limiting the analysis to the case in which the layer is salted from above (

In the case

Let us first observe that

According to the definition given in [

In order to establish if a secondary steady or oscillatory motion arises when (

If either

From (

If

The proof follows since (

In this section we investigate for the influence of

If

From (

If (

From (

From Lemmas

If (

In this paper we analyze a vertical fluid motion of a binary mixture saturating a horizontal porous layer with large pores, uniformly heated from below and salted by one salt (either from above or from below). In particular,

the definitely boundedness of solutions (existence of absorbing sets) has been recalled;

the critical Rayleigh numbers at which steady or oscillatory instability occurs have been recovered;

sufficient conditions guaranteeing that a steady or oscillatory secondary motion sets in after the loss of stability have been found.

the onset of “cold” instability, possible only when the layer is salted from above, has been analyzed;

the stabilizing/destabilizing effect of the vertical throughflow has been investigated.

We conclude this section by showing some numerical simulations on the performed analysis. Figure

Behavior of

Behavior of

Behavior of

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This paper has been performed under the auspices of the G.N.F.M. of INdAM.