Soret and Dufour effects on heat and mass transfer in chemically reacting MHD flow through a wavy channel

ABSTRACT The problem of coupled heat and mass transfer by free convection of a chemically reacting viscous incompressible and electrically conducting fluid confined in a vertical channel bounded by wavy wall and flat wall in the presence of diffusion-thermo (Dufour), thermal-diffusion (Soret) and internal heat source or sink is studied. The walls are maintained at constant but different temperatures and species concentrations. A uniform magnetic field β0 is acting transversely to the walls which are assumed to be electrically non-conducting. The dimensionless governing equations are perturbed into mean part (zeroth-order) and perturbed part (first-order), using amplitude as a perturbation parameter. The first-order quantities are obtained by long wave approximation. The resulting set of coupled ordinary differential equations are solved numerically using the Adomian decomposition method. Some of the results indicating the influence of various parameters on the zeroth-order and first-order fluid flow, heat and mass transfer characteristics are presented graphically.


Introduction
Recently, there has been some renewed attentions in the study of free convective heat and mass transfer in fluid flow through a wavy or irregular channel. This is because the analysis of such flows find applications in various industrial and engineering problems such as nuclear reactor, heat ex-changers, transpiration cooling of re-entry vehicles and rocket boosters, film vapourization in combustion chambers and cross-hatching on ablative surfaces [1,2]. Additional significant applications include the use of rough (wavy) walls in medical apparatus to increase the mass transfer in blood (blood oxygenator). Rough surfaces are often used as flow passages in several applications for controlling the rate of heat transfer, cooling of electronic components and the designing of ventilating-heating building [3,4].
Some of the mentioned studies include, for example, the work of Vajraelu and Sastri [1] who examined the influence of waviness of one of the walls on the flow and heat transfer characteristics of an incompressible viscous fluid confined between two long vertical walls. Fasogbon [2] discussed analytically, the studies of heat and mass transfer by free convection in a two-dimensional irregular channel. Recently, the effects of chemical reaction and heat source on twodimensional free convection magneto-hydrodynamics (MHD) flow in an irregular channel with porous medium were investigated by Davika et al. [3]. Most recently, two-dimensional heat transfer of a free convection MHD flow with radiation and temperature-dependent heat source of a viscous incompressible fluid in a porous medium within a wavy channel were discussed by Dada and Disu [4]. Fasogbon and Omolehin [5] examined the radiation effect on natural convection in an irregular channel. Abubakar [6] studied the effect of the wall slip on a laminar two-dimensional free convective flow of fluid confined between an irregular wall and a flat wall. He discovered that increase in the slip parameter at the flat wall leads to a decrease in fluid velocity. Kumar [7] studied two-dimensional heat transfer of a free convective MHD flow with radiation and temperaturedependent heat source of a viscous incompressible fluid in a vertical irregular channel.
Meanwhile, it is well known that for a simultaneous occurrence of heat and mass transfer in a moving fluid, the relationships between the driven potential and the corresponding fluxes are of important. Also it is noticed that the energy flux (rate of energy transfer per unit area) and mass flux (rate of mass flow per unit area) can be generated by temperature gradients as well as composition gradients. The energy caused or generated by composition gradients is known as Dufour or diffusion-thermo effect and is considered useful in isotope separation. The mass flux created or generated by temperature gradients is known as Soret or thermal-diffusion effect which is consid-ered useful in mixture of gases with very light molecular weight (hydrogen-helium) and medium molecular weight (nitrogen-air) Kafoussias and William [8]. In view of the technological important applications of Soret and Dufour effects in sciences and engineering, these terms are considered in energy and concentration equations respectively in the present work.
In view of these applications, Gbadeyan et al. [9] examined the effects of thermal-diffusion and diffusionthermo on combined heat and mass transfer on mixed convective boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid in the presence of magnetic field. Makinde and Olanrewaju [10] carried out the study of an unstable mixed convection with Soret and Dufour effects past a porous plate moving through a binary mixture of chemically reacting fluid. Olanrewaju and Gbadeyan [11] analysed a mathematical model in order to study the effect of Soret, Dufour, chemical reaction, thermal radiation and volumetric heat generation/absorption on mixed convection stagnation point flow on an isothermal vertical plate in a porous media. Alam et al. [12] studied Dufour and Soret effects on steady free convection and mass transfer flow past a semi-infinite vertical porous plate in a porous medium. Olanrewaju et al. [13] analysed the influence of chemical reaction, thermal radiation, thermal-diffusion, diffusion-thermo on hydromagnetic free convection with heat and mass transfer past a vertical plate with suction/injection. Also, Olanrewaju and Makinde [14] investigated free convective heat and mass transfer of an incompressible electrically conducting fluid past a moving vertical plate in the presence of suction and injection with thermal-diffusion (Soret) and diffusion-thermo (Dufour) effects.
MHD (magneto fluid dynamics or hydromagnetics) is the study of the dynamics of electrically conducting fluids. Example of such fluid include plasma, liquid metal and salt water or electrolytes.The fundamental concept behind MHD is that magnetic field can induce electrical currents in a moving conductive fluid, which in turn creates forces on the fluid and also changes the magnetic field itself, this has led to extensive studies of MHD fluid flow. For example, Das and Ahmed [15] studied the problem of free convective MHD flow and heat transfer in a viscous incompressible fluid confined between a long wavy wall and a parallel flat wall. Disu and Dada [16] examined two-dimensional heat transfer of a free convective-radiative MHD flows with variable viscosity and heat source of a viscous incompressible fluid in a porous medium between two vertical wavy walls.
Also chemical reaction is a process that leads to transformation of one set of chemical substances to another. The chemical reactions are central to chemical engineering where they are used for synthesis of new compound from natural raw materials such as petroleum, mineral ores and thermite reaction to generate light and heat in pyrotechnics and welding. The presence of a foreign mass in air or water causes some kind of chemical reaction. Some foreign mass may be present either naturally or mixed with air or water. It is enormous practical consequence to engineers and scientists, because of the study of heat and mass transfer with chemical reaction is almost universal occurrence in many branches of science and engineering (Loganathan et al. [17]). In this article [17], the authors examined the influence of chemical reaction on unsteady free convective and mass transfer flow past a vertical plate with variable viscous and thermal conductivity. Also, the natural convective power-law fluid flow past a vertical plate embedded in a non-darcian porous medium in the presence of a homogeneous chemical reaction is studied in Chamka et al. [18].
Furthermore, heat transfer is a process by which internal energy from one substance transfers to another and mass transfer is the transport of a substance (mass) in liquid or gaseous media. Heat and mass transfer occur simultaneously in many processes, such as drying evaporation at the surface of the wet body, energy transfer in a wet cooling tower, flow in a desert cooler, polymer production and food processing. In view of these applications, therefore, it is of interest to combine heat and mass transfer with chemical reaction, MHD, Dufour and Soret effects because of their applications in many processes occurring both in nature and industries involving fluid flow.
However, to the best of our knowledge, no work has been done on the problem of free convection heat and mass transfer in a viscous fluid flowing over a wavy wall or confined between two walls one or both of which are wavy, involving chemical reaction in the presence of both Soret and Dufour. Thus, there is a definite need for the investigation of such a problem. This fact along with the potential application of this problem represent the significance of the present work. Hence, the goal of the present work is, therefore, to investigate the combined effects of Soret and Dufour on heat and mass transfer in chemically reacting MHD flow through a rough (wavy) channel. The non-dimensional coupled boundary value problem governing the fluid flow was perturbed and the resulting zeroth-and first-order boundary value problem were solved using the Adomian decomposition method with MAPLE 14 software.

Formulation of the research problem
The channel considered is made up of a finitely long irregular (wavy) wall at one end and a flat wall at the other. The X -axis is taken to be vertically upward and parallel to the flat wall in the direction of bouyancy, while Y -axis is perpendicular to the X -axis in such a way that the position of irregular wall is represented by Y = * cos kX , | * | < 1, and that of the flat wall is represented by Y = d (Figure 1). substantial magnitudes such that they cannot be neglected. (VII) The electric field is assumed to be zero, the induced magnetic field is negligible compared to the applied magnetic field, since the magnetic Reynolds number is very small for most fluid used in industrial application. (VIII) The reaction is of first-order, the rate of reaction is directional proportional to concentration differences.
Based on these assumptions with the usual Boussinesq approximation, the governing equations of steady twodimensional heat and mass transfer are made up of the following dimensional continuity, momentum, energy and species equations. Continuity equation Momentum equations (3) Energy equation and Concentration equation The corresponding boundary conditions of the problem are taken as Furthermore, it is remarked at this juncture, that for free convection flows, the associated velocities are considerably small while the presence of viscous energy dissipation term usually calls for high-speed flows. Hence, the above assumption that the influence of the viscous dissipation term in the energy equation may be considered negligible is practically reasonable, on the other hand, the inclusion of Joule heating's (magnetic dissipation) effect is of importance for fluid whose medium possesses very high electrical conductivity. However, fluid whose medium is of low electrical conductivity are considered in this problem. Hence, the effect of the magnetic dissipation may be safely regarded negligible. Similar argument hold for the above assumptions.
The following non-dimensional variables were introduced into equations (1)-(7) to obtain the following non-dimensional equations.
Continuity equation Momentum equations Energy equation (12) and Concentration equation The corresponding boundary conditions are In the static fluid (subscripts), we have so that equation (10) becomes All the physical variables are as defined in the nomenclature.

Method of solution
The irregularity of the wall is assumed small, hence, it is appropriate to seek a perturbation solution for small . The limit = 0 is, of course, the limit of a smooth flat wall, for which the solution is well known. Thus we take the flow field variables, velocity, pressure, temperature and concentration as Putting equation (19) into equations (9)-(13), we obtained zeroth-and first-order set of equations where the perturbed quantities u 1 , v 1 , θ 1 and c 1 are small compared with mean or the zeroth-order quantities u 0 , θ 0 and c 0 . For zeroth-order (by equating the coefficients of 0 ), we have with the following boundary conditions where c p = (∂(p 0 − p s )/∂x), and is taken to be equal to zero, following [1,2]. And for first-order (by equating coefficients of 1 ), we have with the boundary conditions where prime denotes differentiation with respect to Y. To solve equations (24)-(29), we introduce stream function ψ 1 (x, y) defined as So that (24) is satisfied and eliminating the dimensionless pressure p 1 , we obtain while the corresponding boundary conditions become Due to the nature of the wall, we assume wave-like solutions of the form perturbation series expansion for small wavelength in which terms of exponential order arise from which we deduce that where i is the complex unit (see [2]).
Substituting (35) into (31)-(34), we have with the boundary conditions (39) By considering series expansion for small λ (or k < < 1) of the form putting (40) into equations (36)-(39) on restriction to the real parts of the solutions for the perturbed quantities and retaining up to order λ 2 , we obtained the following sets of differential equations.
with the corresponding boundary conditions  (44) and (45) are solved, using the Adomian decomposition method programme written by [19] with MAPLE 14 software and obtained the expression for (u 0 , θ 0 and c 0 ) and (u 1 , v 1 , θ 1 and c 1 ) as the zeroth-order and first-order solutions, respectively, where λx is taken to be (π/2) following [2]. The solutions are not presented here for the sake of brevity.

Discussion of the zeroth-order and first-order results
The primary objective of this paper is to investigate the effects of Soret S r , Dufour D u and chemical reaction γ on heat and mass transfer by free convection flow through a wavy/irregular channel under the influence of an externally applied magnetic field and heat source/sink. This objective is accomplished by evaluating numerically, the expressions for the zeroth-order and the first-order velocity, temperature and concentration. The solutions are presented through graphs.
In the numerical discussion that follows, for physical reality, we take Prandtl number to be 0.71 which corresponds to the air at 20 0 C. Schmidt number is chosen to represent the presence of diffusing chemical species of most common interest in air, e.g. carbon dioxide CO 2 at 25 0 C at one atmosphere pressure (S c = 1.0). The values of (G1, G2) take positive value, which corresponds to cooling the wall or heating the fluid by free convection currents. The parameter α and (m1, m2) are varied to stimulate physically realistic situations. α is the heat sink (α < 0 = −5)/source (α > 0 = 5). In the absence of heat source, we have α = 0. Physically, m1,m2 = −1 means that the temperatures and concentrations of the two walls are equal to that of the static fluid while m1,m2 = 2 means equal wall temperatures and concentrations. In the absence of chemical reaction, we have γ = 0 while γ < 0 corresponds to exothermic chemical reaction and γ > 0 corresponds to endothermic chemical reaction. The geometric parameters representing the waviness/roughness of the wall (the amplitude , the wavelength λ and λx). The amplitude > 0 = 0.25, characteristic of dilated channel and λx = π/2 have been fixed while λ = 0.001, 0.002 is varied. The values of the other parameters are chosen arbitrarily.

Velocity profiles
In order to get a clear insight of the physical problem, the velocity, temperature and concentration profiles have been discussed by assigning numerical values to various parameters encountered in the problem. The results are presented graphically and conclusions are drawn for the flow field and other physical quantities of interest that have significant effects. To test the validity of our results, a comparison of the present results with those of the previous works [1,2] are performed by setting the introduced new parameters (i.e D u , S r , H, G2 and γ ) to zero and excellent agreements were obtained (see Figures 2 and 3).
The zeroth-order velocity (u 0 ) profiles when m1 = −1 are plotted against y in Figure 2. It is observed that in the presence of heat source (α > 0) with an increase in the free convection parameter G1, the velocity u 0 increases across the entire channel width (curves III and VI ). When there is heat sink (α < 0) and the free convection parameter G1 increases, the velocity (u 0 ) decreases across the entire channel width (curves I and IV). In the absence of heat generation or absorption (α = 0), the fluid velocity increases in the first half of the channel width (y = 0.5) and then decreases with an increase in the free convection parameter G1 (curves II and V ). However, when the heat source parameter α increases with constant free convection parameter G1, the velocity u 0 increases considerably (curves I, II, III) and (curves IV, V, VI ). Figure 3 depicts the zeroth-order velocity u 0 profile when m1 = 2. It is noticed that the fluid velocity u 0 increases generally as the free convection parameter G1 increases for all the value of heat generation parameter α (curves I and IV, II and V, III and VI). On fixing the    free convection parameter G1 and increasing the heat source parameter α, it is found that the velocity (u 0 ) also increases across the channel width (curves I, II and III ) and (curves IV, V and VI ). Figure 4 shows the effect of MHD (H) on the fluid flow when m1 = m2 = −1. It is clearly seen that when there is an increase in H with constant free convection parameters G1and G2, the velocity u 0 decreases across the entire channel width (curves I, II, III) and (curves IV, V, VI). When the free convection parameters G1 and G2 increase with a fixed magnetic parameter H = 0.1, the velocity increases to a certain point y = 0.5 on the channel width and then decreases (curves I and IV). For H = 1 and H = 3, the velocity increases up to y = 0.4 (curves II and V) and y = 0.28 (curves III and VI), respectively, and then decreases. Figure 5 represents the effects of magnetic field (H) when m1 = m2 = 2. It is observed that an increase in H with constant free convection parameters G1 and G2 leads to a decrease in velocity u 0 (curves I, II, III) and  (curves IV, V, VI). We also noticed that for an increase in free convection parameters G1 and G2 with a fixed magnetic parameter H, the velocity u 0 increases (curves I and IV, II and V, III and VI).
The effects of chemical reaction (γ ) is presented in Figure 6 for m1 = m2 = −1. It is clearly seen that an increase in γ with a fixed free convection parameters G1 and G2 leads to a decrease in velocity (u 0 ) profile (curves I, II, III) and (curves IV, V, VI). An increase in G1 and G2 with a fixed chemical reaction parameter, leads to an increase in the velocity (u 0 ) profile to a particular point on the width of the channel and then decreases (curves I and IV, II and V, III and VI).
For m1 = m2 = 2. Figure 7 described the effects of chemical reaction on the velocity(u 0 ) profiles. It is observed that when the free convection parameters G1 and G2 are fixed, the velocity u 0 decreases with an increase in chemical reaction parameter γ (curves I, II, III) and (curves IV, V, VI). Also, when free convection parameter increases the velocity increases with constant chemical reaction parameter (curves I and IV, II and V, III and VI). Figure 8, when m1 = m2 = −1, shows that an increase in Dufour parameter D u with constant free convection parameters leads to a decrease in velocity profile (curves I, II, III) and (curves IV, V, VI). It is also noticed that an increase in free convection parameters with constant Dufour parameter increases the velocity (u 0 ) profile to a particular point on the channel width and then decreases (curves I and IV, II and V, III and VI).
In Figure 9, when m1 = m2 = 2, it is observed that an increase in Dufour parameter with constant free convection parameters decreases the velocity (u 0 ) profiles (curves I, II, III) and (curves IV, V, VI). Reverse is the case when Dufour parameter is kept constant with an increase in free convection parameters.
It is clearly seen in Figure 10 that the velocity (u 0 ) profiles increase across the entire channel width with an increase in Soret parameter when the free convection parameter is kept constant for m1 = m2 = −1 (curves I, II, III) and (curves IV, V, VI). Reverse is the case when free convection parameters increases with a constant Soret parameter.
For m1 = m2 = 2, Figure 11 shows that when the free convection parameters are fixed, the velocity (u 0 ) profiles increase with an increase in Soret parameter (curves I, II, II) and (curves IV, V, VI). The same behaviour is observed for velocity when there is an increase in free   convection parameters with constant Soret parameter (curves I and IV, II and V, III and VI).

Temperature profiles
As it was done in the discussion on the zeroth-order velocity profiles, we begin our zeroth-order temperature profiles by carrying out a comparison of the present and previous results. Figure 12 depicts the results of the variation of zeroth-order temperature with y when the new parameters (i.e S r , D u , H and γ ) are zero. It is seen clearly that when m1 = −1, the temperature (θ 0 ) profiles increase with an increase in heat parameter α. It is also noticed that in the absence of heat sink/source (α = 0), the temperature (θ 0 ) profile is a linearly decreasing function of y while in the presence of heat sink or source it is parabolic in nature. When m1 = 2, the temperature (θ 0 ) profile behaved in the opposite way for all values of heat parameter α. Figure 13 shows the effects of Soret on the temperature θ 0 profiles. It is clearly seen that an increase in Soret number with constant wall temperature ratios   (m1 = m2 = −1), the temperature θ 0 profiles decrease across the entire channel width (curves I, II, III) and when m1 = m2 = 2, the temperature (θ 0 ) profiles remain the same in behaviour as we have when m1 = m2 = −1 (curves IV, V, VI). Figure 14 deals with the effects of chemical reaction on temperature (θ 0 ). It is observed that when m1 and m2 are fixed, temperature (θ 0 ) profiles increase with an increase in the chemical reaction parameter, γ . In both cases the curves are parabolic in nature (curves I, II, III) and (curves IV, V, VI). Figure 15 depicts the effects of Dufour, on the temperature θ 0 . We noticed that the temperature θ 0 profile increases with an increase in Dufour number D u . In both cases the curves are parabolic in nature (curves I, II, III) and (curves IV, V, VI).    Figure 16 shows that on fixing wall concentration ratio m2 = −1, the concentration c 0 profiles decrease with an increase in heat source parameter α (curves I, II, III). The concentration profiles behaved the same way as when m2 = −1 for m2 = 2 with an increase in heat source parameter α (curves IV, V, VI). The curves I, II, III are parabolic functions of y while curves IV, V, VI are parabolic increasing functions of y. Figure 17 reveals the effects of Soret number on the concentration (c 0 ) profiles. We noticed that when concentration ratio m2 is constant, there is an increase in  concentration (c 0 ) profiles with an increase in Soret number (curves I, II, III) and (curves IV, V, VI). The curves are parabolic in nature. Figure 18 shows the effects of chemical reaction (γ ) on the concentration (c 0 ) profiles. It is observed that for a fixed concentration ratio m2 and an increase in chemical reaction parameter (γ ) leads to a decrease in concentration (c 0 ) profiles, which are parabolic in nature (curves I, II, III) and (curves IV, V, VI).  Figure 19 presents the behaviour of the perturbed velocity quantity u (1) when m1 = −1 and the embedded parameters are set to zero. It is observed that in the presence of heat sinks, the velocity u(1) increases to a particular point of y (y = 0.5) and then decreases with constant free convection parameter G1 and frequency parameters λ (curves I and II, VII and VIII). On the other hand, when heat source is considered, there exist an increase in the velocity u(1) up to y = 0.3, followed by a decrease up to y = 0.7 and finally an increase for y > 0.7 (curves II and III, VIII and IX). It is also noticed that with an increase in G1 or λ, the velocity increases close to the walls of the channel and decreases in between the walls (curves I and VII, II and VIII, III and IX) or (curves I and IV, II and V, III and VI).

Velocity profiles
The effect of MHD parameter H on the fluid flow when m1 = m2 = −1 is shown in Figure 20. It is clearly seen that when there is an increase in H for constant free convection parameters G1,G2 and frequency parameters λ, the velocity increases to a point (y = 0.25 ), decreases to another point (y = 0.7) and then increases (curves I, II, III and VII, VIII, IX). When there is an increase in the frequency λ or free convection G1,G2 parameters with constant value of H, the velocity increases to certain value of y(y = 0.25) and decreases to y = 0.70 and then increases again (curves I and IV, III and VI ) or (curves I and VII, III and IX). Figure 21 shows the effect of Dufour parameter D u on the fluid flow when m1 = m2 = −1. It is observed that with an increase in the Dufour parameter D u when frequency λ and free convection G1,G2 parameters are constant, there is an increase in velocity up to (y = 0.25 ), a decrease up to (y = 0.7) and then an increase. (curves I, II and III) and (curves VII, VIII and IX). It is also noticed that with constant Dufour parameter D u and an increase in frequency λ or free convection G1,G2 parameters, the   velocity increases to a particular point y = 0.25 on the width of the channel and then decreases to another point y = 0.7 after which it increases (curves I and IV, II and V, III and VI) or (curves I and VII, II and VIII, III and IX).
The effect of chemical reaction parameter γ on the fluid flow when m1 = m2 = −1 is depicted in Figure 22. It is clearly seen that with an increase in chemical reaction parameter when the frequency λ and free convection G1,G2 parameters are kept constant, the velocity increases to a particular point y = 0.25 and leads to a decrease up to y = 0.70 before increasing again along the width of the channel. (curves I, II, III) and (curves VII, VIII IX). It is also observed that with constant chemical reaction parameter γ and an increase in the frequency λ or free convection G1,G2 parameters, there is an increase in velocity up to y = 0.25, a decrease   between y = 0.25 and 0.70 while it then increases along the width of the channel (curves I and IV, II and V, III and VI) or (curves I and VII, II and VIII, III and IX). Figure 23 describe the effect of Soret parameter S r on velocity u(1) when m1 = m2 = −1. It is noticed that with an increase in the Soret parameter when the frequency λ and the free convection G1,G2 parameters are constants, the velocity remains the same up to y = 0.25, increases up to y = 0.70 and then decreases. (curves I, II, III) and (curves VII, VIII, IX). When the frequency or free convection parameters increases with constant Soret parameter S r , it is observed that the velocity increases to a certain point y = 0.25, decreases between y = 0.25 and y = 0.70, then increases (curves I and IV, II and V, III and VI) or (curves I and VII, II and VIII, III and IX).
The behaviour of the fluid velocity v (1) perpendicular to the channel is discussed in Figures 24-29. From Figure 24 it is noticed that when there is an increase in the heat source parameter α, the velocity v (1) profile decreases with constant free convection G1 and frequency λ parameters (curves I, II, III) and (curves VII, VIII, IX). It is also observed that with varying free convection G1 or frequency λ parameter, there is an increase in velocity v (1) profile when the heat source parameter   α < 0 (curves I, IV) and (curves I, VII) and a decrease when α ≥ 0 (curves II and V, III and VI) and (curves II and VIII, III and IX). Figure 25 shows the effect of the embedded parameters on the fluid velocity v (1). It is clearly seen that with an increase in heat source parameter α, there is a decrease in velocity v (1) when the free convection G1,G2 and frequency λ are kept constant (curves I, II, III) and (curves VII, VIII, IX). We also realized that the velocity v (1) decreases with constant heat source parameter α, when there is an increase in free convection G1,G2 or frequency λ parameters (curves I and IV, II and V, III and VI) and (curves I and VII, II and VIII, III and IX).
The effect of MHD parameter H on the fluid velocity v (1) is as shown in Figure 26. It is observed that there is a decrease in velocity v (1) when the MHD parameter increases from H = 0.1 to H = 1.0 with constant free convection G1,G2 and frequency λ parameters (curves I and II) and (curves VII and VIII). There is also a decrease in velocity v (1) profile when there is an increase in MHD parameter from H = 1.0 to H = 3.0 (curves II and III) and (curves VIII and IX). It is, moreover, clearly seen from the  figure that when there is an increase in the free convection parameters G1,G2 or frequency λ parameters, there is a decrease in velocity v (1) profiles with constant MHD parameter H (curves I and IV, II and V, III and VI) and (curves I and VII, II and VIII, III and IX). Figure 27 reveals the effect of Dufour parameter D u on the fluid velocity v (1) profiles. From curves (I, II III) and curves (VII, VIII, IX) of the figure, it is noticed that with increase in Dufour parameter, there is an increase in velocity v (1) profiles when the free convection parameters G1,G2 and frequency λ parameters are constant. But with an increase in frequency parameter λ or free convection parameters G1,G2, it is clearly seen that the velocity v (1) profiles decrease for a fixed value of Dufour parameter (curves I and IV, II and V, III and VI) or (curves I and VII, II and VIII, III and IX). Figure 28 illustrates the effect of chemical reaction parameter γ on the fluid velocity v (1) profiles. It is observed from the figure that the effects of the chemical reaction parameter on the fluid velocity v (1) profile are the same as those of Dufour parameter D u in Figure 27. Figure 29 presents the effects of Soret parameter S r on the fluid velocity v (1)profile. We realized from the figure that with an increase in the Soret parameter, the velocity v (1) profile decreases across the channel width when the free convection parameters G1,G2 and frequency λ parameters are constant (curves I, II, III) and (curves VII, VIII, IX). An increase in each of G1 and G2 and λ leads to a decrease to a particular point of y and then an increase when there is an increase in Soret parameter from S r = 0.1 toS r = 0.4 (curves I and IV, II and V) and (curves I and VII, II and VIII). A decrease in velocity v(1) across the channel width is experienced when there is an increase in Soret parameter from h = 0.4 to S r = 2.0 (curves III and VI) and (curves III and IX).

Temperature profiles
Figures 30-33 depict the behaviour of the fluid temperature θ(1) profiles. Figure 30 describes the effect of MHD parameter H on the fluid temperature θ(1) profile. It is noticed that with an increase in H, there is a decrease in the fluid temperature θ(1) profile when free convection parameters G1,G2 and frequency parameters λ are kept constant, (curves I, II, III) and (curves VII, VIII, IX). We also observed that when there is an increase in free convection parameters G1,G2 with parameters λ, α, H remaining constant or an increase in frequency parameter λ with parameters α, H, G1, G2 being constant, there is a decrease in fluid temperature θ (1) profiles (curves I and VII, II and VIII, III and IX) or (curves I and IV, II and V, III and VI). Figure 31 and 32 show the effects of Dufour parameter D u and chemical reaction (γ ) respectively on the fluid temperature θ(1) profiles. It is observed that their   effects on fluid temperature remain the same as the effect of MHD parameter H in Figure 30. Figure 33 reveals the effect of Soret parameter S r on the fluid temperature θ(1) profiles. It is clearly seen from the figure that with an increase in S r , there is an increase in temperature θ(1) profile of the fluid, when the free convection parameters G1,G2 and frequency λ are kept constant (curves I, II, III) and (curves VII, VIII, IX). However, a decrease in the fluid temperature θ (1) is observed, when there is an increase in the free convection parameters G1,G2 or frequency λ with constant Soret parameter S r (curves I and IV, II and V, III and VI) and (curves I and VII, II and VIII, III and IX).

Concentration profiles
The behaviour of the fluid concentration c (1)profiles is depicted in Figures 34-37. It is observed from Figure 34 that an increase in heat source parameter α, leads to a decrease in fluid concentration c (1) profiles when free convection parameters G1,G2 and frequency parameter λ are constant (curves I, II, III) and (curves VII, VIII, IX). On keeping heat source parameter α constant and increasing the free convection parameters G1,G2 or frequency parameter λ, a decrease in fluid concentration c (1) profiles is noticed (curves I and VII, II and VIII, III and IX) or (curves I and IV, II and V, III and VI).      Figure 35 illustrates the effect of MHD parameter H on the fluid concentration c (1)profiles. It is clearly seen that with an increase in MHD parameter H, there is an increase in fluid concentration c (1) profiles when the free convection G1,G2 and frequency λ are kept constant (curves I, II, III) and (curves VII, VIII, IX). The reverse is the case when there is an increase in free convection parameters G1,G2 or frequency parameter λ with constant MHD parameter H (curves I and IV, II and V, III and VI) and (curves I and VII, II and VIII, III and IX). Figure 36 deals with the effect of chemical reaction parameter γ on the fluid concentration c (1) profiles. It is observed that the behaviour remains the same as that of the effect of MHD in Figure 35 The effect of Soret parameter S r on the fluid concentration c (1) profiles is shown in Figure 37. It is noticed that the fluid concentration c (1) profile decreases across the channel width, with an increase in Soret parameter when free convection parameters G1,G2 and frequency λ are constant (curves I, II, III) and (curves VII, VIII, IX). It is also observed that there is a decrease in fluid concentration c (1) profile up to a particular point on the width of the channel and then an increase, with an increase in frequency parameter λ or free convection parameters G1,G2 when S r = 0.1 and S r = 0.4 (curves I and IV, II and V) or (curves I and VII, II and VIII). While a decrease in fluid concentration c(1) profile is observed when S r = 2.0 for an increase in frequency parameter λ or free convection parameters G1,G2 (curves III and VI) or (curves III and IX).

Conclusions
A numerical study has been conducted on free convective heat and mass transfer of an incompressible electrically conducting fluid in a finitely long vertical wavy channel, considering Soret, Dufour and chemical reaction effects in the presence of constant heat source or sink. Employing the perturbation technique, the solutions of the dimensionless governing equations are assumed to be of a mean part and disturbance (contribution from the waviness of the wall) part and are evaluated numerically using the Adomian decomposition method with MAPLE 14 software.
Numerical results of the fluid flow are presented for different physical parameters and are shown by means of graphs. From the previous results and discussion, the numerical observations are as follows: (i) When H = 0, D u = 0, S r = 0 and γ = 0, the numerical values are the same with that of Vajravelu and Sastri [1] and Fasogbon [2] for u 0 , θ 0 , c 0 , u 1 , v 1 , θ 1 and c 1 for different values of α, G1, G2, m1, m2 and λ which confirm the validity of our numerical simulation. (ii) An increase in the magnetic field parameter H, chemical reaction parameter γ and Dufour number D u considerably reduced the mean velocity, but get enhanced due to increase in Soret number S r . (iii) A rise in the Soret number S r depreciates the mean temperature profiles of the flow while an increase in Dufour number D u and chemical reaction parameter γ enhanced it. (iv) Heat source/sink parameter α and chemical reaction parameter γ have tendency to decrease the mean concentration profile and Soret number S r accelerates it. (v) It is found, in general, that G1,G2,m1,m2 and α strongly enhanced the mean velocity, temperature and concentration.