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The study of chemical effect, heat source/sink, and Joule effect on heat transfer and mass transfer (HAMT) in Magneto hydrodynamics (MHD) mixed convection flow through an infinite vertical plate along with viscous dissipation and ohmic heating is of greater importance nowadays and has been examined. The aim of the current work is to include Dufour effect and to study its result along with various parameters. The emphasis is on the flow of a viscous, electrically conducting and incompressible fluid which is mixed convective liquid. The xaxis is chosen along the plate in vertical direction and yaxis is perpendicular to xaxis. A crosswise magnetic field is enforced in the direction of yaxis in a constant manner. For flow on the cooled plate for which “Gr” is greater than 0, and for flow on the heated plate for which “Gr” is lesser than 0, the fluid velocity, fluid temperature and fluid concentration profiles are compared between air (Pr = 0.71) and water (Pr = 7). In the current study, the concurrent occurrence of radiation, heat absorption and dufour effect in the magnetic field has been considered. The numerical solution obtained by software system Matlab and the analytic solution obtained by Perturbation method are studied.
MHD, heat and mass transfer, dufour effect, convective fluid
The simultaneous occurrence of HAMT in a flowing fluid, affecting each other creates a cross diffusion effect. The transfer of mass acquired by temperature gradient is termed as Soret effect. The transfer of heat caused by concentration effect is called as Dufour effect. Dufour and Soret effects are important phenomena in hydrology, geosciences and petrology. Diffusion is the net movement of an object from a higher concentration area to a lower concentration area. Distribution is regulated by a concentration curve. Numerous disciplines, including physics (more precisely, diffusion), chemistry, genetics, economics, and finance, regularly employ the diffusion idea. Chemical reactors are the primary focus of chemical reaction engineering, a subfield of industrial chemistry or chemical engineering. The expression frequently only refers to catalytic reaction systems when a homogeneous or heterogeneous catalyst is present in the reactor. Examples include photo catalytic surfaces, reactive separations, containers, retorts, certain fuel cells, and so forth. Additionally, it is believed that the subject of solvent effects on kinetic parameters is vital. Reactors are built using principles from several different scientific disciplines, including thermodynamics, chemical kinetics, fluid mechanics, heat and mass transport, and economics. Chemical reaction engineering seeks to synthesize each of these components in order to successfully build a chemical reactor. According to Srinivas Reddy [1], thermal diffusion and diffusion thermo effects were taken into account when analyzing the effect of chemical reactions on MHD free convection HAMT from vertical surfaces in porous media. Effects of diffusionthermo and chemical reactions on an unstable MHD free convection flow in a micropolar fluid were examined by Sheri and Shamshuddin [2]. Krishna Murthy and Kumar [3] studied the MHD double diffusive free convection operation in addition to a normal “wavy surface embedded in a doubly stratified, fluidsaturated medium that is porous in nature after learning the results of Dufour and Soret effect” [3]. Umamaheshwar et al. [4] carefully observed the impact of chemicals on MHD fluid flow that is doubly diffusive past a rotating porous plate. Agarwalla and Ahmed [5] looked into the MHD transfer of mass flow via an inclined plate that is embedded in a penetrable medium and has variable velocity and temperature. Lorenzini et al. [6] examined MHD HAMT steady flow of a convective fluid past a permeable plate in the existence of various parameters. Kafoussias and Williams [7] investigated the impact of diffusionthermo and thermal diffusion on mixed freeforced convective and mass transfer boundary layer flow with temperaturedependent viscosity in light of the significance of the diffusionthermo effect. Raju et al. [8] have conducted a thorough investigation on MHD fluid with viscoelastic flow across a limitless perpendicular plate while taking into account chemical reaction and radiation reaction. Seth et al. [9] reveals the theoretical analysis of convective fluid flow under thermal diffusion, heat increase and MHD transfer in a conducting field. Krishna Murthy et al. [10] used soret and dufour effects to analyse the HAMT in two dimensional MHD free convective boundary layer flow along a straight semi infinite flat surface submerged in a thermally and mass stretched Darcy porous media. Ibrahim [11] investigated the influence of thermal radiation, heat generation and chemical reaction on unstable free convection on the unsteady free convection flow of viscous incompressible fluid through a porous medium with high porosity that is contained by a perpendicular endless moving plate. Postelnicu [12] studied the dufour and soret effects on the properties of natural convection around a vertical plane contained in a saturated porous media exposed to a magnetic field.
In the presence of a magnetic field, there are numerous instruments that measure thermal energy diffusion through mercury, electrolyte liquids, water, and air as well as absorption, thermal diffusivity and chemical effects. The effects of mass and heat transfer of a natural convection fluid flow through a plate that is porous in nature are therefore suggested to be studied in this work in the appearance of various parameters along with Dufour effect.
In this present problem, the fluid under consideration is viscous, incompressible, radiating in nature and electrically conducting that progress through a permeable medium in a space especially semiinfinite which is bordered by an erect and infinite surface. The xaxis is in a rising direction and normal to it is the yaxis. A uniform magnetic field, radiation and Joule heating are imposed transversely along yaxis. The properties of the fluid are supposed to be fixed keeping its density factored out in the term that involves body force. Species of chemically reactive quality are thought of to be secreted from the surface toward a hydrodynamic field where they diffuse into the liquid and undergo a chemically homogenous reaction. Throughout in the stream, the reaction is assumed to take place.
Figure 1. Configuration of the problem
The flow past a porous medium is governed by the consequent equations:
Eq. (1) implies Equation of Continuity: $\nabla \cdot v=0$, (since fluid is incompressible).
Eq. (2) is Equation of Motion.
Eq. (3) implies Equation of Energy.
Eq. (4) shows Equation of Mass transfer.
$\frac{\partial v^{\prime}}{\partial y^{\prime}}=0$ (1)
$\begin{aligned} & v^{\prime} \frac{\partial u^{\prime}}{\partial y^{\prime}}=v \frac{\partial^2 u^{\prime}}{\partial y^{\prime 2}}+g \beta\left(T^{\prime}T_{\infty}^{\prime}\right)+g \beta *\left(C^{\prime}C_{\infty}{ }^{\prime}\right) \frac{\sigma B_0{ }^2}{\rho} u^{\prime}\frac{v u^{\prime}}{K_p}\frac{\partial p}{\partial x}\left(\frac{1}{\rho}\right)\end{aligned}$ (2)
$\begin{aligned} & v^{\prime} \frac{\partial T^{\prime}}{\partial y^{\prime}}=\frac{k}{\rho C_p} \frac{\partial^2 T^{\prime}}{\partial y^{\prime 2}}+\frac{v}{C_p}\left(\frac{\partial u^{\prime}}{\partial y^{\prime}}\right)^2\frac{1}{\rho C_p} \frac{\partial q_r^{\prime}}{\partial y^{\prime}} +\frac{\sigma B_0^2}{\rho C_p} u^{\prime}\frac{Q_1}{\rho C_p}\left(T^{\prime}T_{\infty}\right)+\frac{D_{K_t}}{\rho C_p C_s} \frac{\partial^2 C^{\prime}}{\partial y^{\prime 2}}\end{aligned}$ (3)
$v^{\prime} \frac{\partial C^{\prime}}{\partial y^{\prime}}=D \frac{\partial^2 C^{\prime}}{\partial y^{\prime 2}}K\left(C^{\prime}C_{\infty}^{\prime}\right)$ (4)
The boundary conditions (B.C) are:
$\begin{aligned} & u^{\prime}=0, T^{\prime}=T_w, C^{\prime}=C_w a t y^{\prime}=0 \\ & u^{\prime} \rightarrow 0, T^{\prime} \rightarrow T_{\infty}, C^{\prime} \rightarrow C_{\infty} a s y^{\prime} \rightarrow \infty\end{aligned}$ (5)
The Eq. (1) shows,
$v^{\prime}=$ constant $=v_0\left(v_0>0\right)$ (6)
The dimensionless quantities are as mentioned below:
$\begin{aligned} & u=\frac{u^{\prime}}{v_0}, y=\frac{v_0 y^{\prime}}{v}, \theta=\frac{T^{\prime}T_{\infty}{ }^{\prime}}{T_wT_{\infty}^{\prime}}, \varphi=\frac{C^{\prime}C_{\infty}^{\prime}}{C_wC_{\infty}{ }^{\prime}}, \\ & P r=\frac{\mu C_p}{K}, S c=\frac{v}{D}, M=\frac{\sigma B_0^2 v}{\rho v_0{ }^2}, \\ & G r=\frac{v g \beta\left(T_wT_{\infty}{ }^{\prime}\right)}{v_0{ }^3}, G m=\frac{v g \beta *\left(C_wC_{\infty}{ }^{\prime}\right)}{v_0^3}, \\ & E=\frac{v_0{ }^2}{C_p\left(T_wT_{\infty}{ }^{\prime}\right)}, K=\frac{v_0^2 K_p}{v^2}, K_0=\frac{v K c}{v_0^2}, \\ & F=\frac{4 I_1 v^2}{K v_0^2}, Q=\frac{Q_1 v^2}{K v_0^2}, D_f=\frac{D_{K T}\left(C_wC_{\infty}{ }^{\prime}\right)}{C_s v\left(T_wT_{\infty}{ }^{\prime}\right)}\end{aligned}$ (7)
The dimension less form of the governing Eqns. (2)(4) are written in the following form as:
$u^{\prime \prime}+u^{\prime}=G r \thetaG m \varphi+M_1 u$ (8)
$\begin{aligned} \theta^{\prime \prime}+\operatorname{Pr} \theta^{\prime}(F & +Q) \theta=\operatorname{Pr} u^{\prime}2 E\operatorname{Pr} M u^2 E+D f \varphi^{\prime \prime} \operatorname{Pr}\end{aligned}$ (9)
$\varphi^{\prime \prime}+S c \varphi^{\prime}S c K_0 \varphi=0$ (10)
where, $M_1=M+1 / K$.
B.C are:
$\begin{aligned} & u=0, \theta=1, \varphi=1 \text { aty }=0 \\ & u \rightarrow 0, \theta \rightarrow 0, \varphi \rightarrow 0.5 \text { asy } \rightarrow \infty\end{aligned}$ （11）
The Eqns. (8)(10) are solved with B.C (11) by simple perturbation method. The governing Eqns. (8)(10) expanded in terms of powers of “E” whose value is very small.
$\begin{aligned} & u=u_0+E u_1+O\left(E^2\right) \\ & \theta=\theta_0+E \theta_1+O\left(E^2\right) \\ & \varphi=\varphi_0+E \varphi_1+O\left(E^2\right)\end{aligned}$ （12）
On substituting the Eq. (12) into Eqns. (8)(10) and expanding by neglecting the higher powers of “E”, the following equations are obtained by equating the like terms.
Zero order terms:
$u_0^{\prime \prime}+u_0^{\prime}=G r \theta_0G m \varphi_0+M_1 u_0$ (13)
$\theta_0^{\prime \prime}+P r \theta_0^{\prime}(F+Q) \theta_0=D f \varphi_0^{\prime \prime} P r$ (14)
$\varphi_0^{\prime \prime}+S c \varphi_0^{\prime}S c K_0 \varphi_0=0$ (15)
First order terms:
$u_1^{\prime \prime}+u_1^{\prime}=G r \theta_1G m \varphi_1+M_1 u_1$ (16)
$\begin{aligned} & \theta_1^{\prime \prime}+\operatorname{Pr} \theta_1^{\prime}(F+Q) \theta_1=\operatorname{Pr} u_0^{\prime 2}\operatorname{Pr} M u_0^2 \\ & +D f \operatorname{Pr} \varphi_1^{\prime \prime}\end{aligned}$ (17)
$\varphi_1^{\prime \prime}+S c \varphi 1^{\prime}S c K_0 \varphi 1=0$ (18)
B.C are:
$\begin{aligned} & u_0=0, u_1=0, \theta_0=1, \theta_1=0, \varphi_0=1, \varphi_1 =0.5 a t y=0 \\ & u_0 \rightarrow 0, u_1 \rightarrow 0, \theta_0 \rightarrow 1, \theta_1 \rightarrow 0, \varphi_0 \rightarrow 1, \varphi_1 \rightarrow 0 \text { asy } \rightarrow \infty\end{aligned}$ (19)
By using the boundary conditions in Eq. (19), the Eqns. (13)(18) are solved and the equations mentioned below are obtained.
$\varphi_0=e^{m_2 y}$ (20)
$\theta_0=B_2 e^{m_4 y}+C_2 e^{m_2 y}$ (21)
$u_0=B_3 e^{m_6 y}C_3 e^{m_4 y}\left(D_3+E_3\right) e^{m_2 y}$ (22)
$\varphi_1=(0.5) e^{m_8 y}$ (23)
$\begin{aligned} & \theta_1=B_6 e^{m 10 y}H_1 e^{2 m_6 y}H_2 e^{2 m_4 y}H_3 e^{2 m_2 y} +H_4 e^{\left(m_4+m_6\right) y}+H_5 e^{\left(m_6+m_2\right) y}H_6 e^{\left(m_4+m_2\right) y} +H_{11} e^{m_8 y}\end{aligned}$ （24）
$\begin{gathered}u_1=B_7 e^{m_{12} y}S_1 e^{m_{10} y}S_2 e^{2 m_6 y}S_3 e^{2 m_4 y} S_4 e^{2 m_2 y} +S_5 e^{\delta_1 y}+S_6 e^{\delta_2 y}S_7 e^{\delta_3 y}S_8 e^{m_8 y}S_9 e^{m_8 y}\end{gathered}$ （25）
On substituting the Eqns. (20)(25) in the Eq. (12), we get the velocity (u), temperature (θ) and concentration field (φ) as mentioned below:
$\begin{aligned} & u=B_3 e^{m_6 y}C_3 e^{m_4 y}\left(D_3+E_3\right) e^{m_2 y} +E\left(\begin{array}{l}B_7 e^{m_{12} y}S_1 e^{m_{10} y}S_2 e^{2 m_6 y}S_3 e^{2 m_4 y} S_4 e^{2 m_2 y}+S_5 e^{\delta_1 y}+ \\ S_6 e^{\delta_2 y}S_7 e^{\delta_3 y}S_8 e^{m_8 y}S_9 e^{m_8 y}\end{array}\right)\end{aligned}$ （26）
$\begin{aligned} & \theta=B_2 e^{m_4 y}+C_2 e^{m_2 y} \\ & +E\left(\begin{array}{l}B_6 e^{m 10 y}H_1 e^{2 m_6 y}H_2 e^{2 m_4 y}H_3 e^{2 m_2 y}+ \\ H_4 e^{\left(m_4+m_6\right) y}+H_5 e^{\left(m_6+m_2\right) y} \\ H_6 e^{\left(m_4+m_2\right) y}+H_{11} e^{m_8 y}\end{array}\right)\end{aligned}$ （27）
$\varphi=e^{m_2 y}+E\left((0.5) e^{m_8 y}\right)$ （28）
Skin Friction (SF):
The skin friction at the surface is obtained by
$\begin{aligned} & \tau=\left(\frac{\partial u}{\partial y}\right)_{y=0} \\ & \tau=\left[B_3 m_6C_3 m_4\left(D_3+E_3\right) m_2\right] +E\left[\begin{array}{l}B_7 m_{12}S_1 m_{10}S_2\left(2 m_6\right)S_3\left(2 m_4\right) S_4\left(2 m_2\right)+S_5 \delta_1+S_6 \delta_2S_7 \delta_3 \\ S_8 m_8S_9 m_8\end{array}\right] \\ & \end{aligned}$ （29）
Nusselt Number (Nu):
The heat transfer rate by means of Nusselt number, is got by
$\begin{aligned} & N u=\left(\frac{\partial \theta}{\partial y}\right)_{y=0} \\ & N u=\left\{\begin{array}{l}{\left[B_2 m_4+C_2 m_2\right]} \\ +E\left[\begin{array}{l}B_6 m_{10}2 H_1 m_62 H_2 m_4 \\ 2 H_3 m_2+\left(m_6+m_4\right) H_4 \\ +\left(m_6+m_2\right) H_5\left(m_4+m_2\right) H_6 \\ +H_{11} m_8\end{array}\right]\end{array}\right\}\end{aligned}$ （30）
In this work, a detailed study in velocity, temperature and concentration field concerning to the effects of MHD convective flow on HAMT of a viscous, conducting, incompressible fluid through a porous and infinite plate is considered. Along with this, the effects of magnetic field, radiation, chemical effects and absorption are also taken into account. Two versions of ‘Gr’ are presented numerically during heating and cooling of the plate. The Prandtl number (Pr) values are chosen as 0.025, 0.71, 1 and 7, which imply mercury, air, electrolytic friction, water, heat and mass transfer respectively. At 20℃ temperature, 1 atmospheric pressure and at the choice of Eckert number as 0.02 the mercury level is taken into consideration. The values of Schmidt number Sc are taken to be 0.22, 0.30, 0.60 and 0.78 that represents hydrogen, helium, water vapour and ammonia respectively. The values Gr=5, Gm=5, K_{0}=1, K=1, M=2, F=1, Q=1 is chosen for consideration. By the aid of following figures, the obtained results are explained.
On choosing different values of the parameters, the outcome of parameters on velocity, temperature and concentration can be studied. The influence of parameters on SF and Nusselt number are given in Tables 1 and 2.
Figure 2. Mon VP (Gr=5)
Figure 3. M on VP (Gr=5.0)
Figure 4. Sc on VP (Gr=5)
Figure 5. Sc on VP (Gr=5.0)
Figure 6. K on VP (Gr=5)
Figure 7. K on VP (Gr=5.0)
Figure 8. Gm on VP (Gr=5)
Figure 9. Gm on VP (Gr=5.0)
Figure 10. Df on VP (Gr=5)
Figure 11. Df on VP (Gr=5.0)
Figure 12. Pr on TP (Gr=5)
Figure 13. Pr on TP (Gr=5.0)
Figure 14. F on TP (Gr=5)
Figure 15. F on TP (Gr=5.0)
Figure 16. Q on TP (Gr=5)
Figure 17. Q on TP (Gr=5.0)
Figure 18. K_{0} on CP (Gr=5)
Figure 19. K_{0} on CP (Gr=5.0)
Figure 20. Sc on CP (Gr=5)
Figure 2 to Figure 5 imply the effect of M and Sc on the velocity profile (VP) of fluid for Pr = 0.71 and Pr = 7. They show velocity drops with the increase in the values of M and Sc. The application of magnetic field repels fluid transportation; hence velocity diminishes if M boosts up. The effect of K and Gm for Pr = 0.71 and Pr = 7 for fluid VP are depicted in Figure 6 to Figure 9. The figures expose that velocity rises with the boom in the values of K and Gm. Figure 10 exhibits VP falls with rise in Df for cooled plate and reverse effect is produced in Figure 11 for heated plate. The influence of Pr, F and Q on temperature profiles (TP) are depicted in Figure 12 to Figure 17 for cooled and heated plates. The figures show temperature lowers with enlargement in the values of Pr, F and Q. As Pr increases, thermal boundary layer thickness reduces, hence temperature falls for the rise in value of Pr. Figure 18 to Figure 21 exhibit the consequence of K_{0} and Sc on concentration profile (CP) for Gr > 0 and Gr < 0. They exhibit that with the boost in K_{0} and Sc, concentration reduces. The molecular diffusivity decreases as Sc value rises; hence concentration diminishes for enlarged Sc values.
The influences of Gm, M, K, Df, F and Q on SF and heat transfer rate (Nu) are validated in Table 1 and Table 2.
Figure 21. Sc on CP (Gr=5.0)
Table 1. Variations in SF
Gm 
M 
K 
SF τ (Gr = 5) 
SF τ (Gr =  5) 




Pr =0.025 
Pr =1.0 
Pr =0.71 
Pr =7 
Pr =0.025 
Pr =1.0 
Pr =0.71 
Pr =7 



Mercury 
Electrolytic solution 
Air 

Mercury 
Electrolytic solution 
Air 

5 


4.4980 
5.3031 
4.6095 
2.2723 
0.8460 
1.5427 
1.3879 
3.4911 
10 


7.1590 
8.7363 
7.5563 
4.3470 
3.5289 
4.9555 
4.4387 
7.1799 
15 


9.8128 
12.1762 
10.4683 
5.8836 
6.2191 
8.3614 
7.5241 
11.4067 

2 

4.4980 
5.3031 
4.6095 
2.2723 
0.8460 
1.5427 
1.3879 
3.4911 

3 

4.0170 
4.0141 
3.9215 
2.0905 
0.6813 
1.2995 
1.1113 
2.9204 

4 

3.6710 
3.5210 
3.5296 
1.9556 
0.5728 
1.0814 
0.9273 
2.5431 


2 
4.8239 
8.9072 
5.4406 
2.3724 
0.9682 
0.9233 
1.6265 
4.0231 


3 
4.9517 
14.0595 
5.9566 
2.4077 
1.0181 
1.2855 
1.7031 
4.2552 


4 
5.0201 
20.8213 
6.3144 
2.4256 
1.0453 
5.0997 
1.7291 
4.3854 
Table 2. Variations in Nusselt number
Df 
F 
Q 
Nusselt Number Nu (Gr = 5) 
Nusselt Number Nu (Gr =  5) 




Pr =0.025 
Pr =1.0 
Pr =0.71 
Pr = 7 
Pr =0.025 
Pr =1.0 
Pr =0.71 
Pr = 7 



Mercury 
Electrolytic solution 
Air 

Mercury 
Electrolytic solution 
Air 

1 


1.4132 
2.3975 
0.5092 
10.5812 
1.4326 
0.9311 
1.8053 
10.4532 
2 


1.4176 
2.9224 
0.5160 
13.0101 
1.4370 
0.7803 
1.9179 
13.5243 
3 


1.4219 
3.4945 
0.5193 
15.4989 
1.4415 
0.5824 
2.0269 
16.6552 

4 

0.7042 
3.2930 
2.3439 
11.9028 
1.0163 
0.5952 
0.0813 
11.6852 

6 

2.6576 
3.5268 
3.1926 
11.5652 
2.6011 
2.9513 
3.0521 
11.1549 


3 
1.9356 
1.0954 
16.1709 
12.1600 
1.9925 
3.5705 
24.3061 
12.1009 


5 
2.3685 
3.1634 
3.2622 
11.7106 
2.2567 
3.3657 
0.4935 
11.3809 


6 
2.6576 
3.5268 
3.1926 
11.5652 
2.6011 
2.9513 
3.0521 
11.1549 
In this paper, discussions on mixed MHD convection flow of HAMT through an infinite plate that is vertical and porous with effects of Ohmic heating and viscosity dissipation have been explained in the existence of the Dufour effect, absorption and radiation, heat source or sink, and chemical reactions. The governing equations are modified into dimensionless form and then are explained by the method perturbation technique. The outcomes are represented in a graphical manner for various range of values of the parameters. The current analysis can be summarized as mentioned below:
The results of the present study can be applied effectively to the development of many chemical engineering processes, including aeration, material vaporisation and condensation, crystal evolution and sublimation.
B_{0} 
Magnetic field strength 
C_{p} 
Specific heat at constant pressure 
D 
Chemical molecular diffusivity 
D_{Kt} 
Dufour coefficient 
F 
Radiation parameter 
Gm 
Mass Grashof number 
Gr 
Grashof number 
K 
Chemical reaction parameter 
K_{0} 
Chemical reaction rate constant 
M 
Magnetic parameter 
Df 
Dufour number 
Nu 
local Nusselt number along the heat source 
Pr 
Prandtl number 
Q 
Heat sink 
Sc 
Schmidt number 
Greek symbols 

$\beta$ 
thermal expansion coefficient 
$\beta^*$ 
Solute expansion coefficient 
θ 
Dimensionless fluid temperature 
φ 
Dimensionless species concentration of the fluid 
ρ 
Fluid density 
σ 
Electrical conductivity 
µ 
dynamic viscosity 
Subscripts 

T_{w} 
Temperature at the stationary plate 
T_{∞} 
Temperature of the free flowing fluid 
C_{w} 
Concentration at the stationary plate 
C_{∞} 
Concentration of the free flowing fluid 
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