# Analytical and Numerical Study of Soret and Dufour Effects on Double Diffusive Convection in a Shallow Horizontal Binary Fluid Layer Submitted to Uniform Fluxes of Heat and Mass

Authors: A. Lagra; M. Bourich; M. Hasnaoui; A. Amahmid; M. Er-Raki
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Combined Soret and Dufour effects on thermosolutal convection induced in a horizontal layer filled with a binary fluid and subject to constant heat and mass fluxes are investigated analytically and numerically. The thresholds marking the onset of supercritical and subcritical convection are predicted analytically and explicitly versus the governing parameters. The present investigation shows that different regions exist in theN-Du plane corresponding to different parallel flow regimes. The number, the extent, and the locations of these regions depend on whether SrDu>-(1+Le2)/2Le2=f(Le) or SrDu<-(1+Le2)/2Le2. Conjugate effects of cross-phenomena on thresholds of fluid flow and heat and mass transfer characteristics are illustrated and discussed.


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## Body

## 1. Introduction

Great interest in the study of thermosolutal convection in fluid and porous media has been motivated by its presence in many engineering applications, such as in hydrology, petrology, geophysics, and material processing technology where melting and solidification of binary alloys are involved [1]. More specifically, such flows are encountered in nature (lakes, solar ponds, and atmosphere). In the industrial field, examples include food processing, chemical processes, crystal growth, energy storage, material processing, and many other examples. Important experimental, analytical, and numerical results on convective heat and mass transfer are documented in earlier books of Nield and Bejan [2] and De Groot and Mazur [3]. Most studies on this topic are concerned with double diffusive convection in vertical/horizontal cavities for which the flows induced by the buoyancy forces result from the imposition of both thermal and solutal boundary conditions on the vertical/horizontal walls [4–6].The diffusion of mass due to temperature gradient is called Soret or thermodiffusion effect that often counts among the main drivers of various convective phenomena occurring within thermal stratified media. Recently, Rahman and Saghir [7] proposed a detailed historical review of works, focusing on different aspects of Soret effect. Examples of interesting phenomena resulting from the coupling between double diffusive convection and Soret effect are available in the paper by Bourich et al. [8] who investigated analytically and numerically Soret-driven thermosolutal convection within a shallow porous or fluid layer subject to a vertical gradient of temperature, using a Brinkman-Hazen-Darcy model in its transient form. The critical Rayleigh numbers for the onset of subcritical, oscillatory, and stationary convection were determined explicitly as functions of the governing parameters for infinite and finite layers.Generally, Soret and Dufour effects are assumed negligible in problems related to double diffusive convection. However, such effects could be of significant effect when density differences exist in the flow regime. In fact, energy flux can be generated by composition of gradients (Dufour or diffusion-thermoeffect). Similarly, mass fluxes can be created by temperature gradients (Soret or thermodiffusion effect). In an earlier study, Malashetty [9] investigated the effect of anisotropic thermoconvective currents, in the presence of Soret and Dufour effects, on the critical Rayleigh number for both marginal and overstable motions. Mortimer and Eyring [10] used the elementary transition state approach to obtain a simple model theory for the Soret and Dufour effects. They found that the results of the theory conform to the Osanger reciprocal relationship. Gaikwad et al. [11] studied the onset of double diffusive convection in a two-component couple stress fluid layer with Soret and Dufour effects using both linear and nonlinear stability analysis. The effects of Soret and Dufour parameters together with the couple stress parameter on the stationary and oscillatory convection are graphically illustrated and discussed. In the presence of Soret and Dufour effects, Nithyadevi and Yang [12] presented numerical results on natural convection in a square enclosure filled with water, partially heated from one vertical wall, and totally cooled from the opposite vertical wall. The study was conducted around the maximum density for three different combinations of the heating element location. Makinde et al. [13] described a theoretical study used to analyze the hydromagnetic flow and mass diffusion of chemical reactive species with first- and higher-order reactions of an electrically conducting fluid over a moving vertical plate. The study was conducted in the presence of Soret and Dufour effects with convective heat exchange at the plate surface. Pal and Mondal [14] considered the problem of steady laminar, hydromagnetic two-dimensional mixed convection flow due to stretching sheet in the presence of Soret and Dufour effects. Cheng [15] examined the Dufour and Soret effects on the steady boundary layer flow due to natural convection heat and mass transfer over a vertical cone embedded in a porous medium with constant wall temperature and concentration. The results presented show that the effects of Dufour and Soret parameters on the local surface temperature are increased by increasing the Lewis number. Tsai and Huang [16] investigated heat and mass transfer from natural convection flow along a vertical surface with variable heat fluxes embedded in a porous medium due to Soret and Dufour effects. They concluded that Soret and Dufour effects could play a significant role. Soret and Dufour effects have been also considered by Hayat et al. [17] who studied mixed convection boundary layer flow about a linearly stretching vertical surface in a porous medium filled with a viscoelastic fluid and, more recently, by Wang et al. [18] who studied the onset of double diffusive convection in a horizontal cavity.The main purpose of the present investigation is to study analytically and numerically the combined effects of Soret and Dufour parameters on double diffusive convection developed in a horizontal layer filled with a binary fluid. This paper is an extended version of preliminary results presented in a conference [19]. Analytical predictions are developed and validated numerically for shallow enclosures. The Dufour parameter effects on thresholds of stationary convection, subcritical convection, flow structure, and heat and mass transfer are also discussed.


## 2. Mathematical Formulation

The system under study is a two-dimensional shallow cavity of lengthL′ and height H′, filled with a binary fluid. The vertical end-walls of the layer are adiabatic and impermeable to mass transfer while its horizontal walls are subject to uniform fluxes of heat, q′, and mass, j′. The flow is assumed to obey the Boussinesq approximation. Using the vorticity and the stream function formulation and taking into account the cross-phenomena (Soret and Dufour effects), the dimensionless governing equations are obtained as follows:(1)1Pr∂∇2ψ∂t+u∂∇2ψ∂x+v∂∇2ψ∂y=∇4ψ-RT∂T∂x+N∂S∂x,(2)∂T∂t+u∂T∂x+v∂T∂y=∇2T+DuN∇2S,(3)∂S∂t+u∂S∂x+v∂S∂y=SrN∇2T+1Le∇2S,(4)∇2ψ=-ζ.The associated hydrodynamic, thermal, and solutal boundary conditions are(5a)x=±Ar2:ψ=0,∂T∂x=0,∂S∂x=0,(5b)y=±12:ψ=0,∂T∂y=-1-NDu1-LeSrDu=-φT,∂S∂y=-1-LeSr/N1-LeSrDu=-φS.The parameters governing the problem are the thermal Darcy-Rayleigh number,RT=gβTΔT′H′3/(ҡ11ν), the Prandlt number, Pr⁡=ν/ҡ11, the Lewis number, Le=ҡ11/ҡ22, the buoyancy ratio, N=βSΔS′/(βTΔT′), and the aspect ratio of the cavity, Ar=L′/H′. The parameters Sr and Du are, respectively, the Soret and Dufour parameters expressed as Sr=βSҡ21/(βTҡ11) and Du=βTҡ12/(βSҡ11), with ҡ11 being the thermal diffusivity,ҡ12 being cross-diffusion due to solute concentration component,ҡ21 being cross-diffusion due to temperature component, and ҡ22 being solute diffusivity.In the presence of the cross-diffusion phenomena, the Nusselt and Sherwood numbers are defined as follows:(6)Nu=1ΔT+NDuΔS,Sh=1ΔS+Sr·Le/NΔT,where ΔT=T(0,-1/2)-T(0,1/2) and ΔS=S(0,-1/2)-S(0,1/2) are the temperature and concentration differences, evaluated at x=0 with the origin of the coordinate system being taken at the center of the cavity.


## 3. Methods

### 3.1. Numerical Solution

The numerical solution of (1) to (3) was obtained using a finite-difference method, described in detail by Bourich et al. [8]. A second-order scheme was used for the discretization of the spatial derivatives. Equations (2)-(3) were marched in time using the Alternating Direction Implicit (ADI) method. The stream function equation (4) was solved at each time step with the Point Successive Overrelaxation (PSOR) method with an optimum overrelaxation coefficient calculated for the used grid. In addition, a convergence criterion was adopted for the stream function to satisfy a variation by less than 10−5 for each time step. A second criterion ∑∑Γi,jn+1-Γi,jn/∑∑Γi,jn+1≤10-6 was used to check the convergence of the numerical code. Here, Γ stands for any of the variables T, S, or ψ. The superscripts n and (n+1) indicate the iterations numbers, and the subscripts i and j indicate locations in the grid system. For large aspect ratio enclosures, nonuniform grid was used in the x-direction near the short walls to capture the flow details near the enclosures end-walls and also in the y-direction to obtain a finer grid in the close vicinity of the horizontal walls.


### 3.2. Analytical Solution

For a shallow enclosure with constant heat and mass flux boundary conditions, an approximate analytical solution based on the parallel flow concept is possible, which renders the problem amenable to a parametric study while retaining the essential physics of the problem. The analytical solution is developed for steady-state flows using the parallel flow approximation (see, e.g., Bourich et al. [8]), which leads to the following simplifications (justified by examining the streamlines, isotherms, and iso-concentration lines obtained numerically). These simplifications lead to ψ(x,y)=ψ(y), T(x,y)=CTx+θT(y), and S(x,y)=CSx+θS(y), where CT and CS are, respectively, unknown constant temperature and concentration gradients in the x-direction (the direction of the long sides of the cavity). Using these approximations together with the boundary conditions (5a) and (5b), (1)–(3) reduce to a set of ordinary differential equations for which the solution is obtained as follows:(7)ψy=ψ04y2-12,Tx,y=CTx+CT-CSDuNLe1-LeSrDuψ015y48y4-40y2+15-φTy,Sx,y=CSx+CS-CTSr/N1-LeSrDu·Leψ015·y8y4-40y2+15-φSy,where ψ0 is the stream function value at the midheight of the layer, given by(8)ψ0=RT384CT+NCS.The analytical expressions of CT and CS were determined by using thermal and solutal balances in the layer, which leads to the following expressions:(9a)CT=-815ψ02aψ02φT+φSNDuLe2+AφT-φSLeNDu2aψ022+2aψ02B+A2,(9b)CS=-815ψ02aψ02φTLeSr/N+φSLe+ALeφS-φTSr/N2aψ022+2aψ02B+A2,where a=64/315,  A=1-LeSrDu, and B=Le2+2Le2DuSr+1.Introducing the expressions ofCT and CS into (8) yields a fourth-order polynomial in terms of ψ0 for which the following solutions are obtained:(10)ψ0=0,ψ0=∓12-d1∓d12-d21/2,with(11)d1=12aLe2B-RT720F1,d2=1a2Le2A2-RT720F2,F1=LeN+Le,F2=NC1+C2,where C1=DuLeSr-1-1+Le and C2=SrLeLeDu-1-1+1.From a mathematical point of view, (10) may exhibit, in addition to the rest state solution, two types of bifurcations depending on the sign within the square root. Although it should be mentioned that several numerical tests were performed, only the solutions corresponding to positive sign were obtained numerically. Hence, the convective solution with the negative sign within the square root is termed “unstable solution” and the convective solution corresponding to positive sign within the square root is similarly qualified “stable solution.”The new expressions of Nusselt and Sherwood numbers are obtained as follows:(12)Nu=11-8/15ψ0CT,Sh=11-8/15ψ0LeCS.Foremost, it may be remarked from (10) that the parallel flow solutions exist only when the following two conditions are satisfied:(13)-d1∓d12-d2>0,d12-d2>0.The resolution of the inequalities of (13) is performed in the N-Du plane with Sr, Le, and RT as parameters. Depending of the sign of B (parameter in the expression of d1 given before), two main cases are possible.


#### 3.2.1. The ParameterB>O

This condition is satisfied ifSrDu>-(1+Le2)/2Le2. Depending on the signs of F1 and F2, four cases are to be distinguished.Case 1 (F1≤0 and F2≤0).
For this case,F1≤0⇒d1≥0 and F2≤0⇒d2≥0, which means that the parallel flow solution is not existing regardless the values of Sr, Le and RT. In Figure 1, the domain in the N-Du plane where the convective parallel flow is not possible is denoted as region 1. This region is defined by F1≤0 and F2≤0.Figure 1
Regions corresponding to different bbehaviours forSr=0.1 and Le=1.1.Case 2 (F1≤0 and F2>0).
For this case,F1≤0⇒d1≥0 and F2>0⇒d2<0 for RT≥RTC=R0A2/F2. Thus, the unstable solution is not possible and the stable solution is possible only when RT exceeds the critical value RTC. The convective flow bifurcates from the rest state through a zero amplitude convection which indicates that the solution corresponds to a supercritical bifurcation. The critical value of RTC represents the supercritical Rayleigh number, RTCsup, marking the onset of the supercritical convection. The expression of this threshold value is given by(14)RTCsup=R0A2F2.In Figure1, the domain in the N-Du plane for which the conditions of Case 2 are satisfied is denoted as region 2 and it is defined by F1≤0 and F2>0.Case 3 (F1>0 and F2>0).
For these conditions, the signs ofd1 and d2 depend on the Rayleigh number as follows:(15)d1≥0forRT≤RTC1=R0BF1,d2≥0forRT≤RTC2=R0A2F2.As a result, two subcases are to be considered.Case 3.1 (
R
T
C
2
≤
R
T
C
1
). The development of the above inequality leads to(16)F3=NA2Le-BC1+A2Le2-BC2≤0.Note that, for RT≤RTC2,  d1≥0 and d2≥0 which means that there is no parallel flow solution. On the other hand, for RT>RTC2,  d1≥0 and d2<0 indicating the existence of a supercritical bifurcation and the corresponding supercritical Rayleigh number is given by (14). The domain corresponding to these conditions in the N-Du plane is denoted region 3 in Figure 1. This region is defined by F1>0,  F2>0, and F3=NA2Le-BC1+A2Le2-BC2≤0.Even if regions 2 and 3 seem to have similar characteristics, they are presented separately because they differ in terms of some asymptotical behaviors at large values ofRT.Case 3.2 (
R
T
C
2
>
R
T
C
1
). For this case F3=NA2Le-BC1+A2Le2-BC2>0). Hence, for RT<RTC1,  d1≥0 and d2≥0. It follows that there is no parallel flow solution. For RTC1<RT≤RTC2,  d1<0 and d2≥0 which means that both stable and unstable solutions are existing within this range of RT. The stable solution corresponds to a subcritical bifurcation, which occurs through finite amplitude convection at a saddle node point. The subcritical threshold, RTCsub, is given by(17)RTCsub=R0BF1-2Le2F2+2LeA2F12+Le2F22-BF1F2F12.In addition, forRT>RTC2,  d1<0 and d2<0 which implies that the unstable solution disappears for this range of RT. The domain corresponding to these conditions in the N-Du plane is termed region 4 in Figure 1. This region is defined by F1>0,  F2>0, and F3=NA2Le-BC1+A2Le2-BC2>0.Case 4 (F1>0 and F2<0).
For these conditions,d1≥0 for RT≤RTC1=R0B/F1 and d2≥0 independently of the values of the Rayleigh number. It follows that two subcases are possible.Case 4.1 (
R
T
≤
R
T
C
1
). Note that, for this range of RT,  d1≥0 and d2≥0 which means that there is no parallel flow solution.Case 4.2 (
R
T
>
R
T
C
1
). This condition is satisfied for d1<0 and d2≥0. This means that both stable and unstable solutions are existing for any value of RT greater than the subcritical threshold, RTCsub, given by the same expression as that for region 4. The corresponding domain in the N-Du plane is termed region 5 in Figure 1 and it is defined by F1>0 and F2<0.


#### 3.2.2. The ParameterB<O

This inequality is satisfied forSrDu<-(1+Le2)/2Le2. For this case also, four cases are possible depending of the signs of F1 and F2.Case 1 (F1>0 and F2<0).
These conditions are the same as those defining region 5. The corresponding domain in theN-Du plane is termed region 5 in Figure 1 and it is defined by F1>0 and F2<0.Case 2 (F1>0 and F2≥0).
For these conditions,d1<0 independently of the values of the Rayleigh number and d2≥0 for RT≤RTC2=R0A2/F2. This means that both stable and unstable solutions are existing in this range for RT≥RTCsub. The supercritical threshold Rayleigh number is given by (17). In addition, for RT>RTC2, d1<0, and d2<0 which implies that the unstable solution disappears for this range of RT. The corresponding domain in the N-Du plane is termed region 4 in Figure 1, defined by F1>0 and F2≥0.Case 3 (F1≤0 and F2≥0).
For these conditions, the signs ofd1 and d2 depend on the Rayleigh number as follows:(18)d1≥0forRT≥RTC1=R0BF1,d2≥0forRT≤RTC2=R0A2F2.Two subcases emerge from these conditions.Case 3.1 (
R
T
C
2
≤
R
T
C
1
). This inequality is verified for F3=NA2Le-BC1+A2Le2-BC2≥0. Thus, for RT≤RTC2,  d1≤0 and d2≥0, which means that both stable and unstable solutions are existing. The bifurcation is of a subcritical nature and the subcritical Rayleigh number is given by (17). For RT>RTC2, the unstable solution disappears within this range of RT. The domain in the N-Du plane satisfying these conditions is indicated by region 6 in Figure 1. This region is defined by F1≤0, F2≥0, and F3≥0. It is to underline that, even if regions 5 and 6 seem to have the same characteristics, they are presented separately because they differ in terms of some asymptotical behaviors at large values of RT.Case 3.2 (
R
T
C
2
>
R
T
C
1
). For this case, F3<0. Then, for RT<RTC1,  d1≥0 and d2≥0 and the parallel flow solution is not existing. On the other side, for RT>RTC2,  d1≥0 and d2<0 indicating the existence of a supercritical bifurcation and the corresponding supercritical Rayleigh number is given by (14). The domain corresponding to these conditions in the N-Du plane is termed region 2 in Figure 1 and it is defined by F1≤0,  F2≥0, and F3<0.Case 4 (F1≤0 and F2≤0).
For this case, the inequalitySrDu<-(1+Le2)/2Le2 is verified and there is no corresponding domain in the N-Du plane.In summary, theN-Du plane can be divided into five regions for SrDu>-(1+Le2)/2Le2 and into four regions for SrDu<-(1+Le2)/2Le2. In region 1, the parallel flow is not possible while in regions 2 and 3, the only possible flow is the supercritical stationary one. For the remaining regions (regions 4, 5, and 6) both subcritical and supercritical convections are possible.


## 3.1. Numerical Solution

The numerical solution of (1) to (3) was obtained using a finite-difference method, described in detail by Bourich et al. [8]. A second-order scheme was used for the discretization of the spatial derivatives. Equations (2)-(3) were marched in time using the Alternating Direction Implicit (ADI) method. The stream function equation (4) was solved at each time step with the Point Successive Overrelaxation (PSOR) method with an optimum overrelaxation coefficient calculated for the used grid. In addition, a convergence criterion was adopted for the stream function to satisfy a variation by less than 10−5 for each time step. A second criterion ∑∑Γi,jn+1-Γi,jn/∑∑Γi,jn+1≤10-6 was used to check the convergence of the numerical code. Here, Γ stands for any of the variables T, S, or ψ. The superscripts n and (n+1) indicate the iterations numbers, and the subscripts i and j indicate locations in the grid system. For large aspect ratio enclosures, nonuniform grid was used in the x-direction near the short walls to capture the flow details near the enclosures end-walls and also in the y-direction to obtain a finer grid in the close vicinity of the horizontal walls.


## 3.2. Analytical Solution

For a shallow enclosure with constant heat and mass flux boundary conditions, an approximate analytical solution based on the parallel flow concept is possible, which renders the problem amenable to a parametric study while retaining the essential physics of the problem. The analytical solution is developed for steady-state flows using the parallel flow approximation (see, e.g., Bourich et al. [8]), which leads to the following simplifications (justified by examining the streamlines, isotherms, and iso-concentration lines obtained numerically). These simplifications lead to ψ(x,y)=ψ(y), T(x,y)=CTx+θT(y), and S(x,y)=CSx+θS(y), where CT and CS are, respectively, unknown constant temperature and concentration gradients in the x-direction (the direction of the long sides of the cavity). Using these approximations together with the boundary conditions (5a) and (5b), (1)–(3) reduce to a set of ordinary differential equations for which the solution is obtained as follows:(7)ψy=ψ04y2-12,Tx,y=CTx+CT-CSDuNLe1-LeSrDuψ015y48y4-40y2+15-φTy,Sx,y=CSx+CS-CTSr/N1-LeSrDu·Leψ015·y8y4-40y2+15-φSy,where ψ0 is the stream function value at the midheight of the layer, given by(8)ψ0=RT384CT+NCS.The analytical expressions of CT and CS were determined by using thermal and solutal balances in the layer, which leads to the following expressions:(9a)CT=-815ψ02aψ02φT+φSNDuLe2+AφT-φSLeNDu2aψ022+2aψ02B+A2,(9b)CS=-815ψ02aψ02φTLeSr/N+φSLe+ALeφS-φTSr/N2aψ022+2aψ02B+A2,where a=64/315,  A=1-LeSrDu, and B=Le2+2Le2DuSr+1.Introducing the expressions ofCT and CS into (8) yields a fourth-order polynomial in terms of ψ0 for which the following solutions are obtained:(10)ψ0=0,ψ0=∓12-d1∓d12-d21/2,with(11)d1=12aLe2B-RT720F1,d2=1a2Le2A2-RT720F2,F1=LeN+Le,F2=NC1+C2,where C1=DuLeSr-1-1+Le and C2=SrLeLeDu-1-1+1.From a mathematical point of view, (10) may exhibit, in addition to the rest state solution, two types of bifurcations depending on the sign within the square root. Although it should be mentioned that several numerical tests were performed, only the solutions corresponding to positive sign were obtained numerically. Hence, the convective solution with the negative sign within the square root is termed “unstable solution” and the convective solution corresponding to positive sign within the square root is similarly qualified “stable solution.”The new expressions of Nusselt and Sherwood numbers are obtained as follows:(12)Nu=11-8/15ψ0CT,Sh=11-8/15ψ0LeCS.Foremost, it may be remarked from (10) that the parallel flow solutions exist only when the following two conditions are satisfied:(13)-d1∓d12-d2>0,d12-d2>0.The resolution of the inequalities of (13) is performed in the N-Du plane with Sr, Le, and RT as parameters. Depending of the sign of B (parameter in the expression of d1 given before), two main cases are possible.


### 3.2.1. The ParameterB>O

This condition is satisfied ifSrDu>-(1+Le2)/2Le2. Depending on the signs of F1 and F2, four cases are to be distinguished.Case 1 (F1≤0 and F2≤0).
For this case,F1≤0⇒d1≥0 and F2≤0⇒d2≥0, which means that the parallel flow solution is not existing regardless the values of Sr, Le and RT. In Figure 1, the domain in the N-Du plane where the convective parallel flow is not possible is denoted as region 1. This region is defined by F1≤0 and F2≤0.Figure 1
Regions corresponding to different bbehaviours forSr=0.1 and Le=1.1.Case 2 (F1≤0 and F2>0).
For this case,F1≤0⇒d1≥0 and F2>0⇒d2<0 for RT≥RTC=R0A2/F2. Thus, the unstable solution is not possible and the stable solution is possible only when RT exceeds the critical value RTC. The convective flow bifurcates from the rest state through a zero amplitude convection which indicates that the solution corresponds to a supercritical bifurcation. The critical value of RTC represents the supercritical Rayleigh number, RTCsup, marking the onset of the supercritical convection. The expression of this threshold value is given by(14)RTCsup=R0A2F2.In Figure1, the domain in the N-Du plane for which the conditions of Case 2 are satisfied is denoted as region 2 and it is defined by F1≤0 and F2>0.Case 3 (F1>0 and F2>0).
For these conditions, the signs ofd1 and d2 depend on the Rayleigh number as follows:(15)d1≥0forRT≤RTC1=R0BF1,d2≥0forRT≤RTC2=R0A2F2.As a result, two subcases are to be considered.Case 3.1 (
R
T
C
2
≤
R
T
C
1
). The development of the above inequality leads to(16)F3=NA2Le-BC1+A2Le2-BC2≤0.Note that, for RT≤RTC2,  d1≥0 and d2≥0 which means that there is no parallel flow solution. On the other hand, for RT>RTC2,  d1≥0 and d2<0 indicating the existence of a supercritical bifurcation and the corresponding supercritical Rayleigh number is given by (14). The domain corresponding to these conditions in the N-Du plane is denoted region 3 in Figure 1. This region is defined by F1>0,  F2>0, and F3=NA2Le-BC1+A2Le2-BC2≤0.Even if regions 2 and 3 seem to have similar characteristics, they are presented separately because they differ in terms of some asymptotical behaviors at large values ofRT.Case 3.2 (
R
T
C
2
>
R
T
C
1
). For this case F3=NA2Le-BC1+A2Le2-BC2>0). Hence, for RT<RTC1,  d1≥0 and d2≥0. It follows that there is no parallel flow solution. For RTC1<RT≤RTC2,  d1<0 and d2≥0 which means that both stable and unstable solutions are existing within this range of RT. The stable solution corresponds to a subcritical bifurcation, which occurs through finite amplitude convection at a saddle node point. The subcritical threshold, RTCsub, is given by(17)RTCsub=R0BF1-2Le2F2+2LeA2F12+Le2F22-BF1F2F12.In addition, forRT>RTC2,  d1<0 and d2<0 which implies that the unstable solution disappears for this range of RT. The domain corresponding to these conditions in the N-Du plane is termed region 4 in Figure 1. This region is defined by F1>0,  F2>0, and F3=NA2Le-BC1+A2Le2-BC2>0.Case 4 (F1>0 and F2<0).
For these conditions,d1≥0 for RT≤RTC1=R0B/F1 and d2≥0 independently of the values of the Rayleigh number. It follows that two subcases are possible.Case 4.1 (
R
T
≤
R
T
C
1
). Note that, for this range of RT,  d1≥0 and d2≥0 which means that there is no parallel flow solution.Case 4.2 (
R
T
>
R
T
C
1
). This condition is satisfied for d1<0 and d2≥0. This means that both stable and unstable solutions are existing for any value of RT greater than the subcritical threshold, RTCsub, given by the same expression as that for region 4. The corresponding domain in the N-Du plane is termed region 5 in Figure 1 and it is defined by F1>0 and F2<0.


### 3.2.2. The ParameterB<O

This inequality is satisfied forSrDu<-(1+Le2)/2Le2. For this case also, four cases are possible depending of the signs of F1 and F2.Case 1 (F1>0 and F2<0).
These conditions are the same as those defining region 5. The corresponding domain in theN-Du plane is termed region 5 in Figure 1 and it is defined by F1>0 and F2<0.Case 2 (F1>0 and F2≥0).
For these conditions,d1<0 independently of the values of the Rayleigh number and d2≥0 for RT≤RTC2=R0A2/F2. This means that both stable and unstable solutions are existing in this range for RT≥RTCsub. The supercritical threshold Rayleigh number is given by (17). In addition, for RT>RTC2, d1<0, and d2<0 which implies that the unstable solution disappears for this range of RT. The corresponding domain in the N-Du plane is termed region 4 in Figure 1, defined by F1>0 and F2≥0.Case 3 (F1≤0 and F2≥0).
For these conditions, the signs ofd1 and d2 depend on the Rayleigh number as follows:(18)d1≥0forRT≥RTC1=R0BF1,d2≥0forRT≤RTC2=R0A2F2.Two subcases emerge from these conditions.Case 3.1 (
R
T
C
2
≤
R
T
C
1
). This inequality is verified for F3=NA2Le-BC1+A2Le2-BC2≥0. Thus, for RT≤RTC2,  d1≤0 and d2≥0, which means that both stable and unstable solutions are existing. The bifurcation is of a subcritical nature and the subcritical Rayleigh number is given by (17). For RT>RTC2, the unstable solution disappears within this range of RT. The domain in the N-Du plane satisfying these conditions is indicated by region 6 in Figure 1. This region is defined by F1≤0, F2≥0, and F3≥0. It is to underline that, even if regions 5 and 6 seem to have the same characteristics, they are presented separately because they differ in terms of some asymptotical behaviors at large values of RT.Case 3.2 (
R
T
C
2
>
R
T
C
1
). For this case, F3<0. Then, for RT<RTC1,  d1≥0 and d2≥0 and the parallel flow solution is not existing. On the other side, for RT>RTC2,  d1≥0 and d2<0 indicating the existence of a supercritical bifurcation and the corresponding supercritical Rayleigh number is given by (14). The domain corresponding to these conditions in the N-Du plane is termed region 2 in Figure 1 and it is defined by F1≤0,  F2≥0, and F3<0.Case 4 (F1≤0 and F2≤0).
For this case, the inequalitySrDu<-(1+Le2)/2Le2 is verified and there is no corresponding domain in the N-Du plane.In summary, theN-Du plane can be divided into five regions for SrDu>-(1+Le2)/2Le2 and into four regions for SrDu<-(1+Le2)/2Le2. In region 1, the parallel flow is not possible while in regions 2 and 3, the only possible flow is the supercritical stationary one. For the remaining regions (regions 4, 5, and 6) both subcritical and supercritical convections are possible.


## 3.2.1. The ParameterB>O

This condition is satisfied ifSrDu>-(1+Le2)/2Le2. Depending on the signs of F1 and F2, four cases are to be distinguished.Case 1 (F1≤0 and F2≤0).
For this case,F1≤0⇒d1≥0 and F2≤0⇒d2≥0, which means that the parallel flow solution is not existing regardless the values of Sr, Le and RT. In Figure 1, the domain in the N-Du plane where the convective parallel flow is not possible is denoted as region 1. This region is defined by F1≤0 and F2≤0.Figure 1
Regions corresponding to different bbehaviours forSr=0.1 and Le=1.1.Case 2 (F1≤0 and F2>0).
For this case,F1≤0⇒d1≥0 and F2>0⇒d2<0 for RT≥RTC=R0A2/F2. Thus, the unstable solution is not possible and the stable solution is possible only when RT exceeds the critical value RTC. The convective flow bifurcates from the rest state through a zero amplitude convection which indicates that the solution corresponds to a supercritical bifurcation. The critical value of RTC represents the supercritical Rayleigh number, RTCsup, marking the onset of the supercritical convection. The expression of this threshold value is given by(14)RTCsup=R0A2F2.In Figure1, the domain in the N-Du plane for which the conditions of Case 2 are satisfied is denoted as region 2 and it is defined by F1≤0 and F2>0.Case 3 (F1>0 and F2>0).
For these conditions, the signs ofd1 and d2 depend on the Rayleigh number as follows:(15)d1≥0forRT≤RTC1=R0BF1,d2≥0forRT≤RTC2=R0A2F2.As a result, two subcases are to be considered.Case 3.1 (
R
T
C
2
≤
R
T
C
1
). The development of the above inequality leads to(16)F3=NA2Le-BC1+A2Le2-BC2≤0.Note that, for RT≤RTC2,  d1≥0 and d2≥0 which means that there is no parallel flow solution. On the other hand, for RT>RTC2,  d1≥0 and d2<0 indicating the existence of a supercritical bifurcation and the corresponding supercritical Rayleigh number is given by (14). The domain corresponding to these conditions in the N-Du plane is denoted region 3 in Figure 1. This region is defined by F1>0,  F2>0, and F3=NA2Le-BC1+A2Le2-BC2≤0.Even if regions 2 and 3 seem to have similar characteristics, they are presented separately because they differ in terms of some asymptotical behaviors at large values ofRT.Case 3.2 (
R
T
C
2
>
R
T
C
1
). For this case F3=NA2Le-BC1+A2Le2-BC2>0). Hence, for RT<RTC1,  d1≥0 and d2≥0. It follows that there is no parallel flow solution. For RTC1<RT≤RTC2,  d1<0 and d2≥0 which means that both stable and unstable solutions are existing within this range of RT. The stable solution corresponds to a subcritical bifurcation, which occurs through finite amplitude convection at a saddle node point. The subcritical threshold, RTCsub, is given by(17)RTCsub=R0BF1-2Le2F2+2LeA2F12+Le2F22-BF1F2F12.In addition, forRT>RTC2,  d1<0 and d2<0 which implies that the unstable solution disappears for this range of RT. The domain corresponding to these conditions in the N-Du plane is termed region 4 in Figure 1. This region is defined by F1>0,  F2>0, and F3=NA2Le-BC1+A2Le2-BC2>0.Case 4 (F1>0 and F2<0).
For these conditions,d1≥0 for RT≤RTC1=R0B/F1 and d2≥0 independently of the values of the Rayleigh number. It follows that two subcases are possible.Case 4.1 (
R
T
≤
R
T
C
1
). Note that, for this range of RT,  d1≥0 and d2≥0 which means that there is no parallel flow solution.Case 4.2 (
R
T
>
R
T
C
1
). This condition is satisfied for d1<0 and d2≥0. This means that both stable and unstable solutions are existing for any value of RT greater than the subcritical threshold, RTCsub, given by the same expression as that for region 4. The corresponding domain in the N-Du plane is termed region 5 in Figure 1 and it is defined by F1>0 and F2<0.


## 3.2.2. The ParameterB<O

This inequality is satisfied forSrDu<-(1+Le2)/2Le2. For this case also, four cases are possible depending of the signs of F1 and F2.Case 1 (F1>0 and F2<0).
These conditions are the same as those defining region 5. The corresponding domain in theN-Du plane is termed region 5 in Figure 1 and it is defined by F1>0 and F2<0.Case 2 (F1>0 and F2≥0).
For these conditions,d1<0 independently of the values of the Rayleigh number and d2≥0 for RT≤RTC2=R0A2/F2. This means that both stable and unstable solutions are existing in this range for RT≥RTCsub. The supercritical threshold Rayleigh number is given by (17). In addition, for RT>RTC2, d1<0, and d2<0 which implies that the unstable solution disappears for this range of RT. The corresponding domain in the N-Du plane is termed region 4 in Figure 1, defined by F1>0 and F2≥0.Case 3 (F1≤0 and F2≥0).
For these conditions, the signs ofd1 and d2 depend on the Rayleigh number as follows:(18)d1≥0forRT≥RTC1=R0BF1,d2≥0forRT≤RTC2=R0A2F2.Two subcases emerge from these conditions.Case 3.1 (
R
T
C
2
≤
R
T
C
1
). This inequality is verified for F3=NA2Le-BC1+A2Le2-BC2≥0. Thus, for RT≤RTC2,  d1≤0 and d2≥0, which means that both stable and unstable solutions are existing. The bifurcation is of a subcritical nature and the subcritical Rayleigh number is given by (17). For RT>RTC2, the unstable solution disappears within this range of RT. The domain in the N-Du plane satisfying these conditions is indicated by region 6 in Figure 1. This region is defined by F1≤0, F2≥0, and F3≥0. It is to underline that, even if regions 5 and 6 seem to have the same characteristics, they are presented separately because they differ in terms of some asymptotical behaviors at large values of RT.Case 3.2 (
R
T
C
2
>
R
T
C
1
). For this case, F3<0. Then, for RT<RTC1,  d1≥0 and d2≥0 and the parallel flow solution is not existing. On the other side, for RT>RTC2,  d1≥0 and d2<0 indicating the existence of a supercritical bifurcation and the corresponding supercritical Rayleigh number is given by (14). The domain corresponding to these conditions in the N-Du plane is termed region 2 in Figure 1 and it is defined by F1≤0,  F2≥0, and F3<0.Case 4 (F1≤0 and F2≤0).
For this case, the inequalitySrDu<-(1+Le2)/2Le2 is verified and there is no corresponding domain in the N-Du plane.In summary, theN-Du plane can be divided into five regions for SrDu>-(1+Le2)/2Le2 and into four regions for SrDu<-(1+Le2)/2Le2. In region 1, the parallel flow is not possible while in regions 2 and 3, the only possible flow is the supercritical stationary one. For the remaining regions (regions 4, 5, and 6) both subcritical and supercritical convections are possible.


## 4. Results and Discussion

### 4.1. Effect of Du andN on Convection Thresholds

This section is devoted to analyze the combined effects of the Dufour parameter and the buoyancy ratio on thresholds of convection. In addition, some interesting behaviors of the fluid flow are illustrated and discussed.The parameter Du represents the relative importance of the cross-diffusion due to solute concentration component with respect to that due to the thermal gradient. It could be varied by changing the mass flux intensity or by considering various working binary fluids. For a givenN, different regions could be crossed in Figure 1 by incrementing Du, depending on whether N≥NLSup or N<NLSup. The parameter NLSup is calculated analytically and its expression is given by NLSup=-SrLe2/LeSr-1-1. The effect of Du on thresholds of stationary convection is exemplified in Figures 2(a) and 2(b) illustrating, respectively, the cases N≥NLSup (case illustrated with N=1.5,  Le=1.1, and different Sr) and N<NLSup (case illustrated with N=-1.5,  Le=1.1, and different Sr). The value Le=1.1 was chosen within the experimental range of the Lewis number where both cross-phenomena of Soret and Dufour effects are significant.Figure 2
(a)
The evolution ofRTCsup with Du for N=1.5 and Le=1.1 (case of N≥NLSup)
(b)
The evolution ofRTCsup with Du for N=-1.5 and Le=1.1 (case of N<NLSup)ForN≥NLSup, the threshold of stationary convection is characterized by a decrease toward a minimum (well visible in Figure 2(a) for the negative values of Sr but not visible for the positive values of this parameter due to the restricted range of Du). It is seen in Figure 2(a) that the threshold of stationary convection decreases first by increasing Du to reach a minimum value at Du=Dumin and increases afterwards quickly toward a vertical asymptote (infinite value) for Du=DuCsup. However, for the positive values of Sr, the variations of RTCSup versus Du are moderate for negative values of the latter parameter. The increase of RTCSup accompanying the increase of Du in its negative range is characterized by an augmentation which becomes more and more slow by decreasing Sr within its positive range. In the positive range of Du, the fast increase toward the vertical asymptote is also observed for the positive values of Sr. The critical value of Du leading to this asymptote increases by decreasing Sr. For N<NLSup, an opposite behavior to that described in Figure 2(a) is observed. In fact, an increase of Du from the limiting value DuCsup induces first a sharp decrease of RTCsup from infinite toward a minimum value obtained at Du=Dumin. Afterward, RTCsup increases slowly (with different rates depending on Sr). These results show that, for N≥NLSup/(N<NLSup), the Dufour parameter could play a destabilizing effect if Du<DuCsup/(DuCsup<Du<Dumin) or a stabilizing one if Dumin<Du<DuCsup/(Du<Dumin). The analytical expressions of Dumin and DuCsup are, respectively, given by(19)DuCsup=SrLeLe+1-NLe-1SrLeN+Le-NLe+1,Dumin=-LeSr-2SrLeLe+1+N2Le+1+Le+2-NLe+1LeSrLeSrN+Le-NLe+1.The Dufour effect onRTCSub is illustrated for different values of Soret parameter, Sr, for the three possible cases (determined analytically) that lead to different behaviors depending on the crossing region. Thus, Figure 3(a) illustrates the case NLSub≥N>-Le, Figure 3(b) illustrates the case N>NLSub, and the third case illustrated in Figure 3(c) corresponds to N≤-Le and Sr<SrC. The analytical expression of NLSub is given by(20)NLSub=Sr·Le2Le·2-Sr+2.Figure 3
(a)
The evolution ofRTCSub versus Du for N=-1 and Le=1.1 (case of -Le<N≤NLSub)
(b)
The evolution ofRTCSub versus Du for N=1.5 and Le=1.1 (case of N>NLSub)
(c)
The evolution ofRTCSub with Du for N=-1.2 and Le=1.1 (case of N≤-Le)For the first case, Figure3(a) shows that the subcritical Rayleigh number, RTCSub, decreases linearly by increasing Du and vanishes when Du approaches a critical value DCSub. This critical value of Du increases by the increase of Sr. For the second case, Figure 3(b) indicates that the subcritical convection starts from DCSub. More increase of Du from this threshold leads to a linear increase of RTCSub and stabilizes more the system. Note that for the case of Figure 3(b), the effect of Sr on RTCSub is seen to be limited, but its effect of the limit DuCsub is important. The analytical expression of DuCsub is given by DuCsub=2+β1∓β12-4β2/2SrLe, where β1 and β2 are functions of the governing parameters. The opposite tendencies observed by increasing the Dufour parameter in Figures 3(a) and 3(b) are explained by the fact that the increase of Du leads to the successive cross of regions 5 and region 4 in the case of Figure 3(a), while in the case of Figure 3(b) it leads to the successive cross of regions 4 and 5. For N≤-Le, the subcritical convection corresponding to region 6 appears only for Sr<SrC as shown in Figure 1. For this region, the subcritical convection starts from Dud and increases linearly with Du inducing a stabilizing effect. The Soret parameter has an important effect on the subcritical Rayleigh number in this region as indicated in Figure 3(c).The buoyancy ratioN measures the relative importance of solutal buoyancy force (due to combined contributions of imposed flux of mass and Soret effect) and thermal buoyancy force (induced by the imposed flux of heat and Dufour effect). It could be varied by changing the imposed flux of heat and/or mass or by considering various fluids with different properties.The influence of the buoyancy ratioN on the thresholds of stationary convection RTCSup is illustrated in Figures 4(a) and 4(b) for Le = 1.1 and various Sr. In Figure 4(a), Du=-1.5 to illustrate the case Du<DuLsup, while in Figure 4(b)  Du=1.5 to illustrate the case Du≥DuLsup. As shown from Figure 4(a), the evolution of RTCsup with N changes drastically depending on whether Du<DuLsup or Du≥DuLsup. For Du≥DuLsup, an increase of N from a supercritical value NCsup induces a destabilizing effect characterized first by a sharp decrease of RTCsup followed by a monotonous and moderate decrease. The trend is inverted for Du≥DuLsup as it can be seen in Figure 4(b). In fact, RTCsup increases first monotonously and in a moderate way. A change in the tendency of the increase occurs when the value of N becomes close to a critical value NCsup, leading to a quick increase toward the infinite value of the asymptote. For this case, it is clear that the increase of N has a stabilizing role. This behavior can be explained by the fact that, for Du=-1.5, case corresponding to Du<DuLsup, the increase of N in Figure 1 leads to cross region 1 first which requires an infinite Rayleigh number to trigger the flow. However, for Du=1.5, case corresponding to Du≥DuLsup, region 3 is crossed first, then region 1. The latter corresponds to the case where the parallel flow is impossible, which explains why an infinite Rayleigh number is required to start the flow. The critical value NCsup can be calculated from the following analytical expression:(21)NCsup=-SrLeLeDu-Le2-Le+1DuLeSr-Le2-Le+Le.The expression of DuLSup is also calculated analytically to obtain(22)DuLSup=-Le1-LeSr-1.Figure 4
(a)
The evolution ofRTCsup with N for Du=-1.5 and Le=1.1 (case of Du<DuLsup)
(b)
The evolution ofRTCsup with N for Du=1.5 and Le=1.1 (case of Du≥DuLsup)The evolution of the subcritical Rayleigh number,RTCSub, versus N is depicted in Figures 5(a) and 5(b). The case of Du<DuLsub is illustrated in Figure 5(a) for Du=-1.5, Le = 1.1, and different values of Sr. It is to note that, for this case, the subcritical flow exists only for negative values of N. By increasing N from a subcritical value NCsub, the subcritical Rayleigh number RTCSub undergoes a sharp decline from its infinite value, indicating that the increase of N has a destabilizing effect. For this case, by increasing N, we cross first region 1 (rest state region) in Figure 1 which explains why an infinite Rayleigh number is required to start the flow for N=NCSub. The case Du≥DuLSub is exemplified in Figure 5(b) for Du=2, Le = 1.1, and different values of Sr. The examination of this figure shows a big change in the behavior compared with the first case illustrated in Figure 5(a). In fact, for Sr=-0.8, an increase of N induces first an increase of RTCSub toward a maximum value and undergoes after that a decrease in the remaining range of N. Note that, for Sr = 0.1, the fluid remains at rest as long as N<NCSub.Figure 5
(a)
The evolution ofRTCsub with N for Du=-1.5 and Le=1.1 (case of Du<DuLsub)
(b)
The evolution ofRTCsub versus N for Du=2 and Le=1.1 (case of Du≥DuLsub)The analytical expressions ofNCSub and DuLSub are, respectively, given by(23)NCSuB=BSrLeLeDu-Le2-Le+B-A2Le2A2Le-BDuSrLe-Le-1-BLe,DuCsub=2+α1∓α12-4α22SrLe,where α1=LeSr-Le-1Le2+4Le+1+2SrLe3/Le(2Le+2-SrLe) and α2=-Le+13(SrLe-1)/Le(2Le+2-SrLe).


### 4.2. Effect of Du on the Flow Intensity and Heat and Mass Transfer

In this subsection, the influence of the Dufour effect on the flow intensity and heat and mass transfer is examined. The effect of Du onψ0, Nu, and Sh is depicted in Figures 6(a)–6(c) for Le=1.1, Ar=10, different values of Sr, and two cases: RT=2730 and N=1.2 (case I) and RT=889.3, N=-1.2 (case II). The choice of these two cases was dictated by the fact that, by varying Du, two types of traversed regions are possible depending on whether N≥-Le (case I) or N<-Le (case II). In these figures, it can be seen that the numerical results are in excellent agreement with the analytical ones corresponding to the stable branches.Figure 6
(a)
Influence of the Dufour number onψ0 for Le = 1.1 and Ar = 10: (I) RT=2730 and N=1.2 (case of N≥-Le); (II) RT=889 and N=-1.2 (case of N<-Le)
(b)
Influence of the Dufour number on Nu for Le = 1.1 and Ar = 10: (I)RT=2730 and N=1.2 (case of N≥-Le); (II) RT=889 and N=-1.2 (case of N<-Le)
(c)
Influence of the Dufour number on Sh for Le = 1.1 and Ar = 10: (I)RT=2730 (case of N≥-Le); (II) RT=890 (case of N<-Le)For case I, both stable and unstable branches are existing regardless of the Soret parameter. For the stable branch, the evolutions ofψ0 and Nu versus Du are characterized by a monotonous decrease (the decrease occurs with a higher rate for the negative value of Sr) and vanishes when Du exceeds a critical value Duc which depends on Sr. The unstable branch exists only between two critical values of Du, that is, for Duu<Du≤Duc. Within this range of Du, the evolution of ψ0 is characterized by an important increase toward a maximum reached for Du=Duc. This critical value Duc marks also the limit of the existence of the unstable branch. This means that, before their disappearance, the unstable solutions become identical to the ones corresponding to the stable branches.For case II, the evolutions ofψ0 and Nu versus Du depend on the Soret parameter. In fact, for Sr=-1 (for this value of Sr the corresponding regions in N-Du plane are not presented here), both stable and unstable solutions exist and the quantities ψ0 and Nu increase with Du to reach a maximum and decrease afterward before vanishing. The unstable branch exists for Duu<Du≤Duc. For Sr=0.1, only the stable solution exists and the quantities ψ0 and Nu exhibit monotonous increases with Du. The dependence of the flow intensity vis-à-vis Sr can be explained by the fact that, by varying Du, the crossed regions are region 2 and then region 6 for Sr=-1 while for Sr=0.1 the crossed regions are region 1 and then region 2. From these observations, we can conclude that the Dufour effect enhances/reduces the flow intensity and heat transfer for N<-Le/N≥-Le.The evolution of Sh versus Du exhibits a different tendency in comparison withψ0 and Nu. In fact, the Sh variations with Du show similar tendencies independently of the crossed regions for both cases I and II and Sr=-1. For this case, Sh varies slightly with Du. The behavior changes for Sr=0.1 since the evolution of Sh is characterized by a monotonous increase with Du to reach a maximum marking the limit of the stable branch. For case I, by increasing Du the crossed regions are successively regions 3, 4, and 5 while for case II the crossed regions are 2 (crossed first) and 6.


## 4.1. Effect of Du andN on Convection Thresholds

This section is devoted to analyze the combined effects of the Dufour parameter and the buoyancy ratio on thresholds of convection. In addition, some interesting behaviors of the fluid flow are illustrated and discussed.The parameter Du represents the relative importance of the cross-diffusion due to solute concentration component with respect to that due to the thermal gradient. It could be varied by changing the mass flux intensity or by considering various working binary fluids. For a givenN, different regions could be crossed in Figure 1 by incrementing Du, depending on whether N≥NLSup or N<NLSup. The parameter NLSup is calculated analytically and its expression is given by NLSup=-SrLe2/LeSr-1-1. The effect of Du on thresholds of stationary convection is exemplified in Figures 2(a) and 2(b) illustrating, respectively, the cases N≥NLSup (case illustrated with N=1.5,  Le=1.1, and different Sr) and N<NLSup (case illustrated with N=-1.5,  Le=1.1, and different Sr). The value Le=1.1 was chosen within the experimental range of the Lewis number where both cross-phenomena of Soret and Dufour effects are significant.Figure 2
(a)
The evolution ofRTCsup with Du for N=1.5 and Le=1.1 (case of N≥NLSup)
(b)
The evolution ofRTCsup with Du for N=-1.5 and Le=1.1 (case of N<NLSup)ForN≥NLSup, the threshold of stationary convection is characterized by a decrease toward a minimum (well visible in Figure 2(a) for the negative values of Sr but not visible for the positive values of this parameter due to the restricted range of Du). It is seen in Figure 2(a) that the threshold of stationary convection decreases first by increasing Du to reach a minimum value at Du=Dumin and increases afterwards quickly toward a vertical asymptote (infinite value) for Du=DuCsup. However, for the positive values of Sr, the variations of RTCSup versus Du are moderate for negative values of the latter parameter. The increase of RTCSup accompanying the increase of Du in its negative range is characterized by an augmentation which becomes more and more slow by decreasing Sr within its positive range. In the positive range of Du, the fast increase toward the vertical asymptote is also observed for the positive values of Sr. The critical value of Du leading to this asymptote increases by decreasing Sr. For N<NLSup, an opposite behavior to that described in Figure 2(a) is observed. In fact, an increase of Du from the limiting value DuCsup induces first a sharp decrease of RTCsup from infinite toward a minimum value obtained at Du=Dumin. Afterward, RTCsup increases slowly (with different rates depending on Sr). These results show that, for N≥NLSup/(N<NLSup), the Dufour parameter could play a destabilizing effect if Du<DuCsup/(DuCsup<Du<Dumin) or a stabilizing one if Dumin<Du<DuCsup/(Du<Dumin). The analytical expressions of Dumin and DuCsup are, respectively, given by(19)DuCsup=SrLeLe+1-NLe-1SrLeN+Le-NLe+1,Dumin=-LeSr-2SrLeLe+1+N2Le+1+Le+2-NLe+1LeSrLeSrN+Le-NLe+1.The Dufour effect onRTCSub is illustrated for different values of Soret parameter, Sr, for the three possible cases (determined analytically) that lead to different behaviors depending on the crossing region. Thus, Figure 3(a) illustrates the case NLSub≥N>-Le, Figure 3(b) illustrates the case N>NLSub, and the third case illustrated in Figure 3(c) corresponds to N≤-Le and Sr<SrC. The analytical expression of NLSub is given by(20)NLSub=Sr·Le2Le·2-Sr+2.Figure 3
(a)
The evolution ofRTCSub versus Du for N=-1 and Le=1.1 (case of -Le<N≤NLSub)
(b)
The evolution ofRTCSub versus Du for N=1.5 and Le=1.1 (case of N>NLSub)
(c)
The evolution ofRTCSub with Du for N=-1.2 and Le=1.1 (case of N≤-Le)For the first case, Figure3(a) shows that the subcritical Rayleigh number, RTCSub, decreases linearly by increasing Du and vanishes when Du approaches a critical value DCSub. This critical value of Du increases by the increase of Sr. For the second case, Figure 3(b) indicates that the subcritical convection starts from DCSub. More increase of Du from this threshold leads to a linear increase of RTCSub and stabilizes more the system. Note that for the case of Figure 3(b), the effect of Sr on RTCSub is seen to be limited, but its effect of the limit DuCsub is important. The analytical expression of DuCsub is given by DuCsub=2+β1∓β12-4β2/2SrLe, where β1 and β2 are functions of the governing parameters. The opposite tendencies observed by increasing the Dufour parameter in Figures 3(a) and 3(b) are explained by the fact that the increase of Du leads to the successive cross of regions 5 and region 4 in the case of Figure 3(a), while in the case of Figure 3(b) it leads to the successive cross of regions 4 and 5. For N≤-Le, the subcritical convection corresponding to region 6 appears only for Sr<SrC as shown in Figure 1. For this region, the subcritical convection starts from Dud and increases linearly with Du inducing a stabilizing effect. The Soret parameter has an important effect on the subcritical Rayleigh number in this region as indicated in Figure 3(c).The buoyancy ratioN measures the relative importance of solutal buoyancy force (due to combined contributions of imposed flux of mass and Soret effect) and thermal buoyancy force (induced by the imposed flux of heat and Dufour effect). It could be varied by changing the imposed flux of heat and/or mass or by considering various fluids with different properties.The influence of the buoyancy ratioN on the thresholds of stationary convection RTCSup is illustrated in Figures 4(a) and 4(b) for Le = 1.1 and various Sr. In Figure 4(a), Du=-1.5 to illustrate the case Du<DuLsup, while in Figure 4(b)  Du=1.5 to illustrate the case Du≥DuLsup. As shown from Figure 4(a), the evolution of RTCsup with N changes drastically depending on whether Du<DuLsup or Du≥DuLsup. For Du≥DuLsup, an increase of N from a supercritical value NCsup induces a destabilizing effect characterized first by a sharp decrease of RTCsup followed by a monotonous and moderate decrease. The trend is inverted for Du≥DuLsup as it can be seen in Figure 4(b). In fact, RTCsup increases first monotonously and in a moderate way. A change in the tendency of the increase occurs when the value of N becomes close to a critical value NCsup, leading to a quick increase toward the infinite value of the asymptote. For this case, it is clear that the increase of N has a stabilizing role. This behavior can be explained by the fact that, for Du=-1.5, case corresponding to Du<DuLsup, the increase of N in Figure 1 leads to cross region 1 first which requires an infinite Rayleigh number to trigger the flow. However, for Du=1.5, case corresponding to Du≥DuLsup, region 3 is crossed first, then region 1. The latter corresponds to the case where the parallel flow is impossible, which explains why an infinite Rayleigh number is required to start the flow. The critical value NCsup can be calculated from the following analytical expression:(21)NCsup=-SrLeLeDu-Le2-Le+1DuLeSr-Le2-Le+Le.The expression of DuLSup is also calculated analytically to obtain(22)DuLSup=-Le1-LeSr-1.Figure 4
(a)
The evolution ofRTCsup with N for Du=-1.5 and Le=1.1 (case of Du<DuLsup)
(b)
The evolution ofRTCsup with N for Du=1.5 and Le=1.1 (case of Du≥DuLsup)The evolution of the subcritical Rayleigh number,RTCSub, versus N is depicted in Figures 5(a) and 5(b). The case of Du<DuLsub is illustrated in Figure 5(a) for Du=-1.5, Le = 1.1, and different values of Sr. It is to note that, for this case, the subcritical flow exists only for negative values of N. By increasing N from a subcritical value NCsub, the subcritical Rayleigh number RTCSub undergoes a sharp decline from its infinite value, indicating that the increase of N has a destabilizing effect. For this case, by increasing N, we cross first region 1 (rest state region) in Figure 1 which explains why an infinite Rayleigh number is required to start the flow for N=NCSub. The case Du≥DuLSub is exemplified in Figure 5(b) for Du=2, Le = 1.1, and different values of Sr. The examination of this figure shows a big change in the behavior compared with the first case illustrated in Figure 5(a). In fact, for Sr=-0.8, an increase of N induces first an increase of RTCSub toward a maximum value and undergoes after that a decrease in the remaining range of N. Note that, for Sr = 0.1, the fluid remains at rest as long as N<NCSub.Figure 5
(a)
The evolution ofRTCsub with N for Du=-1.5 and Le=1.1 (case of Du<DuLsub)
(b)
The evolution ofRTCsub versus N for Du=2 and Le=1.1 (case of Du≥DuLsub)The analytical expressions ofNCSub and DuLSub are, respectively, given by(23)NCSuB=BSrLeLeDu-Le2-Le+B-A2Le2A2Le-BDuSrLe-Le-1-BLe,DuCsub=2+α1∓α12-4α22SrLe,where α1=LeSr-Le-1Le2+4Le+1+2SrLe3/Le(2Le+2-SrLe) and α2=-Le+13(SrLe-1)/Le(2Le+2-SrLe).


## 4.2. Effect of Du on the Flow Intensity and Heat and Mass Transfer

In this subsection, the influence of the Dufour effect on the flow intensity and heat and mass transfer is examined. The effect of Du onψ0, Nu, and Sh is depicted in Figures 6(a)–6(c) for Le=1.1, Ar=10, different values of Sr, and two cases: RT=2730 and N=1.2 (case I) and RT=889.3, N=-1.2 (case II). The choice of these two cases was dictated by the fact that, by varying Du, two types of traversed regions are possible depending on whether N≥-Le (case I) or N<-Le (case II). In these figures, it can be seen that the numerical results are in excellent agreement with the analytical ones corresponding to the stable branches.Figure 6
(a)
Influence of the Dufour number onψ0 for Le = 1.1 and Ar = 10: (I) RT=2730 and N=1.2 (case of N≥-Le); (II) RT=889 and N=-1.2 (case of N<-Le)
(b)
Influence of the Dufour number on Nu for Le = 1.1 and Ar = 10: (I)RT=2730 and N=1.2 (case of N≥-Le); (II) RT=889 and N=-1.2 (case of N<-Le)
(c)
Influence of the Dufour number on Sh for Le = 1.1 and Ar = 10: (I)RT=2730 (case of N≥-Le); (II) RT=890 (case of N<-Le)For case I, both stable and unstable branches are existing regardless of the Soret parameter. For the stable branch, the evolutions ofψ0 and Nu versus Du are characterized by a monotonous decrease (the decrease occurs with a higher rate for the negative value of Sr) and vanishes when Du exceeds a critical value Duc which depends on Sr. The unstable branch exists only between two critical values of Du, that is, for Duu<Du≤Duc. Within this range of Du, the evolution of ψ0 is characterized by an important increase toward a maximum reached for Du=Duc. This critical value Duc marks also the limit of the existence of the unstable branch. This means that, before their disappearance, the unstable solutions become identical to the ones corresponding to the stable branches.For case II, the evolutions ofψ0 and Nu versus Du depend on the Soret parameter. In fact, for Sr=-1 (for this value of Sr the corresponding regions in N-Du plane are not presented here), both stable and unstable solutions exist and the quantities ψ0 and Nu increase with Du to reach a maximum and decrease afterward before vanishing. The unstable branch exists for Duu<Du≤Duc. For Sr=0.1, only the stable solution exists and the quantities ψ0 and Nu exhibit monotonous increases with Du. The dependence of the flow intensity vis-à-vis Sr can be explained by the fact that, by varying Du, the crossed regions are region 2 and then region 6 for Sr=-1 while for Sr=0.1 the crossed regions are region 1 and then region 2. From these observations, we can conclude that the Dufour effect enhances/reduces the flow intensity and heat transfer for N<-Le/N≥-Le.The evolution of Sh versus Du exhibits a different tendency in comparison withψ0 and Nu. In fact, the Sh variations with Du show similar tendencies independently of the crossed regions for both cases I and II and Sr=-1. For this case, Sh varies slightly with Du. The behavior changes for Sr=0.1 since the evolution of Sh is characterized by a monotonous increase with Du to reach a maximum marking the limit of the stable branch. For case I, by increasing Du the crossed regions are successively regions 3, 4, and 5 while for case II the crossed regions are 2 (crossed first) and 6.


## 5. Conclusion

Thermosolutal natural convection within a shallow fluid layer subjected to vertical gradients of temperature and solute is investigated analytically and numerically in the presence of Soret and Dufour effects. The analytical solution based on the parallel flow assumption is found to be in good agreement with the numerical solution based on a finite-difference method used to solve the full governing equations in their transient forms. In view of the results discussed, the main conclusions are as follows:(i)
The thresholds of stationary and subcritical convections are derived analytically versus the governing parameters.(ii)
We have demonstrated analytically that the planeN-Du can be divided into different regions (up to six regions) with specific flow regimes.(iii)
The number of these regions, their extension, and their locations depend on whether the productSrDu=f(Le) is lower or higher than -(1+Le2)/2Le2.(iv)
The influence of the Dufour effect on the thresholds of stationary and subcritical convections may be stabilizing or destabilizing, depending on the buoyancy ratio and the Soret parameter.(v)
The important effect of the thermodiffusion and diffusion-thermo on fluid flow and heat and mass transfer characteristics is well demonstrated.



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