The onset of double diffusive convection in a rotating bi-disperse porous medium

It is well known that when a horizontal layer of fluid is heated from below, a thermal boundary layer of less dense and hot fluid rises up. When this boundary layer becomes unstable, convective motion in the fluid above sets in. Forecasting when instabilities take place is essential. When a salt dissolved in a fluid saturating a porous medium heated from below is considered, simultaneous mass diffusion and thermal diffusion occur. Unlike the diffusion of heat, the diffusion of salt can take place only through the fluid phase, so an additional physical effect has to be considered: the Soret effect, that is the mass flux created by a temperature gradient. In the present paper the onset of convection in a rotating layer of bi-disperse porous medium saturated by a binary fluid mixture, taking into account the Soret effect, is analysed. Linear stability analysis is performed in order to determine the instability thresholds for the onset of convection via a steady state (stationary convection) and via an oscillatory state (oscillatory convection). Nonlinear stability analysis is performed to obtain the global stability threshold with respect to the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm.


Introduction
The onset of thermal convection in horizontal layers of fluids, uniformly heated from below, was studied for the first time by Henri Bénard in 1900 via experiments and, later, was analytically investigated by Lord Rayleigh. In particular, a "threshold phenomenon" was observed: Lord Rayleigh introduced a dimensionless parameter-the Rayleigh number R, depending on the physical features of the layer and of the fluid and on the gravity acceleration-and established that the thermal convection arises when R overcomes a certain threshold R c . From a mathematical point of view, the critical value R c is determined via the instability analysis of the thermal conduction solution. Concerning the linear instability analysis, requiring that all the eigenvalues of the linear operator have negative real part, one can determine a threshold R L such that condition R > R L implies instability of the thermal conduction solution and, consequently, the onset of thermal convection. Concerning the nonlinear stability analysis, introducing a suitable norm and choosing properly a Lyapunov functional, one needs to determine the conditions for which this functional is decreasing along the solutions of the nonlinear system, these conditions give rise to the nonlinear threshold R N (≤ R L ) such that the condition R < R N implies stability, i.e. if R < R N convection cannot occur. In order to have useful results for the applications, the challenge is to find a norm for which R N R L . Furthermore, in many applications, the thermal convection can arise via steady or oscillatory motions and convection is named steady or oscillatory convection, respectively.
The determination of the critical value R c is a challenging task that attracted the interest of many researchers-in the past as nowadays-due to the many phenomena of the real world involving thermal convection in either clear fluids (see [1][2][3][4] and the references therein) or in fluids saturating porous media (materials composed of a solid skeleton containing interconnected voids through which a fluid can flow). The onset of convection in porous media was widely studied in the literature taking into account various effects, such as uniform rotation of the layer [5][6][7], the influence of an external transverse magnetic field on an electrically conducting fluid [8][9][10][11][12][13], the presence of nanoparticles in the saturating fluid [14,15] or the presence of one or more chemicals dissolved in the fluid [16][17][18][19], the combined effects of uniform rotation and the presence of a chemical [20][21][22]. Recently, the attention of many scientists is moving toward bi-disperse porous media. from stability to instability takes place via a marginal state via a steady pattern of motions. If the perturbations are characterized by oscillations of increasing (or decreasing) amplitude, instability occurs through oscillatory motions. If instability arises via stationary motions, the principle of exchange of stabilities holds and the instability sets in as stationary secondary flow. If oscillatory motions prevail, then one has overstability. Hence, in Sects. 3.1 and 3.2 we determine the critical Rayleigh numbers for the onset of steady and oscillatory convection, respectively. In Sect. 3.3 some mathematical aspects shared by the stationary and the oscillatory instability thresholds are discussed. In Sect. 4 the differential constraint approach is utilized to derive the global nonlinear instability threshold and we found that there are regions of subcritical instabilities. In Sect. 5 we perform some numerical simulations in order to analyse the behaviour of the instability thresholds with respect to fundamental physical parameters. The paper ends with a concluding section that summarizes all the obtained results.

Mathematical model
Let us consider a reference frame Oxyz with fundamental unit vectors i, j, k (k pointing vertically upward) and a plane layer L R 2 × [0, d] of saturated bi-disperse porous medium that is uniformly and simultaneously heated and salted from below and that is filled by an incompressible Newtonian fluid. Furthermore, we confine ourselves to the case of a single temperature bi-disperse porous material, so there is thermal equilibrium between the f -phase and the p-phase, i.e. T f T p T . The layer L rotates about the vertical axis with constant angular velocity k, and hence, Darcy's model is extended in order to include Coriolis forces in the momentum equations in the macropores and in the micropores. Moreover, a Boussinesq approximation is applied and the density in the buoyancy force has a linear dependence on temperature and concentration: α and α C being the thermal and chemical expansion coefficients, respectively. The governing equations for the onset of thermal convection in a uniformly rotating bi-disperse porous medium heated and salted from below, taking into account the Soret effect, are (cf. [39,41] ⎧ where the subscript sol is referred to the solid skeleton. Let us point out that in the case of single temperature BDPM, since macropores and micropores are saturated by the same fluid, we expect that [47] ( where n is the unit outward normal to the impermeable horizontal planes delimiting the layer. Moreover, since the layer of BDPM is heated and salted from below, we assume T L > T U and C L > C U . System (1)-(2) admits the stationary conduction solution is the concentration gradient andp s (0) are assigned constants, Let us introduce the following non-dimensional parameters: where the scales are given by , and let us define the Lewis number Le, the non-dimensional Soret number S, the Taylor number T , the Rayleigh number R and the Rayleigh number for the salt field C, respectively, given by The dimensionless equations describing the evolutionary behaviour of the perturbation fields, dropping all the asterisks, are The initial conditions and the boundary conditions appended to system 4 are respectively.

Remark 2.1
With the aim to perform the stability analysis of the conduction solution to system (1)-(2), so of the null solution to (4)-(5), let us denote by , and that f is a periodic function in the x and y directions of period 2π/l and 2π/m, respectively.

Linear instability analysis
In order to study the linear instability of the thermal conduction solution, let us consider the linear version of system (4) and seek for a solution u f , u p , π f , π p , θ, γ with time dependence like e σ t , with σ ∈ C, so we get and taking the third component of curl of (6) 1 and (6) i.e.
where ξ + ξ K r + K r . Now, substituting the derivative with respect to z of (8) in the third component of double curl of (6) 1 and where 1 ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 is the horizontal Laplacian, we finally get the following boundary value problem in According to boundary conditions (5) and to the periodicity of the perturbations fields, being {sin(nπ z)} n∈N a complete orthogonal system for L 2 ([0, 1]), we employ normal modes solutions in (10): where W f 0n , W p 0n , 0n , 0n are real constants. Consequently we get a 2 l 2 + m 2 being the wavenumber and n n 2 π 2 + a 2 . Setting and requiring the determinant of (12) to be zero, we obtain i.e.
that is surely satisfied if 1 Le ≤ 1, since are positive. Therefore

Steady convection threshold
When convective instability occurs via steady motions, the marginal state is characterized by σ 0, so from (14) we derive the critical Rayleigh number for the onset of stationary convection: R S min (n,a 2 ) n a 2 2 n + −1 T 2 n 2 π 2 n A + −1 T 4 n 4 π 4 η 2 n (1+ K r +4ξ )+ −1 T 2 n 2 π 2 [K r +η 2 +ξ (η − 1) 2 ] where A η 2 (ξ + 1) 2 + 2ηξ 2 + (ξ + K r ) 2 . The minimum with respect to n ∈ N is attained at n 1, so the steady critical Rayleigh number is given by Moreover, the minimum (21) is attained at the positive solution of the fourth-order algebraic equations s(x) 0, with x a 2 and and For that matter, the equation s(x) 0 admits at least one positive root since s(0) h 5 < 0 and lim x→∞ s(x) +∞.
Let us finally point out that in the absence of rotation (i.e. for T → 0) from (19) we recover the stationary threshold found in [41], while confining ourselves to the case of a single component fluid (i.e. for C → 0), (19) coincides with the instability threshold found in [39]. When the Soret effect is neglected (i.e. for S → 0), one obtains that R S f (a 2 c ) + C.

Oscillatory convection threshold
In order to determine the instability threshold for the onset of oscillatory convection, the growth rate of the system σ needs to be purely imaginary, hence let us consider σ iσ 1 , with σ 1 ∈ R − {0}, so (14) becomes whit Imposing the imaginary part I m(R) of (22) to vanish, we get Hence, necessary conditions for the onset of oscillatory convection are Consequently, the critical Rayleigh number for the onset of oscillatory convection is n a 2 2 n + −1 T 2 n 2 π 2 n A+ −1 T 4 n 4 π 4 η 2 n (1+ K r +4ξ )+ −1 T 2 n 2 π 2 [K r +η 2 +ξ (η−1) 2 ] , while the frequency of the oscillations is given by Since the minimum of (26) with respect to n ∈ N is attained at n 1, the oscillatory critical Rayleigh number is given by or, equivalently, hence, while the oscillatory Rayleigh number given by (28) does not depend on the Soret number S, i.e. the Soret effect does not directly affect the oscillatory instability threshold, from (29) one gets (30) and convection arises through stationary motions. As shown for the stationary threshold, in the absence of rotation from (26) we recover the same oscillatory instability threshold found in [41]. Let us remark that when a binary mixture and the Soret effect are considered, convection can arise via steady or oscillatory motions, while the principle of exchange of stabilities was proved for the single component case, so, in that case, oscillatory convection cannot occur (see [39].

Remarks
Since the chemical component dissolved at the bottom of the layer has a stabilizing effect on the onset of convection, i.e. it delays the onset of convection through both stationary and oscillatory motions. Moreover, ∂R S ∂T 2 > 0 and ∂R O ∂T 2 > 0, this means that rotation has a stabilizing effect on the onset of both steady and oscillatory convection. As one is expected, both the rotation and the dissolved solute act to stop heat transfer and fluid motion through convection. From (20) and (28) one obtains that is a necessary and sufficient condition for the onset of stationary convection. Moreover, the steady and oscillatory instability thresholds (20) and (28) are straight lines in the (C, R) plane, so for increasing C there is a transition from steady to oscillatory convection in correspondence of the intersection point (32)

Nonlinear stability
Let us consider the nonlinear system (10) 1,2 and (4) 5, The threshold for the nonlinear stability of the conduction solution will be determined employing the differential constraint approach [18]. Therefore, denoting by (·, ·) and · inner product and norm on the Hilbert space L 2 (V ), respectively, let us set    Table 3 Steady instability thresholds at low concentration (C 1) (a) and at high concentration (C 5) (b) and for quoted values of the Soret number S (b).
0n − μS n 0n + Ra 2 (λ 0n + λ 0n ) − 2 n 0n 0, a 2 l 2 + m 2 being the wavenumber, while n a 2 + n 2 π 2 . Hence, one obtains the stability condition where the critical nonlinear Rayleigh number is found to be where . The maximum of (46) with respect to μ is attained at In conclusion, we obtained that R E f (a 2 c ) and  therefore, there are regions of possible subcritical instabilities. However, for S → 1 the coincidence between the stationary threshold R S and the global nonlinear threshold R E is achieved, even though the dependence of the instability thresholds on the concentration field is lost. Let us observe that the nonlinear stability threshold R E coincides with the stability threshold obtained when the Soret effect is not taken into account, therefore, if R < R E , the thermal conduction solution is unconditionally stable, regardless of what value C has and no matter of whether the Soret effect is taken into account or not, hence, the global nonlinear stability threshold obtained by the energy (34) is affected only by rotation and the stabilizing effect of the concentration gradient on the onset of convection is not achieved. Let us remark that condition (31) becomes

Numerical simulations
The purpose of this section is to numerically investigate the asymptotic behaviour of steady and oscillatory thresholds (20) and (28) with respect to the meaningful parameters of the model. Since our thresholds are consistent with those ones found in [41], let us fix {ξ 0.1, K r 1.5, η 1.5, 1 Le 55.924} in the following simulations. For high concentrations of the dissolved chemical component, convection sets in through oscillatory motions, indeed in Fig. 1a the linear dependence of the Rayleigh number on the chemical Rayleigh number is depicted, and, for the chosen set of parameters, the critical value of the concentration Rayleigh number for which there is a switch from stationary to oscillatory convection is C * 4.0839.
In Table 1(a) and (b) the stabilizing effect of rotation on the onset of convection is displayed, for low salt concentration -cfr. 1(a) -and for high salt concentration -cfr. 1(b). Table 2 and Fig.1b show the stabilizing effect of salt concentration on the onset of convection and, in particular, as C increases, the increasing of R O is slower than the increasing of R S and, as already observed, for C * 4.0839 there is the switch from steady to oscillatory convection. While the Soret effect does not directly affect the oscillatory instability threshold, as already pointed out, Eur. Phys. J. Plus (2022) 137:1034 its increasing leads to a decreasing of the thermal critical stationary Rayleigh number, so the Soret effect has a inhibiting effect on the onset of stationary convection (see Table 3 and Fig. 2b). On the other hand, 1 Le does not directly affect the steady instability threshold but has a destabilizing effect on the onset of oscillatory convection (see Fig. 2a).
The condition (31) gives us a prediction of the type of motions through which convection will arise and it is tested in Table 4, in particular Table 4(a), shows that for 1 Le ≤ 1 the necessary conditions for the onset of oscillatory convection are not satisfied, so convection can arise only via stationary motions.
In Fig. 3a, b the steady and oscillatory thresholds and the global nonlinear threshold are depicted for quoted values of the Soret number, in particular for S 1 the coincidence between R E and R S is depicted in (3)b.

Conclusions
In the present paper the onset of convection in a bi-disperse porous layer, filled by an incompressible fluid mixture, that uniformly rotates about a vertical axis and that is simultaneously heated and salted from below, was studied. Taking into account the Soret effect, via linear instability analysis it was determined that convection can set in via stationary or oscillatory motions and that the critical thermal Rayleigh numbers for the onset of convection and the concentration Rayleigh number have a linear dependence on each other. Moreover, both rotation and salt concentration have a stabilizing effect on the onset of steady and oscillatory convection.
Through differential constraint approach, the global nonlinear stability threshold was determined and it was found that regions of possible subcritical instabilities are present. Numerical simulations were performed in order to test the theoretical proven results.