Convection driven by density variations caused by two different components which have different rates of diffusion plays an important role in fluid dynamics since such flows occur naturally in many physical and engineering processes. Heat and salt in sea water provide perhaps the best known example of double-diffusive convection, Stern 1. Other examples of double diffusive convection are encountered in diverse applications such as in chemical and petroleum industries, filtration processes, food processing, geophysics and in the modeling of solar ponds and magma chambers. A review of the literature in this subject can be found in Nield and Bejan 2.

One of the earliest studies of double diffusive convection was by Nield 3. Baines and Gill 4 investigated linear stability boundaries, while Rudraiah et al. 5 used the nonlinear perturbation theory to investigate the onset of double diffusive convection in a horizontal porous layer. Poulikakos 6 presented the linear stability analysis of thermosolutal convection using the Darcy-Brinkman model. Bejan and Khair 7 presented a multiple scale analysis of heat and mass transfer about a vertical plate embedded in a porous medium. They considered concentration gradients which aid or oppose thermal gradients. Related studies on double diffusive convection have been undertaken by, among others, Lai 8, Afify 9, and Makinde and Sibanda 10.

Investigations by, among others, Eckert and Drake 11 and Mortimer and Eyring 12 have provided examples of flows such as in the geosciences, where diffusion-thermo and thermal-diffusion effects are quite significant. Anjalidevi and Devi 13 showed that diffusion-thermo and thermal-diffusion effects are significant when density differences exist in the flow regime. In general, Diffusion-thermo and thermal-diffusion effects have been found to be particularly important for intermediate molecular weight gases in binary systems that are often encountered in chemical engineering processes. Theoretical studies of the Soret and Dufour effects on double diffusive convection have been made by many researchers, among them, Kafoussias and Williams 14, Postelnicu 15, Mansour et al. 16, Narayana and Sibanda 17, and Awad et al. 18.

In this article, we investigate convective heat and mass transfer along a vertical flat plate in the presence of diffusion-thermo, thermal diffusion effects, and an external magnetic field. The governing momentum, heat and mass transfer equations are, in general, strongly coupled and highly nonlinear. Apart from numerical methods, a number of semi-analytical techniques have in recent years been proposed to find approximate solutions of nonlinear boundary value problems. Some of these techniques, such as the Adomian decomposition method, the variational iteration method, the homotopy analysis method, the homotopy perturbation method, the differential transform method, etc, are now well known and their strengths and weaknesses well understood. Some of these weaknesses include, for example, small regions of convergence and the use of artificially inserted parameters (Liao 19, Geng 20). The recent spectral homotopy analysis method (see Motsa et el. 2122) sought to improve the accuracy and efficiency of the homotopy analysis method while retaining its essence. The use of spectral methods also provided greater flexibility in the choice of basis functions. Nonetheless, the challenge to find more accurate, robust and computationally efficient solution techniques for nonlinear problems in engineering and science still remains.

A recent method that remains to be generalized and whose robustness remains to be tested in the case of highly nonlinear equations with a strong coupling is the successive linearization method (Makukula et al. 2324). This method has been used in a limited number of studies by, for example, Awad et al. 25) and Motsa et al. 26 to solve fluid flow problems. In this study the coupled set of differential equations that describe convective heat and mass transfer flow along a vertical flat plate in the presence of diffusion-thermo, thermal diffusion effects and an external magnetic field are solved using the successive linearization method. A non-similarity technique is used to validate the linearization method.

We consider the problem of double diffusive convection along a vertical plate with an external magnetic field imposed along the y-axis. The induced magnetic field is assumed to be negligible. The fluid temperature and solute concentration in the ambient fluid are T_∞, C_∞ and those at the surface are T_w and C_w, respectively. The coordinates system and the flow configuration are shown in Figure 1.

Under the usual boundary layer and Boussinesq approximations the governing equations for a viscous incompressible fluid may be written as (Kafoussias and Williams 14);

∂ u ∂ x + ∂ v ∂ y = 0 ,

u ∂ u ∂ x + v ∂ u ∂ y = v ∂ 2 u ∂ y 2 - σ B 0 2 ρ u + g β T ( T - T ∞ ) + g β C ( C - C ∞ ) ,

u ∂ T ∂ x + v ∂ T ∂ y = α ∂ 2 T ∂ y 2 + D m k T c s c p ∂ 2 C ∂ y 2 ,

u ∂ C ∂ x + v ∂ C ∂ y = D m ∂ 2 C ∂ y 2 + D m k T T m ∂ 2 T ∂ y 2 ,

subject to the boundary conditions

u = 0 , v = 0 , T = T w , C = C w on y = 0 ,

u = U ∞ , T = T ∞ , C = C ∞ when y → ∞ .

where u and v are the velocity components along the x- and y- axes, respectively T and C are the fluid temperature and solute concentration across the boundary layer, ν is the kinematic viscosity, ρ is the fluid density, σ is the electrical conductivity, B₀ is the uniform magnetic field, β_T and β_C are the coefficients of thermal and solutal expansions, D_m is the thermal diffusivity, k_T is the thermal diffusion ratio, c_s is the concentration susceptibility, c_p is the fluid specific heat capacity, T_m is the mean fluid temperature, U_∞ is the free stream velocity and g is the gravitational acceleration.

To satisfy the continuity equation (1), we define the stream function ψ in terms of the velocity by

u = ∂ ψ ∂ y and v = - ∂ ψ ∂ x ,

and introduce the following dimensionless variables

ξ = x ℓ , η = U ∞ ν ξ 1 / 2 y , ψ = ( U ∞ ν ξ ) 1 / 2 f ( ξ , η ) , θ ( ξ , η ) = T - T ∞ T w - T ∞ , ϕ ( ξ , η ) = C - C ∞ C w - C ∞ ,

where, without loss of generality, we take the constant length ℓ to be unity in the subsequent analysis. Using equations (7) in (2)-(4), we get the transformed equations

f ‴ + 12 f f ″ − H a x f ′ + G r x θ + G c x ϕ = ξ ( f ′ ∂ f ′ ∂ ξ − f ″ ∂ f ∂ ξ ) ,

1 P r θ ″ + 1 2 f θ ′ + D f ϕ ″ = ξ f ′ ∂ θ ∂ ξ - θ ′ ∂ f ∂ ξ ,

1 S c ϕ ″ + 1 2 f ϕ ′ + S r θ ″ = ξ f ′ ∂ ϕ ∂ ξ - ϕ ′ ∂ f ∂ ξ ,

with corresponding boundary conditions

f = 0 , f ′ = 0 , θ = 1 , ϕ = 1 , on η = 0 , f ′ = 1 , θ = 0 , ϕ = 0 , when η → ∞ .

The fluid and physical parameters in equations (8)-(10) are the local thermal and solutal Grashof numbers Gr_x and Gc_x, the local magnetic field parameter Ha_x, the Prandtl number Pr, the Dufour number D_f, the Soret number S_r, and the Schmidt number Sc. These parameters are defined in equations (12), (13) below.

G r x = g β T ( T w - T ∞ ) x ν 2 , G c x = g β C ( C w - C ∞ ) x ν 2 , H a x = σ B 0 2 x ρ U ∞ , P r = ν α ,

D f = D m K T ( C w - C ∞ ) α C s C p ( T w - T ∞ ) , S r = K T D m ( T w - T ∞ ) α T m ( C w - C ∞ ) , S c = ν D m .

The parameters of engineering interest in heat and mass transport problems are the skin friction coefficient C_fx, the Nusselt number Nu_x and the Sherwood number Sh_x. These parameters characterize the surface drag, the wall heat and mass transfer rates, respectively, and are defined by

C f x = μ ρ U 2 ∂ u ∂ y y = 0 = f ″ ( ξ , 0 ) R e x ,

N u x = - x T w - T ∞ ∂ T ∂ y y = 0 = - R e x θ ′ ( ξ , 0 ) ,

and

S h x = - x C w - C ∞ ∂ C ∂ y y = 0 = - R e x ϕ ′ ( ξ , 0 ) ,

where Re_x = U_∞x/ν.

The main challenge in using the linearization method as described in Makukula et al. 2324 is how to generalize the method so as to find solutions of partial differential equations of the form (8)-(10). It is certainly not clear how the method may be applied directly to the terms on the right hand side of equations (8)-(10). For this reason, equations (8)-(10) are first simplified and reduced to sets of ordinary differential equations by assuming regular perturbation expansions for f, θ, and ϕ in powers of ξ (which is assumed to be small) as follows

f = f M 1 ( η , ξ ) = ∑ i = 0 M 1 ξ i f i ( η ) , θ = θ M 1 ( η , ξ ) = ∑ i = 0 M 1 ξ i θ i ( η ) , ϕ = ϕ M 1 ( η , ξ ) = ∑ i = 0 M 1 ξ i ϕ i ( η )

where M₁ is the order of the approximate solution. Substituting (17) into equations (8)-(10) and equating the coefficients of like powers of ξ, we obtain the zeroth order set of ordinary differential equations

f 0 ‴ + 1 2 f 0 f 0 ″ - H a x f 0 ′ + G r x θ 0 + G c x ϕ 0 = 0 ,

1 P r θ 0 ″ + 1 2 f 0 θ 0 ′ + D f ϕ 0 ″ = 0 ,

1 S c ϕ 0 ″ + 1 2 f 0 ϕ 0 ′ + S r θ 0 ″ = 0 ,

with corresponding boundary conditions

f 0 = 0 , f 0 ′ = 0 , θ 0 = 1 , ϕ 0 = 1 , at η = 0 f 0 ′ = 1 , θ 0 = 0 , ϕ 0 = 0 , as η = ∞ .

The O(ξ¹) equations are

f 1 ‴ + 1 2 f 0 f 1 ″ - ( H a x + f 0 ′ ) f 1 ′ + 3 2 f 0 ″ f 1 + G r x θ 1 + G c x ϕ 1 = 0 ,

1 P r θ 1 ″ + 1 2 f 0 θ 1 ′ - f 0 ′ θ 1 + 3 2 θ 0 ′ f 1 + D f ϕ 1 ″ = 0 ,

1 S c ϕ 1 ″ + 1 2 f 0 ϕ 1 ′ - f 0 ′ ϕ 1 + 3 2 ϕ 0 ′ f 1 + S r θ 1 ″ = 0 ,

with boundary conditions

f 1 = 0 , f 0 ′ = 0 , θ 1 = 0 , ϕ 1 = 0 , at η = 0 f 1 ′ = 0 , θ 1 = 0 , ϕ 1 = 0 , as η = ∞ .

Finally, the O(ξ²) equations are

f 2 ‴ + 1 2 f 0 f 2 ″ - ( H a x + 2 f 0 ′ ) f 2 ′ + 5 2 f 0 ″ f 2 + G r x θ 2 + G c x ϕ 2 = f 1 ′ f 1 ′ - 3 2 f 1 f 1 ″ ,

1 P r θ 2 ″ + 1 2 f 0 θ 2 ′ - 2 f 0 ′ θ 2 + 5 2 θ 0 ′ f 2 + D f ϕ 2 ″ = f 0 ′ θ 1 - 3 2 f 1 θ 1 ′ ,

1 S c ϕ 2 ″ + 1 2 f 0 ϕ 2 ′ - 2 f 0 ′ ϕ 2 + 5 2 ϕ 0 ′ f 2 + S r θ 2 ″ = f 1 ′ ϕ 1 - 3 2 f 1 ϕ 1 ′ .

These equations have to be solved subject to boundary conditions

f 2 = 0 , f 2 ′ = 0 , θ 2 = 0 , ϕ 2 = 0 , at η = 0 f 2 ′ = 0 , θ 2 = 0 , ϕ 2 = 0 , as η = ∞ .

The coupled system of equations (18)-(20), (22)-(24), and (26)-(28) together with the associated boundary conditions (21), (25), and (29) may be solved independently pairwise one after another. These equations may now be solved using the successive linearization method in the manner described in Makukula et al. 2324). We begin by solving equations (18)-(20) with boundary conditions (21).

The successive linearization method is a non-perturbation method requiring neither the presence of an embedded perturbation parameter nor the addition of an artificial parameter. The method is therefore free of the major limitations associated with other perturbation methods. In the SLM algorithm assumption is made that the functions f₀(η), θ₀(η), and ϕ₀(η) may be expressed as

f 0 ( η ) = F i ( η ) + ∑ m = 0 i - 1 F m ( η ) , θ 0 ( η ) = Θ i ( η ) + ∑ m = 0 i - 1 Θ m ( η ) , ϕ ( η ) = Φ i ( η ) + ∑ m = 0 i - 1 Φ m ( η ) ,

where F_i, Θ_i, and Φ_i (i ≥ 1) are unknown functions and F_m, Θ_m, and Φ_m are successive approximations which are obtained by recursively solving the linear part of the system that is obtained from substituting equations (30) in (18)-(20). In choosing the form of the expansions (30), prior knowledge of the general nature of the solutions, as is often the case with perturbation methods, is not necessary.

Suitable initial guesses F₀(η), Θ₀(η), and Φ₀(η) which are selected to satisfy the boundary conditions (21) are

F 0 ( η ) = η + e - η - 1 , Θ 0 ( η ) = e - η , Φ 0 ( η ) = e - η ,

The subsequent solutions F_i, Θ_i, and Φ_i are obtained by iteratively solving the linearized form of the equations that are obtained by substituting equation (30) in the governing equations (18)-(20). The linearized equations to be solved are

F i ‴ + a 1 , i - 1 F i ″ - H a x F i ′ + a 2 , i - 1 F i + G r x Θ i + G c x Φ i = r 1 , i - 1 ,

1 P r Θ i ″ + b 1 , i - 1 Θ i ′ + b 2 , i - 1 F i + D f Φ i ″ = r 2 , i - 1 ,

1 S c Φ i ″ + c 1 , i - 1 Φ i ′ + c 2 , i - 1 F i + S r Θ i ″ = r 3 , i - 1 ,

subject to the boundary conditions

F i ( 0 ) = F i ′ ( 0 ) = F i ′ ( ∞ ) = Θ i ( 0 ) = Θ i ( ∞ ) = Φ i ( 0 ) = Φ i ( ∞ ) ,

where the coefficients parameters a_{k, i-1}, b_k,i-1, c_k,i-1, d_k,i-1, and r_k,i-1 are defined by

a 1 , i - 1 = 1 2 ∑ m = 0 i - 1 F m , a 2 , i - 1 = 1 2 ∑ m = 0 i - 1 F m ′ ′ , b 1 , i - 1 = 1 2 ∑ m = 0 i - 1 F m b 2 , i - 1 = 1 2 ∑ m = 0 i - 1 Θ m ′ , c 1 , i - 1 = 1 2 ∑ m = 0 i - 1 F m c 2 , i - 1 = 1 2 ∑ m = 0 i - 1 Φ m ′ , r 1 , i - 1 = H a x ∑ m = 0 i - 1 F m ′ - 1 2 ∑ m = 0 i - 1 F m ∑ m = 0 i - 1 F m ′ ′ - ∑ m = 0 i - 1 F m ′ ′ ′ - G r x ∑ m = 0 i - 1 Θ m - G c x ∑ m = 0 i - 1 Φ m , r 2 , i - 1 = - 1 P r ∑ m = 0 i - 1 Θ m ′ ′ - 1 2 ∑ m = 0 i - 1 F m ∑ m = 0 i - 1 Φ m ′ - D f ∑ m = 0 i - 1 Φ m ′ ′ , r 3 , i - 1 = - 1 S c ∑ m = 0 i - 1 Φ m ′ ′ - 1 2 ∑ m = 0 i - 1 F m ∑ m = 0 i - 1 Φ m ′ - S r ∑ m = 0 i - 1 Φ m ′ ′ ,

Once each solution F_i, Θ_i, and Φ_i (i ≥ 1), has been found from iteratively solving equations (32)-(34), the approximate solutions for the system (18)-(20) are obtained as

f 0 ( η ) ≈ ∑ i = 0 M 2 F i ( η ) , θ 0 ( η ) ≈ ∑ i = 0 M 2 Θ i ( η ) , ϕ 0 ( η ) ≈ ∑ i = 0 M 2 Φ i ( η ) ,

where M₂ is the order of the SLM approximations. In this study, we used the Chebyshev spectral collocation method to solve equations (32)-(34). The physical region [0, ∞) is first transformed into the spectral domain [-1,1] using the domain truncation technique in which the problem is solved on the interval [0, L], where L is a scaling parameter used to invoke the boundary condition at infinity. This is achieved by using the mapping

η L = ς + 1 2 , - 1 ≤ ς ≤ 1 ,

We discretise the spectral domain [-1,1] using the Gauss-Lobatto collocation points given by

ς j = cos π j N , j = 0 , 1 , … , N ,

where N is the number of collocation points used. The unknown functions F_i, Θ_i , and Φ_i are approximated at the collocation points by

F i ( ς ) ≈ ∑ k = 0 N F i ( ς k ) T k ( ς j ) , Θ i ( ς ) ≈ ∑ k = 0 N Θ i ( ς k ) T k ( ς j ) Φ i ( ς ) ≈ ∑ k = 0 N Φ i ( ς k ) T k ( ς j ) , j = 0 , 1 , … , N

where T_k is the kth Chebyshev polynomial defined as

T k ( ς ) = cos [ k cos - 1 ( ς ) ] .

The derivatives of the variables at the collocation points are represented as

F i ( r ) = ∑ k = 0 N D k j r F i ( ς k ) , Θ i ( r ) = ∑ k = 0 N D k j r Θ i ( ς k ) , Φ i ( r ) = ∑ k = 0 N D k j r Φ i ( ς k ) , j = 0 , 1 , … , N

where r is the order of differentiation and D = 2 L D with D being the Chebyshev spectral differentiation matrix whose entries are defined as;

D j k = c j ( - 1 ) j + 1 c k ς j - ς k j ≠ k ; j , k = 0 , 1 , … , N , D k k = - ς k 2 ( 1 - ς k 2 ) k = 1 , 2 , … , N - 1 , D 00 = 2 N 2 + 1 6 = - D N N .

Substituting equations (37)-(41) into (32)-(34) gives the following linear system of equations

A i - 1 X i = R i - 1

subject to the boundary conditions

F i ( ς 0 ) = ∑ k = 0 N D 0 k F i ( ς k ) = g i ( ς 0 ) = Θ i ( ς 0 ) = Φ i ( ς 0 ) = F i ( ς N ) = g i ( ς N ) = Θ i ( ς N ) = Φ i ( ς N ) = 0 .

Here A_i-1 is a 3(N + 1) × 3(N + 1) square matrix, while X_i and R_i-1 are 3(N + 1) × 1 column vectors defined by

A i - 1 = A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 , X i = F ̃ i Θ ̃ i Φ ̃ i , R i - 1 = r 1 , i - 1 r 2 , i - 1 r 3 , i - 1

where

F ̃ i = [ F i ( ς 0 ) , F i ( ς 1 ) , … , F i ( ς N - 1 ) , F i ( ς N ) ] T ,

Θ ̃ i = [ Θ i ( ς 0 ) , Θ i ( ς 1 ) , … , Θ i ( ς N - 1 ) , Θ i ( ς N ) ] T ,

Φ ̃ i = [ Φ i ( ς 0 ) , Φ i ( ς 1 ) , … , Φ i ( ς N - 1 ) , Φ i ( ς N ) ] T ,

r 1 , i - 1 = [ r 1 , i - 1 ( ς 0 ) , r 1 , i - 1 ( ς 1 ) , … , r 1 , i - 1 ( ς N - 1 ) , r 1 , i - 1 ( ς N ) ] T

r 2 , i - 1 = [ r 2 , i - 1 ( ς 0 ) , r 2 , i - 1 ( ς 1 ) , … , r 2 , i - 1 ( ς N - 1 ) , r 2 , i - 1 ( ς N ) ] T

r 3 , i - 1 = [ r 3 , i - 1 ( ς 0 ) , r 3 , i - 1 ( ς 1 ) , … , r 3 , i - 1 ( ς N - 1 ) , r 3 , i - 1 ( ς N ) ] T

and

A 11 = D 3 + a 1 , i - 1 D 2 - H a x D + [ a 3 , i - 1 ] , A 12 = [ G r x ] , A 13 = [ G c x ] , A 21 = [ b 2 , i - 1 ] , A 22 = P r - 1 D 2 + b 1 , i - 1 D , A 23 = D f D 2 A 31 = [ c 2 , i - 1 ] , A 32 = S r D 2 , A 33 = S c - 1 D 2 + c 1 , i - 1 D ,

where [·] is a diagonal matrix of size (N + 1) × (N + 1) and a_k,i-1, b_k,i-1, c_k,i-1 are diagonal matrices of size (N + 1) × (N + 1) and T is the transpose. After modifying the matrix system (43) to incorporate the boundary conditions (44), the solution is obtained as

X i = A i - 1 - 1 R i - 1

Equation (53) gives a solution of (18)-(20) for f₀, θ₀ and ϕ₀. The procedure is repeated to obtain the O(ξ¹) and O(ξ²) solutions using equations (22)-(24) and (26)-(28), respectively.

In generating the results in this article, we determined through numerical experimentation that L = 15, N = 40, and M₂ = 5 gave sufficient accuracy for the linearization method. The value of the Prandtl number used is Pr = 0.71 which physically corresponds to air. The Schmidt number used Sc = 0.22 is for hydrogen at approximately 25° and one atmospheric pressure (see Afify 27). In concert with previous related studies, the Dufour and Soret numbers are chosen in such a way that their product is constant, provided the mean temperature T_m is also kept constant.

To determine the accuracy and validate the linearization method, equations (8)-(10) were further solved using a local non-similarity method (LNSM) developed by Sparrow and Yu 28 and Sparrow et al. 29. Previous studies have consistently used the Matlab bvp4c solver to evaluate the accuracy of the successive linearization method. However, as with other BVP solvers, the accuracy and convergence of the bvp4c algorithm depends on a good initial guess and works better for systems involving few equations (Shampine et al. 30).

Tables 1, 2, and 3 show, firstly the effects of various parameters on the skin-friction and the local heat and mass transfer coefficients at different values of ξ and, second, give a sense of the accuracy and convergence rate of the linearization method. The results from the two methods are in excellent agreement with the second order SLM series giving accuracy of up to five significant figures.

Table 1 shows the effect of increasing the magmatic field parameter Ha_x on the local skin friction, heat and mass transfer coefficients. We observe that increasing the magnetic field parameter reduces the local skin friction as well as the heat and mass transfer coefficients. In Table 2, we present the effect of the Soret parameter on f"(0), -θ'(0), and -ϕ'(0) which are, respectively, proportional to the local skin friction coefficient, the local Nusselt number and Sherwood number. We observe that f"(0) and -θ'(0) increase with increases in Sr, while -ϕ'(0) decreases as Sr increases. These results are confirmed in Table 3. Here the Nusselt number increases as the Soret number increases, while the opposite trend occurs as the Dufour number increases. The recent study by El-Kabeir 31 shows that these results may be modified by injection, suction or the presence of a chemical reaction.

The influence of the various fluid and physical parameters on the fluid properties is given qualitatively in Figures 2, 3, 4, 5, 6 and 7. Figures 2 and 3 illustrate the effect of the magnetic filed parameter on the velocity f'(η), temperature θ and concentration ϕ profiles within the boundary layer. We observe that, as expected, strengthening the magnetic field slows down the fluid motion due to an increasing drag force which acts against the flow if the magnetic field is applied in the normal direction. We also observe that the magnetic field parameter enhances the temperature and concentration profiles. The effect of broadening both the temperature and concentration distributions is to reduce the wall temperature and concentration gradients thereby reducing the heat and mass transfer rates at the wall.

Figure 5 shows the effect of increasing the Soret parameter (reducing the Dufour parameter) on the fluid velocity f'(η). The fluid velocity is found to increase with the Soret parameter.

The effect of Soret parameter on the temperature within the thermal boundary layer and the solute concentration is shown in Figures 6 and 7, respectively. An increase in the Soret effect reduces the temperature within the thermal boundary layer leading to an increase in the temperature gradient at the wall and an increase in heat transfer rate at the wall. On the other hand, increasing the Soret effect increases the concentration distribution which reduces the concentration gradient at the wall. These results are similar to the earlier findings by El-Kaberir 31 and Alam and Rahman 32, although the latter studies were subject to injection/suction.

In this article, we have investigated MHD and cross-diffusion effects on double-diffusive convection from a vertical flat plate in a viscous incompressible fluid. Numerical approximations for the governing equations were found using a combination of a regular perturbation expansion and the successive linearization method. The solutions were validated by using a local similarity, non-similarity method. We determined the effects of various parameters on the fluid properties as well as on the skin-friction coefficient, the heat and the mass transfer rates. We have shown that the magnetic field parameter enhances the temperature and concentration distributions within the boundary layer. The effect of thermo-diffusion is to reduce the temperature and enhance the velocity and the concentration profiles. The diffusion-thermo effect enhances the velocity and temperature profiles while reducing the concentration distribution. The skin-friction, heat and mass transfer coefficients decrease with an increase in the magnetic field strength. The skin-friction and heat transfer coefficients increase whereas the mass transfer coefficient decreases with increasing Soret numbers.

The authors declare that they have no competing interests.

The problem was formulated following discussions by all three authors. Numerical simulations were carried out by AAK and FGA while the discussion was authored principally by PS. All authors read and approved the final manuscript.