1 Introduction

The unusual thermal conductivity enhancement for nanoparticles in a fluid at very low particle loading has drawn world attention. Choi was the first person to use the term nanofluid for the colloidal suspension of nanometer sized particles in ordinary liquids^[1]. A large number of experimental and theoretical studies have shown that nanofluids surpass other fluids in the heat exchange efficiency. Masuda et al.^[2] first showed that the thermal conductivity of nanofluids was enhanced due to the presence of nanoparticles. Eastman et al.^[3] revealed that if 0.3% of copper nanoparticles were added in ethylene glycol,the thermal conductivity of ethylene glycol would increase 40%. Das and Choi^[4] surveyed the heat transfer in nanofluids in detail. Buongiorno^[5] proposed a model based on the laws of fluid dynamics and heat transport for the nanofluids incorporating the important effects of thermophoresis and Brownian diffusion. Tzou^[6] was the first scientist to use the model of Buongiorno^[5] to investigate the instability problem in nanofluids. He used the method of eigen function expansions,and observed that regular fluids were more stable than nanofluids. Kuznetsov and Nield^[7] considered the thermal instabilityproblem for the nanofluid layer,and found that the stability of the nanofluids depended on the distribution of the nanoparticles on the boundaries of the layer. Bhadauria and Agarwal^[8] investigated the effects of rotation on the instability of nanofluids,and found that the rotating nanofluid layer became more stable.

Temperature and salinity produce opposing effects on the density of sea water since the molecular conductivity of heat is very high when it is compared with that of salt. This results in the convective motion known as the double-diffusive/binary convection. This type of convection occurs widely in nature and industry,e.g.,oceans,lakes,shallow coastal water,chemical processes,crystal growth,solidification,and food processing. Kim et al.^[9] studied the binary nanofluid convection under the Soret and Dufour effects. Kuznetsov and Nield^[10] and Nield and Kuznetsov^[11] studied the thermosolutal nanofluid convection in porous and non-porous media. Gupta et al.^[12] studied the binary nanofluid convection independent of the restrictions on the parameters,and found that the instability mode was through oscillatory motion. Yadav et al.^[13] considered the instability problem of top heavy binary nanofluids in porous media.

Magnetic field imparts the fluid certain rigidity and the fluid certain elasticity properties which are related to the stability of the fluid. Gupta et al.^[14] studied the effects of magnetic field on the thermal instability of bottom heavy nanofluids,and found that the magnetic field increased the thermal stability of the nanofluid layer. The convection in the binary nanofluids in magnetic field has important and great applications in geophysics,e.g.,enhanced oil recovery from underground reservoirs,which gives us the motivation to investigate the thermosolutal instability of the nanofluids in a magnetic field. Since the diffusivity of solute is much lower than that of nanoparticles or heat,the stability of the fluid layer must get influenced by the addition of the solute in it. Therefore,the present convection problem is a triple diffusion process. It will be solved by the normal mode analysis and weighted residual method. Due to the inclusion of magnetic field,the Lorentz force term is added in the momentum equation, and the Maxwell equations are introduced. As a result,the eighth-order linear differential equation is obtained,which leads to a complex expression for the thermal Rayleigh number. To analytically examine the results,we approximate the expression for the Rayleigh number by taking negligible values of the Dufour parameter and the Soret parameter and limiting to the large values of the nanofluid Lewis number and the Prandtl number. These approximations on the Rayleigh number for the stationary convection motion and the oscillatory motion are released when the problem is analyzed numerically for the alumina-water nanofluid with the MATHEMATICA software.

2 Problem formulation and conservation equations

An infinite horizontal binary nanofluid layer with nonmagnetic nanoparticles,which is both incompressible and electrically conducting,is taken under a vertical magnetic field H^∗ = (0,0,H₀ ^∗) (see Fig. 1). The asterisks represent the dimensional variables,and distinguish them from the non-dimensional ones. We take T₁^∗ and T₀^∗ as the temperatures,ϕ₁^∗ and ϕ₀^∗ as the nanoparticle volume fractions,and C₁^∗ and C₀^∗ as the concentrations of the solute at the bottom and the top of the fluid layer,respectively.

The conservation equations given by Buongiorno^[5],Nield and Kuznetsov^[11],and Chandrasekhar^[15] along with Maxwell’s equations are

where v∗ is the fluid velocity defined by

ϕ^∗ is the volume fraction of nanoparticles. t^∗ is the time. D_B is the coefficient of diffusion due to the Brownian motion. D_T is the coefficient of diffusion due to the thermophoresis. T^∗ is the fluid temperature. ρ0 is the density of the fluid at the temperature T₀^∗. p^∗ is the fluid pressure. μ is the fluid viscosity. βT is the thermal volumetric coefficient. β_C is the solutal volumetric coefficient. g is the acceleration due to gravity. H^∗ is the magnetic field defined by

μ_e is the fluid magnetic permeability. ρ is the fluid density. c is the fluid specific heat. k is the fluid conductivity. ρ_p is the nanoparticles density. c_p is the nanoparticle specific heat. D_TC is the diffusivity of the Dufour type. C^∗ is the solute concentration. D_S is the diffusivity of solute. D_CT is the diffusivity of the Soret type. η is the fluid electrical resistivity. Let us introduce the dimensionless variables as follows: where

With the introduction of these non-dimensional variables,Eqs. (1)-(7) take the forms as follows: where P_r1 is the Prandtl number. P_r2 is the magnetic Prandtl number. R_m is the basicdensity Rayleigh number. R_t is the thermal Rayleigh number. R_n is the nanoparticle Rayleigh number. R_s is the solutal Rayleigh number. Q is the Chandrasekhar number. L_n is the fluid Lewis number. L_e is the solutal Lewis number. N_A is the diffusivity ratio. N_B is the particle density increment. N_TC is the regular Dufour parameter. N_CT is the regular Soret parameter. They are defined as follows: Therefore,we have the thermal Rayleigh number,the analogous solutal Rayleigh number, and the nanoparticle concentration Rayleigh numbers due to the presence of three diffusing components,i.e.,heat,solute,and nanoparticles,respectively.

3 Primary flow and perturbation equations

Initially,the nanofluid layer is at rest. Therefore,the physical quantities,i.e.,the temperature,the concentration of solute,and the nanoparticle volume fraction,vary in the vertical direction only,and are given by

The subscript p represents the primary variable. Applying Eq. (17) to Eqs. (10)-(13) and using the fact that,for most nanofluids,the Lewis number is large,and the diffusivity ratio is small^[5],we have

Let us introduce small perturbations to the primary flow,and write

Let us substitute these perturbations to Eqs. (9)-(15),neglect the products of the perturbed quantities,and use Eq. (18). Then,we get

where

Equations (27) and (28) are derived by the identity curl curl ≡ graddiv −∇² on Eqs. (20),(21), (25),and (26). The parameter R_m is just a basic density Rayleigh number,and it does not appear in these subsequent equations. 4 Normal modes and Galerkin weighted residual method

The differential equations (22)-(24) and (28) form an eigenvalue problem,which will be solved by the normal modes analysis. Let

From the above-mentioned equations,we get where Let us write s = iω for the neutral stability,where ω is the dimensionless frequency. For the case of two free boundaries, The equations do not contain any variable related to the magnetic field. Therefore,the equations can be solved without any boundary condition on the magnetic field. This agrees with the results obtained by Chandrasekhar^[15].

Equations (29)-(32) can be solved by the one-term Galerkin approximation^[11] to obtain the eigenvalue equation as follows:

where

5 Results and discussion

5.1 Stationary motion

For the stationary motion ω = 0,from Eq. (35),we have

Here,it is worthwhile mentioning that the expression for Rt is independent of both the Prandtl numbers,and the parameters containing the Brownian effects and the thermophoretic effects are presented in the thermal energy equation and the conservation equation for nanoparticles.

Take x = in Eq. (36). Then,we have

The thermal Rayleigh number Rt given by Eq. (37) takes its minimum value when Therefore,the critical wave number xc shows a substantial increase when the Chandrasekhar number Q increases,and it is independent of the nanoparticles and solute concentration in the fluid.

5.2 Oscillatory motion

Let us write the growth rate s as

where s_r and ω are real. For the oscillatory convection,s ≠ 0 and s_r = 0,i.e.,s = iω ≠ 0. Therefore,to determine the state in which the convection sets as the oscillatory motion,we separate the real and imaginary parts of the eigenvalue equation (35) by putting s = iω. For the convection through oscillations,we solve Eqs. (39) and (40) to determine the critical Rayleigh number for which ω is real.

5.3 Validation of results and approximate solution

In the absence of nanoparticles and solute,Eqs. (36),(39),and (40) reduce to those given by Chandrasekhar^[15]. Therefore,due to the addition of nanoparticles and solute in the regular fluid,seven additional parameters N_CT,N_CT,L_e,R_s,R_n,L_n,and N_A are introduced in the expression for the thermal Rayleigh number,which must strongly affect the convection of the fluid layer. In the absence of solute,Eqs. (36),(39),and (40) agree with those given by Gupta et al.^[14]. Thus,we have four additional parameters N_CT,N_CT,L_e,and R_s due to the presence of solute in the fluid. It is clear from Eqs. (39) and (40) that R_t (oscillatory) is complex since it is well-known that the triple diffusion problem is much more complicated. Therefore,it is necessary to make some valid approximations in the result in order to make the problem tractable. Let the Lewis number and the Prandtl number approach infinity when the Dufour and Soret parameters are negligible^[11]. Then,Eqs. (39) and (40) can be reduced to

Solving Eqs. (41) and (42),we get Neglecting the Dufour parameter and the Soret parameter in Eq. (36) and comparing the convection,we get For Q = 0,Eqs. (44) and (45) reduce to those given by Nield and Kuznetsov^[11]. Clearly, for oscillations to occur,R_n must be positive (see Eq. (43)). This means that for top heavy configuration of nanoparticles,the convection in the fluid will be through the stationary mode only. For sufficiently large and negative R_N and hence negative N_A (for top heavy nanofluids), R_t (stationary) takes negative values. This means that the destabilizing effect of top heavy arrangement is so great that the applied magnetic field must be increased or the temperature at the lower layer must be decreased in comparison with that at the upper layer to attain the neutral stability. Equation (45) further establishes that the instability sets in through oscillations for bottom heavy configuration of nanoparticles. It is noteworthy that the frequency of oscillation decreases when the Chandrasekhar number increases,increases the concentration Rayleigh number increases,and it is independent of the solute parameter.

As the flow patterns induced by the thermal and solute effects or by the combination of these two are identical,a sketch of R = R_t + R_s versus Rn is shown in Fig. 2. The value of R_c is the thermal Rayleigh number at the instability onset with a fixed Chandrasekhar number Q for the regular fluid. Clearly,the binary nanofluids with bottom heavy arrangement of nanoparticles are more stable than the regular fluids,while the binary nanofluids with top heavy distribution of nanoparticles are far less stable than the regular binary nanofluids. The magnetic field has a strong stabilizing effect on the binary nanofluid layer when R_c increases. The figure also shows that the slopes of the two convective regimes are different. This means that a small increase in R_n leads to a large decrease in R for top heavy nanofluids and a negligible increase in R for bottom heavy nanofluids.

6 Numerical results and discussion

To investigate the complete parameter effects on the stability problem,Eq. (36) for stationary convection and Eqs. (39) and (40) for oscillatory convection are analyzed numerically with the MATHEMATICA sofware. The bottom heavy nanofluids are considered,in which oscillatory convection is possible. The values of the parameter for the alumina-water nanofluid are the same as those given by Buongiorno^[5] and Kuznetsov^[16]. They are as follows:

and the remaining parameters are

The above parameters are fixed except when the variation is considered with respect to that particular parameter.

Figures 3 and 4 show the variations of R_t when N_TC = 0.01,0.03,0.05 and N_CT = 0,2,4, respectively. From the figures,we can see that R_t (oscillatory) < R_t (stationary),and the instability mode is oscillatory for both cases. When the Dufour parameter increases,R_t (stationary) and R_t (oscillatory) decrease. Moreover,the Soret parameter has a largely destabilizing effect and a slight increase effect on R_t (stationary),and R_t (oscillatory) is observed around the critical wave number (see Table 1). As far as the critical wave number is concerned,it is independent of the two parameters.

Let us now consider the effects of the Chandrasekhar number on R_t for Q = 100,200,500 (see Fig. 5). From the figure,we can see that the instability mode is oscillatory. Moreover,the values of R_t (stationary) and R_t (oscillatory) exhibit a significant increase when Q increases. Therefore,the magnetic field parameter has a strong stabilizing effect on the convection in the fluid layer. The figure also shows that the critical wave number increases when the Chandrasekhar number increases. These results are expected as shown in Section 5. It is noteworthy that for regular fluids without solute,overstability cannot occur for P_r2 < P_r1^[15],but this is not for binary nanofluids.

Figures 6 and 7 show the results of Rt when

The instability is through overstability for both cases. The solute Rayleigh number R_s stabilizes the fluid layer system (see Table 2). This means that the effects of solute are reversed when the effects of the Dufour and Soret parameters are considered. Moreover,the critical wave number does not depend on the solute. R_t (stationary) increases as R_n increases,while the increase in R_t (oscillatory) is so small that it does not appear in the figure. This establishes the fact that the stabilizing effects of the concentration Rayleigh number for the present configuration are small,which is in confirmation with the analytical result drawn in Section 5.

To find the Rayleigh number,we consider the fact that for most nanofluids,the Lewis number is large,and the diffusivity ratio is small. Therefore,the variations in these two parameters will have no effect on the stability of the layer. Also,out of all the parameters which appear in the expression for the Rayleigh number,there are only two parameters,i.e.,the Lewis number and the diffusivity ratio,which depend on the nanoparticle properties. Therefore,it hardly matters which nanoparticle is being used for the numerical computation. However,one can definitely conclude that all the results,which hold for the alumina-water nanofluids,are true for almost all nanofluids. To differentiate the present work with the previous works on nanofluids,it is necessary to mention that Yadav et al.^[13] analyzed the binary nanofluid convection only for top heavy distribution of nanoparticles and without introducing the effects of the Lorentz force due to the presence of the magnetic field. The oscillatory motion is not existent for top heavy configuration of nanoparticles. Gupta et al.^[14] considered a thermal convection problem in a magnetic field,and used arbitrary nanofluid parameter values without introducing valid approximations to obtain the analytical results. Nield and Kuznetsov^[11] analytically examined the double diffusive convection problem without any magnetic field parameter. No numerical computation has been carried out for alumina-water or any other nanofluids.

7 Conclusions

The instability of a horizontal layer of a binary nanofluid in a vertical magnetic field is investigated by the normal mode technique and the one-term Galerkin approximation. The results are encapsulated in Eq. (36) for the stationary convection and Eqs. (39) and (40) for the oscillatory motion. Complex expressions for the Rayleigh number are simplified by valid approximations for the analytical study,and numerical investigations are made for the aluminawater nanofluid. Due to the inclusion of the magnetic field,a Lorentz force term is added in the momentum equation,which results in strong stabilizing effects of the magnetic field parameter (the Chandrasekhar number) on the fluid layer. It is established that the critical wave number increases when the Chandrasekhar number increases,and it is independent of the solute and nanoparticles. The results show that bottom heavy binary nanofluids are more stable than regular binary fluids,while top heavy binary nanofluids are less stable than regular binary fluids. For top heavy nanofluids,the oscillatory motion is not possible,and the instability through stationary convection increases with an increase in the nanoparticle concentration at the upper boundary of the fluid layer. The destabilizing effects of large concentration of nanoparticles at the top are so great that the magnitude of the applied magnetic field must be increased so as to neutralize the effects of the nanoparticles,or the temperature at the lower boundary must be decreased. The heat transfer mode is oscillatory for bottom heavy nanofluids. Moreover,the oscillation frequency increases with the increase in the nanoparticle Rayleigh number,decreases when the magnetic field increases,and it is independent of the solute. When the Soret and Dufour effects are negligible,the solute destabilizes the fluid layer. The Soret parameter has largely destabilizing effects except around a small area of the critical wave number whereas the effects of the Dufour parameter are destabilizing.