Double-Diffusive Convection in Bidispersive Porous Medium with Coriolis Effect

Abstract

In this paper, the thermal instability of rotating convection in a bidispersive porous layer is analyzed. The linear stability analysis is employed to examine the stability of the system. The neutral curves for different values of the physical parameters are shown graphically. The critical Rayleigh number is evaluated for appropriate values of the other governing parameters. Among the obtained results, we find: the Taylor number has a stabilizing effect on the onset of convection; the Soret number does not show any effect on oscillatory convection, as the oscillatory Rayleigh number is independent of the Soret number; there exists a threshold, $R_c^{*}$ ∈ (0.45, 0.46), for the solute Rayleigh number, such that, if $R_C$ > $R_c^{*}$, then the convection arises via an oscillatory mode; and the oscillatory convection sets in and as soon as the value of the Soret number reaches a critical value, (∈(0.6, 0.7)), and the convection arises via stationary convection.

1. Introduction

In recent years, great attention has been devoted to the thermal instability in bidispersive porous medium (BDPM). A BDPM is an extension of a regular porous medium. In general, it is considered a regular porous medium where the solid phase is replaced by another porous medium. A BDPM is composed of clusters of large particles that are agglomerations of small particles [1,2]. The voids between the clusters are known as macropores, and the voids within the clusters are known as micropores. In other words, a BDPM is a porous medium in which fractures or tunnels have been introduced. In the present model, the f-phase and p-phase are represented by ’fracture phase’ and ’porous phase’, respectively. Understanding convection in a BDPM is of considerable interest for geophysical applications [3,4]. The theory of thermal convection in a BDPM was developed by Nield and Kuznetsov [5,6,7,8,9,10,11], Kuznestsov and Nield [12], and Sraughan [13,14]. All these authors considered two different velocities and two different temperatures in the macro and micro pores. In their analysis, they found that, in a BDPM, the critical values of Rayleigh numbers are much larger than those in the regular porous medium. Later, much research made an effort to investigate the convective instability in a BDPM.

Very recently, Falsaperla et al. [15] and Gentile and Straughan [16,17] studied the same problem by using a single equation for temperature. In particular, Gentile and Straughan [16,17] analyzed the non-linear stability theory for the problem of thermal convection in a BDPM. They proved that the linear and non-linear stability thresholds coincide. Very recently, Capone et al. [18] have shown that the linear instability and non-linear stability thresholds for the problem of thermal instability in a rotating BDPM are different. Later, Capone and De Luca [19] extended their work by considering inertia terms, and they showed that the effect of the Vadasz number can give rise to an oscillatory mode at the loss of stability of a thermal motionless state.

On the other hand, double-diffusive instability in porous media is an interesting subject of research due to its applications in different industries, such as the migration of solutes in watersaturated soils, the spread of pollutants, drying processes, evaporative cooling of high-temperature systems, and solar ponds [8]. The study of thermosolutal convection of a fluidsaturated porous medium has attracted the attention of many researchers [20,21,22,23,24,25,26,27,28]. In addition, Straughan [29] developed a model for double-diffusive convection in a BDPM. Later, Straughan [30] extended this work by considering the effect of inertia. He showed that the inertia term had a very strong effect on the double-diffusive convection in a BDPM. Badday and Harfash [31] have studied the double-diffusive convection in BDPM with chemical reaction and magnetic field effects.

In this paper, the coriolis effect on thermosolutal convection in a rotating bidispersive porous layer is studied. We reconsider the problem investigated in [18] in light of the Soret effect. The plan of the article is as follows. Section 2 describes the mathematical problem. In Section 3, we describe the linear stability analysis. The critical values of Rayleigh numbers at the onset of stationary and oscillatory convection are determined. The results and discussions are presented in Section 4, which contains a table to provide some examples in which stationary or oscillatory instability sets in, and figures showing the neutral stability curves for steady and oscillatory instability. The paper ends with a conclusion part in Section 5.

2. Mathematical Formulation

Let us consider a horizontal fluid saturated bidisperse porous layer confined between $z=0$ and $z=d$. In this setting, let $V_i^f$ and $V_i^p$ be the velocity of the fluid in the macro pores and the velocity of the fluid in the micro pores, respectively. The fixed temperatures at $z=0$ and at $z=d$ are $T_L^{0}C$ and $T_U^{0}C$, respectively, with $T_L>T_U>0$. It is rotating at a constant rate $\Omega$. The axis of rotation is parallel to z-axis. The Boussinesq approximation is used to account for the density variations.

The hydrodynamic model representing flow behavior in bidisperse porous layer differs from the classical porous layer theory by exhibiting two different pressures in the pores, following the multiporosity model. The flow within each type of pores is determined by its own pressure gradient through Darcy’s law. Hence, four additional equations corresponding to the micro-pores are considered to make the relevant equations for mass and momentum balances closed. The governing equations consist of the momentum and continuity equations (see the references [18,31], and the visual representation in Figure 1). By adopting the Boussinesq approximation in the macro and micro pores, these equations can be written as

$$\begin{array}{c}\nabla ·{\mathbf{V}}^f=0,\nabla ·{\mathbf{V}}^p=0,\end{array}$$

(1)

$$\begin{array}{c}-\frac{\mu }{{\kappa }_f}{\mathbf{V}}^f-\delta \left({\mathbf{V}}^f-{\mathbf{V}}^p\right)-\nabla P^f-\rho g{\widehat{e}}_z-\frac{2{\rho }_0\Omega }{\delta }{\widehat{e}}_z\times {\mathbf{V}}^f=0,\end{array}$$

(2)

$$\begin{array}{c}-\frac{\mu }{{\kappa }_p}{\mathbf{V}}^p-\delta \left({\mathbf{V}}^p-{\mathbf{V}}^f\right)-\nabla P^p-\rho g{\widehat{e}}_z-\frac{2{\rho }_0\Omega }{ϵ}{\widehat{e}}_z\times {\mathbf{V}}^p=0.\end{array}$$

(3)

Figure 1. Physical Configuration.

Then, we consider a linear relation for the density of form

$$\begin{array}{c}\hfill \rho ={\rho }_0\left[1-\alpha \left(T-T_0\right)+{\alpha }_c\left(C-C_0\right)\right].\end{array}$$

(4)

The equation of the energy balance can be written as

$$\begin{array}{c}\hfill {\left(\rho c\right)}_m\frac{\partial T}{\partial t}+{\left(\rho c\right)}_f\left({\mathbf{V}}^f+{\mathbf{V}}^p\right)·\nabla T=k_m{\nabla }^{2}T,\end{array}$$

(5)

where c is the specific heat in the porous medium. The coefficients ${\left(\rho c\right)}_m$ and $k_m$ are given by

$$\begin{array}{c}{\left(\rho c\right)}_m=(1-ϵ)(1-\delta ){\left(\rho c\right)}_s+\left[\delta +ϵ(1-\delta )\right]{\left(\rho c\right)}_f,\end{array}$$

(6)

$$\begin{array}{c}{k}_m=(1-ϵ)(1-\delta )k_s+\left[\delta +ϵ(1-\delta )\right]k_f.\end{array}$$

(7)

The equation for the concentration field taking into account the Soret effect on the diffusion coefficient can be written as

$$\begin{array}{c}\hfill {\epsilon }_1\frac{\partial C}{\partial t}+\left({\mathbf{V}}^f+{\mathbf{V}}^p\right)·\nabla C={\epsilon }_2{\nabla }^{2}C+\widehat{S}{\nabla }^{2}T,\end{array}$$

(8)

where

$$\begin{array}{cc}\hfill & {\epsilon }_1=\delta +ϵ(1-\delta ),\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\epsilon }_2=\delta k_c^f+ϵ(1-\delta )k_c^p,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & \widehat{S}=\varphi S_T^f+ϵ(1-\varphi )S_T^p,\hfill \end{array}$$

(11)

subject to the boundary conditions

$$\begin{array}{cc}\hfill & {\mathbf{V}}^f·{\widehat{e}}_z={\mathbf{V}}^p·{\widehat{e}}_z=0,onz=0,d,\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & T(x,y,0,t)=T_L,T(x,y,d,t)=T_U(T_L>T_U),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & C(x,y,0,t)=C_L,C(x,y,d,t)=C_U(C_L>C_U).\hfill \end{array}$$

(14)

The basic state solution is then

$$\begin{array}{c}\hfill {\mathbf{V}}_b^f=0,{\mathbf{V}}_b^p=0,T_b=T_L-\beta z,C_b=C_L-{\beta }_{c}z,\end{array}$$

(15)

where $\beta =\frac{T_L-T_U}{d}$ and ${\beta }_c=\frac{C_L-C_U}{d}$.

Let ${\mathbf{V}}^f,{\mathbf{V}}^p,P^f,P^p,T$, and C be a perturbation to the steady Equation (15).

The perturbations are non-dimensional, with length scale d, velocity scale V, time scale $\tau$, temperature scale $T*$, and concentration scale $C*$, where

$$\begin{array}{cccc}\hfill & \tau =\frac{{\left(\rho c\right)}_m{d}^2}{k_m},\hfill & \hfill V& =\frac{k_m}{{\left(\rho c\right)}_{f}d},\hfill \\ \hfill & T*=\frac{\beta V{\left(\rho c\right)}_f{d}^2}{k_m},\hfill & \hfill C*& =\frac{{\beta }_{c}V{d}^2}{{\epsilon }_2}.\hfill \end{array}$$

Define the quantities $\gamma ,{\kappa }_r,A,\eta$, R, $R_C$, $Ta$, $Le$, and S by

$$\begin{array}{cccccc}\hfill & \gamma =\frac{\delta {\kappa }_f}{\mu },\hfill & \hfill {\kappa }_r& =\frac{{\kappa }_f}{{\kappa }_p},\hfill & \hfill \varpi & =\frac{{\left(\rho c\right)}_m}{{\left(\rho c\right)}_f},\hfill \\ \hfill & Ta=\frac{2{\rho }_0\Omega {\kappa }_f}{\mu \varphi },\hfill & \hfill R& =\frac{{\rho }_0\beta g\alpha d^2{\left(\rho c\right)}_f{\kappa }_f}{\mu k_m},\hfill & \hfill R_C& =\frac{{\rho }_0{\beta }_{c}g{\alpha }_c{d}^2{\kappa }_f}{\mu {\epsilon }_2},\hfill \\ \hfill & Le=\frac{k_m}{{\left(\rho c\right)}_m{\epsilon }_2},\hfill & \hfill S& =\frac{\widehat{S}T*}{{\epsilon }_{2}C*}.\hfill \end{array}$$

All these quantities have been explained in the nomenclature. The non-dimensional equations (after omitting the asterisks) governing the system are

$$\begin{array}{cc}\hfill & \nabla ·{\mathbf{V}}^f=0,\nabla ·{\mathbf{V}}^p=0,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & -{\mathbf{V}}^f-\gamma \left({\mathbf{V}}^f-{\mathbf{V}}^p\right)-\nabla P^f+\left(R\theta -R_C\varphi \right){\widehat{e}}_z-Ta{\widehat{e}}_z\times {\mathbf{V}}^f=0,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & -{\kappa }_r{\mathbf{V}}^p-\gamma \left({\mathbf{V}}^p-{\mathbf{V}}^f\right)-\nabla P^p+\left(R\theta -R_C\varphi \right){\widehat{e}}_z-\eta Ta{\widehat{e}}_z\times {\mathbf{V}}^p=0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & \frac{\partial \theta }{\partial t}+\left({\mathbf{V}}^f+{\mathbf{V}}^p\right)·\nabla \theta =\left({\mathbf{w}}^f+{\mathbf{w}}^p\right)+{\nabla }^2\theta ,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & {\epsilon }_{1}Le\frac{\partial \varphi }{\partial t}+ALe\left({\mathbf{V}}^f+{\mathbf{V}}^p\right)·\nabla \varphi =\left({\mathbf{w}}^f+{\mathbf{w}}^p\right)+{\nabla }^2\varphi +S{\nabla }^2\theta ,\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & {\mathbf{w}}^f={\mathbf{w}}^p=\theta =\varphi =0\mathrm{on}z=0,1.\hfill \end{array}$$

(21)

By taking the third component of curl of Equations (17) and (18), one obtains

$$\begin{array}{cc}\hfill & {\mathbf{w}}_3^f+\gamma \left({\mathbf{w}}_3^f-{\mathbf{w}}_3^p\right)-Ta\frac{\partial {\mathbf{w}}^f}{\partial z}=0,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & {\kappa }_r{\mathbf{w}}_3^p+\gamma ({\mathbf{w}}_3^p-{\mathbf{w}}_3^f)-\eta Ta\frac{\partial {\mathbf{w}}^p}{\partial z}=0,\hfill \end{array}$$

(23)

where $D=\frac{\partial }{\partial t}$, ${\mathbf{w}}_3^f=\frac{\partial v^f}{\partial x}-\frac{\partial u^f}{\partial y}$.

By taking the third component of double curl of Equations (17) and (18), one has

$$\begin{array}{cc}\hfill & {\nabla }^2{\mathbf{w}}^f+\gamma \left({\nabla }^2{\mathbf{w}}^f-{\nabla }^2{\mathbf{w}}^p\right)-R{\nabla }_h^2\theta +R_C{\nabla }_h^2\varphi +Ta\frac{\partial {\mathbf{w}}_3^f}{\partial z}=0,\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & {\kappa }_r{\nabla }^2{\mathbf{w}}^p+\gamma \left({\nabla }^2{\mathbf{w}}^p-{\nabla }^2{\mathbf{w}}^f\right)-R{\nabla }_h^2\theta +R_C{\nabla }_h^2\varphi +\eta Ta\frac{\partial {\mathbf{w}}_3^p}{\partial z}=0,\hfill \end{array}$$

(25)

where

$${\nabla }_h^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}$$

and

$${\nabla }^2=\frac{{\partial }^2}{\partial x^2}+\frac{{\partial }^2}{\partial y^2}+\frac{{\partial }^2}{\partial z^2}.$$

Solving Equations (22) and (23) with respect to ${\mathbf{w}}_3^f$ and ${\mathbf{w}}_3^p$, respectively, one has

$$\begin{array}{cc}\hfill & {\mathbf{w}}_3^f=\frac{Ta\left(\gamma +{\kappa }_r\right){\mathbf{w}}_z^f+\eta Ta\gamma {\mathbf{w}}_z^p}{\gamma +{\kappa }_r+\gamma {\kappa }_r},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & {\mathbf{w}}_3^p=\frac{Ta\left(\gamma {\mathbf{w}}_z^f\right)+\eta Ta\left(1+\gamma \right){\mathbf{w}}_z^p}{\gamma +{\kappa }_r+\gamma {\kappa }_r}.\hfill \end{array}$$

(27)

Substituting Equations (26) and (27) into Equations (24) and (25), respectively, one obtains

$$\begin{array}{cc}\hfill & {\nabla }^2{\mathbf{w}}^f+\gamma \left({\nabla }^2{\mathbf{w}}^f-{\nabla }^2{\mathbf{w}}^p\right)-R{{\nabla }_h}^2\theta +R_C{{\nabla }_h}^2\varphi +\frac{{Ta}^2\left(\gamma +{\kappa }_r\right){\mathbf{w}}_{zz}^f+\eta {Ta}^2\gamma {\mathbf{w}}_{zz}^p}{\gamma +{\kappa }_r+\gamma {\kappa }_r}=0,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill & {\kappa }_r{\nabla }^2{\mathbf{w}}^p+\gamma \left({\nabla }^2{\mathbf{w}}^p-{\nabla }^2{\mathbf{w}}^f\right)-R{{\nabla }_h}^2\theta +R_C{{\nabla }_h}^2\varphi +\frac{\eta {Ta}^2\gamma {\mathbf{w}}_{zz}^f+{\eta }^2{Ta}^2\left(1+\gamma \right){\mathbf{w}}_{zz}^p}{\gamma +{\kappa }_r+\gamma {\kappa }_r}=0.\hfill \end{array}$$

(29)

Hence, considering Equations (19), (20), (28) and (29), we see the following problem in ${\mathbf{w}}^f,{\mathbf{w}}^p,\theta ,$ and $\varphi$:

$$\begin{array}{cc}\hfill {\nabla }^2{\mathbf{w}}^f+\gamma \left({\nabla }^2{\mathbf{w}}^f-{\nabla }^2{\mathbf{w}}^p\right)& -R{{\nabla }_h}^2\theta +R_C{{\nabla }_h}^2\varphi +\frac{{Ta}^2\left(\gamma +{\kappa }_r\right){\mathbf{w}}_{zz}^f+\eta {Ta}^2\gamma {\mathbf{w}}_{zz}^p}{\gamma +{\kappa }_r+\gamma {\kappa }_r}=0,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill {\kappa }_r{\nabla }^2{\mathbf{w}}^p+\gamma \left({\nabla }^2{\mathbf{w}}^p-{\nabla }^2{\mathbf{w}}^f\right)& -R{{\nabla }_h}^2\theta +R_C{{\nabla }_h}^2\varphi +\frac{\eta {Ta}^2\gamma {\mathbf{w}}_{zz}^f+{\eta }^2{Ta}^2\left(1+\gamma \right){\mathbf{w}}_{zz}^p}{\gamma +{\kappa }_r+\gamma {\kappa }_r}=0,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \frac{\partial \theta }{\partial t}={\mathbf{w}}^f+{\mathbf{w}}^p+{\nabla }^2\theta ,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\epsilon }_{1}Le\frac{\partial \varphi }{\partial t}=& {\mathbf{w}}^f+{\mathbf{w}}^p+{\nabla }^2\varphi +S{\nabla }^2\theta .\hfill \end{array}$$

(33)

3. Linear Stability Analysis

Let us consider the normal mode solutions in the form of

$$\begin{array}{c}\hfill \left({\mathbf{w}}^f,{\mathbf{w}}^p,\theta ,\varphi \right)=\left(w^f,w^p,\theta ,\varphi \right)sin\left(n\pi z\right)e^{i(lx+my)+\sigma t}.\end{array}$$

(34)

Substituting the above normal mode solution into the Equations (30)–(33), we find

$$\begin{array}{cc}\hfill [A\Lambda (1+\gamma )+n^2{\pi }^2{Ta}^{2}B]{\mathbf{w}}^f+[& \eta \gamma n^2{\pi }^2{Ta}^2-\gamma \Lambda A]{\mathbf{w}}^p-a^{2}RA\theta +a^2{R}_{C}A\varphi =0,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \eta \gamma n^2{\pi }^2{Ta}^2-\gamma \Lambda A]{\mathbf{w}}^f+[\Lambda AB& +{\eta }^2{n}^2{\pi }^2{Ta}^2(1+\gamma )]{\mathbf{w}}^p-a^{2}RA\theta +a^2{R}_{C}A\varphi =0,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & {\mathbf{w}}^f+{\mathbf{w}}^p+[\sigma -\Lambda ]\theta =0,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill {\mathbf{w}}^f+{\mathbf{w}}^p& -S\Lambda \theta -[{\epsilon }_{1}Le\sigma +\Lambda ]\varphi =0,\hfill \end{array}$$

(38)

where

$$\left\{\begin{array}{c}{a}^2=l^2+m^2\mathrm{is}\mathrm{the}\mathrm{wave}\mathrm{number},\hfill \\ \sigma =\iota \omega ,\hfill \\ A=\gamma +{\kappa }_r+\gamma {\kappa }_r,\hfill \\ B=\gamma +{\kappa }_r,\hfill \\ \Lambda ={\pi }^2+a^2.\hfill \end{array}\right.$$

Requiring zero determinant of the above system, one has

$$\begin{array}{c}\hfill R=\frac{{\xi }_1+{\omega }^2{\xi }_2+\iota \left({\xi }_3+{\omega }^2{\xi }_4\right)}{{\xi }_5},\end{array}$$

(39)

with

$$\left\{\begin{array}{c}{\xi }_1={\Lambda }^2[-a^2\Lambda R_C(S-1)(x_1+A\Lambda (1+B+3\gamma ))+\Lambda (x_2+x_3\Lambda +x_4{\Lambda }^2)],\hfill \\ {\xi }_2=x[a^{2}A{R}_C(x_1+A\Lambda (1+B+3\gamma ))+x\Lambda (x_2+x_3\Lambda +x_4{\Lambda }^2)],\hfill \\ {\xi }_3=a^{2}A\Lambda R_C(1-x+Sx)[x_1+A\Lambda (1+B+3\gamma )]+{\Lambda }^2[x_2+x_3\Lambda +x_4{\Lambda }^2],\hfill \\ {\xi }_4=x^2(x_2+x_3\Lambda +x_4{\Lambda }^2),\hfill \\ {\xi }_5=a^{2}A({\omega }^2{x}^2+{\Lambda }^2)[{\pi }^2{Ta}^2(B+\eta (\gamma \eta -2\gamma +\eta ))+A\Lambda (1+B+3\gamma )],\hfill \\ x_1={\pi }^2{Ta}^2(B+{\eta }^2+{\eta }^2\gamma -2\eta \gamma ),\hfill \\ x_2={\pi }^4{Ta}^4(B+B\gamma -{\gamma }^2){\eta }^2,\hfill \\ x_3=A{\pi }^2{Ta}^2(B^2+2\eta {\gamma }^2+(1+{\gamma }^2){\eta }^2),\hfill \\ x_4=A^2(B+B\gamma -{\gamma }^2),\hfill \\ x=Le{\epsilon }_1.\hfill \end{array}\right.$$

3.1. Stationary Convection:

Substituting $\omega =0$ in Equation (39), one obtains

$$\begin{array}{c}\hfill R_{T_{sc}}=\frac{{\xi }_6+{\xi }_7\Lambda +{\xi }_8{\Lambda }^2+{\xi }_9{\Lambda }^3}{{\xi }_{10}+{\xi }_{11}\Lambda },\end{array}$$

(40)

where

$$\left\{\begin{array}{c}{\xi }_6=a^2{\pi }^{2}Rc(1-S)T{a}^2({\kappa }_r+\gamma {(-1+\eta )}^2+{\eta }^2),\hfill \\ {\xi }_7=a^{2}ARc(1-S)(1+k+4\gamma )+{\pi }^{4}T{a}^4{\eta }^2,\hfill \\ {\xi }_8={\pi }^{2}T{a}^2({({\kappa }_r+\gamma )}^2+2\eta {\gamma }^2+{(1+\gamma )}^2{\eta }^2),\hfill \\ {\xi }_9=A^2,\hfill \\ {\xi }_{10}=a^2{\pi }^{2}T{a}^2({\kappa }_r+\gamma {(-1+\eta )}^2+{\eta }^2),\hfill \\ {\xi }_{11}=a^{2}A(1+k+4\gamma ).\hfill \end{array}\right.$$

In the absence of rotation and the Soret effect, the above-stationary Rayleigh number reduces to

$$\begin{array}{c}\hfill R{a}_{sc}=\frac{{\delta }^4\left(\gamma +{\kappa }_r+\gamma {\kappa }_r\right)}{q^2\left(1+{\kappa }_r+4\gamma \right)},\end{array}$$

(41)

which, on comparison, satisfies [16] (Equation (31)).

The case of a monodispersive porous layer rotating about a vertical axis with the Darcy model has been considered in Capone and Rionero [32]. As ${\kappa }_r\to \infty$, $\gamma \to 0$$Rc\to 0$, and $\eta \to \infty$ in Equation (40), we find

$$\begin{array}{c}\hfill R{a}_{sc}=\frac{{\delta }^2({\pi }^{2}T{a}^2+{\delta }^2)}{q^2}.\end{array}$$

(42)

After some calculations, we find

$$\begin{array}{c}\hfill R{a}_{scl}={\pi }^2{(1+\sqrt{1+T{a}^2})}^2,\end{array}$$

(43)

which is in good agreement with [32] (Equation (4.24), p. 195).

3.2. Oscillatory Convection

To study the oscillatory stability, we consider the real and imaginary parts of R. The Rayleigh number at the onset of oscillatory convection is

$$\begin{array}{c}\hfill R_{T_{oc}}=\frac{{\xi }_{12}+{\xi }_{13}\Lambda +{\xi }_{14}{\Lambda }^2+{\xi }_{15}{\Lambda }^3}{{\xi }_{16}+{\xi }_{17}\Lambda },\end{array}$$

(44)

where

$$\left\{\begin{array}{c}{\xi }_{12}=a^2{\pi }^{2}RcT{a}^2({\kappa }_r+\gamma {(-1+\eta )}^2+{\eta }^2),\hfill \\ {\xi }_{13}=a^{2}ARc(1+k+4\gamma )+{\pi }^{4}T{a}^4(1+x){\eta }^2,\hfill \\ {\xi }_{14}={\pi }^{2}T{a}^2(1+x)({({\kappa }_r+\gamma )}^2+2\eta {\gamma }^2+{(1+\gamma )}^2{\eta }^2),\hfill \\ {\xi }_{15}=(1+x)A^2,\hfill \\ {\xi }_{16}=x{a}^2{\pi }^{2}T{a}^2({\kappa }_r+\gamma {(-1+\eta )}^2+{\eta }^2),\hfill \\ {\xi }_{17}=x{a}^{2}A(1+k+4\gamma ).\hfill \end{array}\right.$$

4. Discussion

The numerical results and discussions are presented in this section. The critical Rayleigh number at the onset of stationary convection, $R{a}_{T_{SC}}^c$; at the onset of oscillatory convection, $R{a}_{T_{OC}}^c$; the critical wave number at the onset of stationary convection, $q_{sc}^c$; and at the onset of oscillatory convection, $q_{oc}^c$, are obtained for the prescribed values of other parameters. Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8 show the neutral curves in the parametric plane $(q,R_T)$ with different values of the $Ta,S,R_C$, and ${\kappa }_r$.

Figure 2. Neutral curves for the different values of $Ta$ and for the fixed values of $\gamma =0.5,\eta =0.2,{\kappa }_r=1,$$R_C=50$, and $S=0.5$ for the stationary mode.

Figure 3. Neutral curves for the different values of S and for the fixed values of $\gamma =0.5$, $\eta =0.2$, ${\kappa }_r=1$, $R_C=50$, and $Ta=20$ for the stationary mode.

Figure 4. Neutral curves for the different values of $R_C$ and for the fixed values of $\gamma =0.5,\eta =0.2,$${\kappa }_r=1,S=0.2$, and $Ta=50$ for the stationary mode.

Figure 5. Neutral curves for the different values of ${\kappa }_r$ and for the fixed values of $\gamma =0.5,\eta =0.2,$$R_C=50,S=0.2$, and $Ta=50$ for the stationary mode.

Figure 6. Neutral curves for the different values of $Ta$ and for the fixed values of $\gamma =0.5,\eta =0.2,$$R_C=50$, and ${\kappa }_r=1$ for the oscillatory mode.

Figure 7. Neutral curves for the different values of $R_C$ and for the fixed values of $\gamma =0.5,\eta =0.2,$$Ta=50$, and ${\kappa }_r=1$ for the oscillatory mode.

Figure 8. Neutral curves for the different values of ${\kappa }_r$, and for the fixed values of $\gamma =0.5,\eta =0.2,Ta=50$ and $R_C=100$ for the oscillatory mode.

In the stationary mode, the neutral curves are displayed in Figure 2, Figure 3, Figure 4 and Figure 5. Figure 2 shows the neutral curves in the parametric plane $(q,R_T)$ with different values of the Taylor number. From this figure, one can observe that, as $Ta$ increases, the curves shift upward, indicating a delay in the onset of instability. This can be explained as follows: Vorticity is introduced into the fluid when it rotates. As a result, the fluid travels faster in horizontal planes. The velocity of the fluid perpendicular to the planes decreases as a result of this motion, therefore $R{a}_{T_{SC}}^c$ rises with $Ta$.

The effect of the Soret parameter on the onset of instability is shown in Figure 3. In it, we see that $R_{T_{sc}}^c$ decreases with the Soret parameter, which means that the Soret parameter destabilizes the system. For various values of solute Rayleigh number, with changing values of wave number and then Rayleigh numbers, the neutral curves are obtained in Figure 4. We can see from this figure that $R_{T_{sc}}^c$ increases as $R_c$ increases, indicating that the presence of $R_c$ suppresses the onset of convection.

Figure 5 depicts the neutral curves at the onset of stationary convection for various values of ${\kappa }_r$. According to this figure, $R_{T_{sc}}^c$ decreases as ${\kappa }_r$ increases, indicating that the presence of a solute Rayleigh number advances the onset of convection. The neutral curves at the onset of oscillatory convection are displayed in Figure 6, Figure 7 and Figure 8. Figure 6 displays the neutral curves for different values of $Ta$. According to this figure, increasing $Ta$ causes $R_{T_{oc}}^c$ to increase, indicating that $Ta$ has the effect of stabilizing the system.

Figure 7 depicts the neutral curves for different values of $R_C$ at the onset of oscillatory convection, and it is found that the neutral curves move upward with an increase in the value of $R_C$, thus $R_C$ stabilizes the oscillatory convection.

Figure 8 shows the effect of ${\kappa }_r$. In particular, we observe that the effect of ${\kappa }_r$ advances the onset of convection. This can be understandable, mathematically, because ${\kappa }_r=\frac{{\kappa }_f}{{\kappa }_p}$, ${\kappa }_r$ increases as ${\kappa }_p$ decreases (${\kappa }_f$ is assumed to be fixed here). In other words, as micropermeability declines, fluid movement in micropores becomes more difficult. As a result, convective motions become more difficult, yielding more stability to the system.

In Table 1, Table 2 and Table 3, we present some examples in which steady or oscillatory instability sets in for the constant values of physical parameters. According to Table 1, there is a threshold $R_c^{*}(\in 0.45,0.46)$ for the solute Rayleigh number, such that, if $R_c>R_c^{*}$, then the convection arises via an oscillatory mode. According to Table 2, oscillatory convection occurs initially, and as soon as the value of S reaches a critical value (∈(0.6, 0.7)), the convection ceases to be oscillatory, and stationary convection occurs as the first bifurcation. Table 3 shows that, as the value of ${\kappa }_r$ increases, convection always occurs via stationary mode.

Table 1. Critical stationary and oscillatory Rayleigh numbers for different values of $Rc$ and the fixed values of ${\kappa }_r=1$, $Ta=5$, and $S=0.5$.

Rc	Stationary R	Stationary a	Oscillatory R	Oscillatory a	Instability
0	61.6464	3.9578	62.7612	3.9578	Stationary
1	62.1464	3.9578	62.7793	3.9578	Stationary
2	62.6464	3.9578	62.7973	3.9578	Stationary
3	63.1464	3.9578	62.8154	3.9578	Oscillatory
4	63.6464	3.9578	62.8335	3.9578	Oscillatory
5	64.1464	3.9578	62.8516	3.9578	Oscillatory

Table 2. Critical stationary and oscillatory Rayleigh numbers for the different values of S and the fixed values of ${\kappa }_r=1$, $Ta=50$, and $Rc=50$.

S	Stationary R	Stationary a	Oscillatory R	Oscillatory a	Instability
0.1	1018.7706	8.2527	992.2842	8.2527	Oscillatory
0.2	1013.7706	8.2527	992.2842	8.2527	Oscillatory
0.3	1008.7706	8.2527	992.2842	8.2527	Oscillatory
0.4	1003.7706	8.2527	992.2842	8.2527	Oscillatory
0.5	998.7706	8.2527	992.2842	8.2527	Oscillatory
0.6	993.7706	8.2527	992.2842	8.2527	Oscillatory
0.7	988.7706	8.2527	992.2842	8.2527	Stationary
0.8	983.7706	8.2527	992.2842	8.2527	Stationary
0.9	978.7706	8.2527	992.2842	8.2527	Stationary

Table 3. Critical stationary and oscillatory Rayleigh numbers for the different values of ${\kappa }_r$ and the fixed values of $S=0.8$, $Ta=50$, and $Rc=50$.

${\mathit{\kappa }}_{\mathit{r}}$	Stationary R	Stationary a	Oscillatory R	Oscillatory a	Instability
1	983.7706	8.2540	992.2842	8.2540	Stationary
2	691.3454	6.4421	694.5709	6.4421	Stationary
3	588.8136	5.5171	590.1849	5.5171	Stationary
4	546.4455	4.9469	547.0506	4.9469	Stationary
5	531.2631	4.5668	531.5936	4.5668	Stationary

5. Conclusions

In this study, we investigated the onset of rotating convection in a horizontal bidispersive porous layer that is uniformly heated and salted from below. The behaviour of various parameters, such as the $Ta,S,R_C$, and ${\kappa }_r$, has been analysed. The results can be summarized as follows:

• $R_{T_{sc}}^c$ and $R_{T_{oc}}^c$ increase as the Taylor number increases, indicating that $Ta$ has a stabilizing effect on the onset of convection.
• $R_{T_{sc}}^c$ and $R_{T_{oc}}^c$ are increasing functions of $R_c$ and decreasing functions of ${\kappa }_r$.
• S does not show any effect on $R_{T_{oc}}^c$, as $R_{T_{oc}}^c$ is independent of S.
• There exists a threshold $R_c^{*}\in (0.45,0.46)$ for the solute Rayleigh number such that, if $R_c>R_c^{*}$, then the convection arises via an oscillatory mode.
• The oscillatory convection sets in and, as soon as the value of S attains a critical value (∈(0.6, 0.7)), the convection ceases to be oscillatory, and stationary convection occurs as the first bifurcation.