Analysis of Heat and Mass Transfer in MHD Free Convection with Chemical Reaction Effects on a Moving Vertical Porous Plate

Toha François Lihonou^1,2*, Abdelghani Laouer³, Kalil Pierre Mathos¹, Faya Maurice Yombouno¹

¹Department of Physics, Faculty of Sciences and Techniques, University of N’Zérékoré, Republic of Guinea.
²LMFDNMSB Laboratory, IMSP/UAC, Porto-Novo, Benin.
³LPMCN Laboratory, Faculty of ESCS, University of Jijel, Jijel, Algeria.

DOI: 10.4236/ojfd.2025.152006                

Abstract

This study focuses on the numerical investigation of fluid flow, heat transfer, and free MHD convection through a semi-infinite vertical porous plate, considering the effects of chemical reactions. By analyzing the boundary layer, a flow model was developed to represent the governing equations for movement, energy, and concentration, which are time-independent. These equations are expressed as a dimensionless nonlinear system and mathematically transformed into nonlinear ordinary differential equations (ODEs). The ODEs of the model are solved using the BVP4C method within the Matlab R2024B package. Numerical calculations were performed, and the results were analyzed to explore the effects of various parameters such as the magnetic parameter, the permeability of the porous medium, the Eckert number, the Grashof number, the modified Grashof number, the Schmidt number, the heat source parameter, the Prandlt number, the Dufour number, the blowing/suction velocity, the chemical reaction parameter and the Soret number on the velocity, temperature, and concentration profiles, as well as the skin friction coefficient, Nusselt number, and Sherwood number. The study reveals that an increase in magnetic parameters reduces the velocity profiles and increases the temperature and concentration profiles. An increase in the chemical reaction rate decreases the velocity and concentration distributions but increases the temperature profile. Higher injection velocities enhance the profiles, while suction reduces them.

Keywords

MHD Free Convection, Moving Vertical Plate, BVP4C Technical, Injection/Suction

1. Introduction

Heat transfer is a fundamental phenomenon in many scientific fields. In fluids, it refers to the transfer of thermal energy from one location to another. Free convection is a mode of heat transfer in which fluid motion is driven by buoyancy forces rather than by an external source. It is associated with natural convection processes and involves factors such as heat transfer coefficients and thermodynamic laws. Hossain et al. [1] investigated MHD free convection and mass transfer flow through a vertically oscillating porous plate in the presence of Hall and ion slip currents, a heat source, and a rotating system. The results of this investigation are discussed for the different values of the well-known parameters and are shown graphically. The effects of a heat source and stratification phenomena on the magnetohydrodynamic (MHD) Prandtl model were studied by Khan et al. [2]. They demonstrated that an increase in the stratification parameter leads to a reduction in temperature. S. Zeb et al. [3] examined the stagnation point flow of a Prandtl fluid along a stretched sheet in a permeable medium, integrating natural convection, magnetic field effects, heat generation, thermal radiation, and Soret and Dufour phenomena. The main results reveal that porosity significantly increases the wall friction coefficient, while increasing the heat source parameters reduces the Nusselt number. In addition, chemical reaction parameters significantly increase the concentration distribution. The flow and heat transport in an Eyring-Powell fluid under the influence of mixed convection over a graded surface were studied by Bilal and Ashbar [4]. It was found that the velocity and temperature profiles decrease with enhanced thermal stratification, while both increase with the heat generation parameter. The fusion energy phenomena in the Prandtl MHD fluid through an inclined stretched cylinder were solved numerically by Awais et al. [5]. They showed that the temperature increases with an increase in the fusion parameter while it decreases with an increase in the magnetic parameter. The analysis of mass and heat transport in the Prandtl-Eyring fluid in oscillatory flow along a permeable channel was discussed by Khafajy and Kaabi [6]. The results show that the velocity of both types of flow, Poiseuille and Couette, increases as the parameters Reynolds, Darcy, Grashof number, radiation parameter, and static pressure rise. The heat and mass transport of Casson fluid along a cylinder in a permeable medium with Soret-Dufour, stagnation point, and suction/injection effects were studied by Alizadeh et al. [7]. Ibrahim and Hindebu [8] analyzed MHD boundary layer flow of Eyring-Powell nanofluids using the Cattaneo-Christov heat-mass fluxes theories. The flow model equations, induced by a stretching cylinder, were solved numerically using the Keller-Box technique. They reported that the Nusselt number increased with the Prandtl number, curvature parameter, thermal relaxation time, and the Eyring-Powell fluid parameter. Meanwhile, Layek et al. [9] investigated the combined transport of heat and mass transfer for unsteady, incompressible, viscous Eyring-Powell fluid along expanding/shrinking sheets with suction/injection, Dufour and Soret effects. According to their results, the fluid velocity is high for the Eyring-Powell fluid, but Prandtl number and thermal radiation lessen the fluid temperature. Moreover, an analysis of nonlinear stratified convection of Eyring-Powell fluid past a sheet, which is inclined and stretching with Cattaneo-Christov heat-mass flux model is presented by Jabeen et al. [10]. Their analysis revealed that the thermal stratification parameter and Cattaneo-Chiristov time relaxation dampen the distribution of fluid temperature. Salah [11] examined the flow and heat transfer of dissipative and chemically reacting MHD Eyring-Powell fluid past an exponentially stretching sheet under a non-Fourier heat conduction model. The study revealed that both the thermal relaxation time and the Eyring-Powell fluid parameter are inversely related to the temperature profile, while the Eckert number has a positive effect on the temperature profile. Naseem et al. [12] analyzed the convection of MHD Eyring-Powell fluid over an exponentially stretching sheet. Their study incorporated the Cattaneo-Christov heat flux model and concluded that both the temperature field and the thermal boundary layer thickness decrease with an increase in the thermal relaxation time parameter, but increase with the Eckert number. It was also observed that the velocity of the Eyring-Powell fluid is higher than that of a Newtonian (viscous) fluid, whereas the opposite is true for the fluid temperature. Furthermore, the magnetic field exhibited a retarding effect on the velocity field while enhancing the temperature distribution. In the last decade, the study of chemical reactions has been a process that leads to the transformation of one set of chemical substances into another. Conventionally, chemical reactions encompass changes that only involve electron positions in the formation and breaking of chemical bonds between atoms without modification of the nuclei. Nuclear chemistry is a subdiscipline of chemistry that involves the chemical reactions of unstable and radioactive elements where both electronic and nuclear changes can occur. Chemical reactions occur at a characteristic reaction rate at a given temperature and chemical concentration. Heat is always generated during these chemical reactions. Most common fluids, such as water and air, are contaminated with impurities such as CO₂, C₆H₆, and HCl, etc. The chemical reaction parameter exhibits a retarding effect on the concentration distribution as the reaction transitions from a constructive to a destructive state. The heat transfer of the radiated Casson fluid in a stagnation point flow with a magnetic field and a chemical reaction was explored by Anwar et al. [13]. The study highlights the mutually reinforcing impacts of thermal radiations and chemical processes on the heat transfer process. Jalili et al. [14] examined the nonlinear radiative heat transport of Casson fluid flow along a vertical plate with a porous medium, chemical reaction, Joul heating, viscous dissipation, and stratification phenomena. Their analysis showed that increasing the Casson fluid parameter enhances the fluid’s velocity field while reducing its concentration and temperature profiles. Khan et al. [15] studied the sliding flow and heat transport of a tangent hyperbolic fluid along a rotating greasy sheet. They revealed that the temperature profile and Nusselt number exhibit increasing behavior as the radiation parameter increases. The effect of heat generation on boundary layer fluid flow is very significant due to technical applications such as fire and combustion, metal scrap, radioactive materials, reactor safety analysis, spent nuclear fuel, etc. Convective heat transfer is generally classified into two basic types. When no externally induced flow is present and the fluid motion arises solely due to density differences caused by temperature or concentration gradients within a body force field, such as gravity, the process is known as natural convection. On the other hand, if the fluid movement is caused by an external agent, such as the externally imposed flow of a fluid stream over a heated object, the process is called forced convection. In forced convection, the fluid flow is generated by an external source such as a fan, a blower, and wind or the movement of the heated object itself. Such problems are frequently encountered in technological applications, where heat transfer to or from a body often occurs due to an externally imposed flow of fluid at a temperature different from that of the body. In contrast, in natural convection, fluid motion is driven by buoyancy forces that arise from density differences caused by temperature or concentration variations. A heated body that cools in the ambient air generates such a flow in the region around it. Similarly, buoyant flow results from the rejection of heat to the atmosphere and other ambient media, circulations occurring in heated rooms, in the atmosphere and in bodies of water, the rise of buoyant flow causing thermal stratification of the medium, as in temperature inversion and many other heat transfer processes in our natural environment, as well as in many technological applications, are included in the field of natural convection. Flow can also occur due to concentration differences, such as those caused by salinity differences in the sea and composition differences in the chemical treatment unit, and these cause mass transfer by natural convection. The effects of radiation and chemical reaction on unsteady MHD heat and mass transfer of Casson fluid flow past a vertical plate were discussed by Biswas et al. [16]. Finally, they obtained that velocity decreases with an increase in Casson parameter, Permeability of porous medium and Chemical reaction while it increases with increasing values of Radiation parameter and Grashof number. Temperature increases due to Radiation parameter but decreases for Prandtl number. The effects on the magnetic field during compressional flow of a Casson fluid between parallel plates were explored by Ahmed et al. [17]. In their study, it is observed that magnetic field can be used as a control phenomenon in many flows as it normalizes the flow behavior. The influence of the induced magnetic field on the incompressible Prandtl fluid across a stretched plate with homogeneous and heterogeneous reactions was studied by Meenakumari et al. [18]. The authors observed that the large values of the stretching ratio and the induced magnetic parameters are primarily moderate magnetic field, velocity, and temperature. Also, the authors found more velocity and temperatures by boosting the slip parameters. The effect of thermodiffusion, Soret and heat generation effects, radiation and chemical reaction effects on MHD, etc., has been presented by Kataria and Patel [19]-[21]. The effect of the magnetic field through the analysis of the linear temporal stability of the flow of a viscous, incompressible and electrically conductive fluid forming a dynamic laminar boundary layer on an impermeable horizontal flat magnetic plate is presented by Lihonou et al. [22].

The aim of this paper is to study and analyze the effects of various parameters, such as the magnetic parameter, permeability of the porous medium, Eckert number, Prandtl number, Soret number, Schmidt number, heat source parameter, chemical reaction parameter, Grashof number, modified Grashof number, Dufour number, and injection/suction velocity-on particle velocity, temperature and concentration profiles, as well as on the skin friction coefficient, Nusselt number, and Sherwood number for a steady MHD heat and mass transfer flow through a vertical porous plate. To achieve this objective, the paper is organized as follows: Section 2 presents the mathematical formulation of the problem; Section 3 describes the numerical solution method; Section 4 discusses and analyzes the results; and the conclusion is provided in the final section.

2. Mathematical Modeling

2.1. Governing Equations

The continuity, momentum, energy and concentration equations for a viscous, incompressible and electrically conductive fluid are given by:

$\nabla \cdot V=0,$ (1)

$\begin{array}{c}\frac{\partial V}{\partial t}+\left(V\cdot \nabla \right)V=-\frac{1}{\rho }\nabla p+\nu {\nabla }^{2}V+\frac{1}{\rho }J\wedge B-\frac{\nu }{k^{\text{*}}}V\\ +g{\beta }_T\left(T-T_{\infty }\right)+g{\beta }_c\left(C-C_{\infty }\right),\end{array}$ (2)

$\frac{\partial T}{\partial t}+\left(V\cdot \nabla \right)T=\frac{k}{\rho C_p}{\nabla }^{2}T+\frac{Q}{\rho C_p}\left(T-T_{\infty }\right)+\frac{D_m{k}_T}{C_s{C}_p}{\nabla }^{2}C+\frac{\mu }{\rho C_p}{\left(\nabla V\right)}^2,$ (3)

$\frac{\partial C}{\partial t}+\left(V\cdot \nabla \right)C=D_m{\nabla }^{2}C-k_1\left(C-C_{\infty }\right)+\frac{D_m{k}_T}{T_m}{\nabla }^{2}T,$ (4)

$\nabla \wedge B={\mu }_e\left(J+{\epsilon }_{e}E\right),$ (5)

$\nabla \wedge E=-\frac{\partial B}{\partial t},$ (6)

$\nabla \cdot B=0,$ (7)

$\nabla \cdot E=0,$ (8)

$\nabla \cdot J=0,$ (9)

where Equations (1)-(9) are continuity, Newton’s second law, energy, concentration, Ampere’s law, Faraday’s law, Maxwell’s law and Gauss law equations respectively, with

$J=\sigma \left(E+V\wedge B\right),$ (10)

Here $V\left(u,v,w\right)$ is the velocity of fluid, $B$ the magnetic field, $E$ the electric field, $J$ the current density vector, ${\mu }_e$ the magnetic permeability, ${\epsilon }_e$ the absolute permittivity of the fluid, $t$ denotes the time, $\rho$ is the fluid density, $T$ is the fluid temperature, $C$ is the fluid concentration, $\nu$ is the kinematic viscosity, $k$ is the thermal conductivity, $D_m$ is the species concentration diffusivity, $\mu$ is the dynamic viscosity, $g$ is the acceleration due to gravity, $\sigma$ is the Stefan-Boltzmann constant, $k^{\text{*}}$ is the Darcy permeability, $T_m$ is the mean fluid temperature, $k_1$ is the reaction rate constant, ${\beta }_T$ is the coefficient of thermal expansion, ${\beta }_c$ is the coefficient of concentration expansion, $C_p$ is the specific heat at constant pressure, $C_s$ is the concentration susceptibility, and $k_T$ is the thermal diffusion ratio.

We put also

$B=\left(0,B_0,0\right),$ (11)

$E=\left(E_x,E_y,E_z\right),$ (12)

$J=\left(J_x,0,J_z\right),$ (13)

where $B_0$ is a constant. We assumed that no applied polarization voltage exists (i.e., $E=0$ ). Then, Equation (10) and Equation (13) give

$J=\sigma B_0\left(-w,0,u\right)$ (14)

and Equation (9) yields

$\nabla \cdot J=\sigma B_0\left(\frac{\partial u}{\partial z}-\frac{\partial w}{\partial x}\right).$ (15)

2.2. Physical Model

The flow is considered to be two-dimensional and steady, driven by free convection of an incompressible, viscous, and electrically conducting fluid past a semi-infinite vertical porous plate located at $y=0$ . The analysis takes into account thermal diffusion, viscous dissipation, heat absorption, and chemical reaction effects. The coordinate system is defined in such a way that the x-axis runs vertically along the plate, while the y-axis is perpendicular to it. A uniform magnetic field of intensity $B_0$ is applied in the y-direction. At the plate surface, the velocity, temperature, and concentration are held constant at $u=U_0$ , $T=T_w$ , and $C=C_w$ , respectively. Far from the wall, the velocity is zero and the ambient fluid temperature and concentration approach ($T_{\infty }$ ) and ($C_{\infty }$ ), respectively. The physical configuration and coordinate system of the model are illustrated in Figure 1.

Figure 1. Physical model.

2.3. Governing Equations

Under Prandtl’s assumptions and boundary layer approximations, the dimensional equations of continuity, momentum, energy, and concentration for steady fluid flow are given as follows:

$\frac{\partial u}{\partial x}+\frac{\partial \upsilon }{\partial y}=0,$ (16)

$u\frac{\partial u}{\partial x}+\upsilon \frac{\partial u}{\partial y}=\nu \left(\frac{{\partial }^{2}u}{\partial y^2}\right)+g{\beta }_T\left(T-T_{\infty }\right)+g{\beta }_c\left(C-C_{\infty }\right)-\frac{\nu }{k^{\text{*}}}u-\frac{\sigma B_0^2}{\rho }u,$ (17)

$u\frac{\partial T}{\partial x}+\upsilon \frac{\partial T}{\partial y}=\frac{k}{\rho C_p}\left(\frac{{\partial }^{2}T}{\partial y^2}\right)+\frac{Q}{\rho C_p}\left(T-T_{\infty }\right)+\frac{D_m{k}_T}{C_s{C}_p}\left(\frac{{\partial }^{2}C}{\partial y^2}\right)+\frac{\mu }{\rho C_p}{\left(\frac{\partial u}{\partial y}\right)}^2,$ (18)

$u\frac{\partial C}{\partial x}+\upsilon \frac{\partial C}{\partial y}=D_m\left(\frac{{\partial }^{2}C}{\partial y^2}\right)-k_1\left(C-C_{\infty }\right)+\frac{D_m{k}_T}{T_m}\frac{{\partial }^{2}T}{\partial y^2},$ (19)

with the boundary conditions:

$\{\begin{array}{l}u=U_0,\upsilon =v_0,T=T_w,C=C_w\text{at}y=0\\ u=0,\upsilon =0,T\to T_{\infty },C\to C_{\infty }\text{at}y\to \infty .\end{array}$ (20)

The dimensionless governing equations were obtained by applying the following boundary layer approximations and dimensionless variables:

$U=\frac{u}{U_O};V=\upsilon /\left(U_0/\sqrt{R_e}\right);X=\frac{x}{L};Y=\frac{y}{\delta };$

$R_e=\frac{U_{0}L}{\nu };\delta =\sqrt{\nu L/U_0};L=\frac{{\delta }^2{U}_O}{\nu };$

$\frac{\delta }{L}=\frac{1}{\sqrt{R_e}};\frac{U_e}{L}=\frac{\nu }{{\delta }^2};T=T_{\infty }+\tilde{T}\left(T_w-T_{\infty }\right);$

$C=C_{\infty }+\tilde{C}\left(C_w-C_{\infty }\right)$ ;

with $\delta$ the maximum boundary layer thickness, $L$ the plate length, $R_e$ the hydrodynamic Reynolds number, $X$ and $Y$ the dimensionless coordinates.

Thus, we obtain the following dimensionless equations:

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (21)

$\begin{array}{c}U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{{\partial }^{2}U}{\partial Y^2}+\frac{g{\delta }^2{\beta }_T\left(T_w-T_{\infty }\right)}{\nu U_0}\tilde{T}+\frac{g{\delta }^2{\beta }_c\left(C_w-C_{\infty }\right)}{\nu U_0}\tilde{C}\\ -\frac{{\delta }^2}{k^{\text{*}}}U-\frac{\sigma B_O^2{\delta }^2}{\rho \nu }U\end{array}$(22)

$\begin{array}{c}U\frac{\partial \tilde{T}}{\partial X}+V\frac{\partial \tilde{T}}{\partial Y}=\frac{k}{\nu \rho C_p}\frac{{\partial }^2\tilde{T}}{\partial Y^2}+\frac{Q{\delta }^2}{\rho C_p\nu }\tilde{T}+\frac{D_m{k}_T}{C_s{C}_p\nu }\frac{C_w-C_{\infty }}{T_w-T_{\infty }}\frac{{\partial }^2\tilde{C}}{\partial Y^2}\\ +\frac{U_o^2}{C_p\left(T_w-T_{\infty }\right)}{\left(\frac{\partial U}{\partial Y}\right)}^2.\end{array}$(23)

$U\frac{\partial \tilde{C}}{\partial X}+V\frac{\partial \tilde{C}}{\partial Y}=\frac{D_m}{\nu }\frac{{\partial }^2\tilde{C}}{\partial Y^2}-\frac{k_1{\delta }^2}{\nu }\tilde{C}+\frac{D_m{k}_T}{T_m\nu }\frac{T_w-T_{\infty }}{C_w-C_{\infty }}\frac{{\partial }^2\tilde{T}}{\partial Y^2}.$ (24)

Taking the following dimensionless parameters:

$S=\frac{Q{\delta }^2}{\rho C_p\nu };M=\frac{\sigma B_O^2{\delta }^2}{\rho \nu };S_c=\frac{\nu }{D_m};$

$S_r=\frac{D_m{k}_T}{T_m\nu }\frac{T_w-T_{\infty }}{C_w-C_{\infty }};k_p=\frac{{\delta }^2}{k^{\text{*}}};\gamma =\frac{k_1{\delta }^2}{\nu };P_r=\frac{\nu \rho C_p}{k};$

$E_c=\frac{U_o^2}{C_p\left(T_w-T_{\infty }\right)};G_r=\frac{g{\delta }^2{\beta }_T\left(T_w-T_{\infty }\right)}{\nu U_0};G_m=\frac{g{\delta }^2{\beta }_c\left(C_w-C_{\infty }\right)}{\nu U_0};$

$D_u=\frac{D_m{k}_T}{C_s{C}_p\nu }\frac{C_w-C_{\infty }}{T_w-T_{\infty }};$

The dimensionless governing equations therefore become:

$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0$ (25)

$U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=\frac{{\partial }^{2}U}{\partial Y^2}+G_r\tilde{T}+G_m\tilde{C}-k_{p}U-MU$ (26)

$U\frac{\partial \tilde{T}}{\partial X}+V\frac{\partial \tilde{T}}{\partial Y}=\frac{1}{P_r}\frac{{\partial }^2\tilde{T}}{\partial Y^2}+S\tilde{T}+D_u\frac{{\partial }^2\tilde{C}}{\partial Y^2}+E_c{\left(\frac{\partial U}{\partial Y}\right)}^2.$ (27)

$U\frac{\partial \tilde{C}}{\partial X}+V\frac{\partial \tilde{C}}{\partial Y}=\frac{1}{S_c}\frac{{\partial }^2\tilde{C}}{\partial Y^2}-\gamma \tilde{C}+S_r\frac{{\partial }^2\tilde{T}}{\partial Y^2}.$ (28)

with the boundary conditions:

$\{\begin{array}{l}U=1,V=V_0,\tilde{T}=1,\tilde{C}=1\text{at}Y=0\\ U=0,V=0,\tilde{T}\to 0,\tilde{C}\to 0\text{at}Y\to \infty ,\end{array}$(29)

where $U$ , $\tilde{T}$ and $\tilde{C}$ represent the dimensionless velocity, temperature and concentration respectively, $E_c$ is the Eckert number, $G_r$ is the Grashof number, $G_m$ is the modified Grashof number, $S_c$ is the Schmidt number, $S$ is the heat source parameter, $P_r$ is the Prandlt number, $k_p$ is the permeability of the porous medium, $M$ is the magnetic parameter, $D_u$ is the Dufour number, $\gamma$ is the chemical reaction parameter and $S_r$ is the Soret number.

2.4. Ordinary Differential Equations

The previous equations admit for the variables $U$ , $\tilde{T}$ and $\tilde{C}$ , the solutions, $U\left(X,Y\right)=f\left(\theta \right)$ , $\tilde{T}\left(X,Y\right)=\tilde{T}\left(\theta \right)$ and $\tilde{C}\left(X,Y\right)=\tilde{C}\left(\theta \right)$ with $\theta =\frac{Y}{\sqrt{X}}$ .

So we have:

$\{\begin{array}{l}\frac{\partial U}{\partial X}=\frac{\partial f\left(\theta \right)}{\partial X}=\frac{\partial \theta }{\partial X}\frac{\text{d}f\left(\theta \right)}{\text{d}\theta }=-\frac{Y}{2X\sqrt{X}}\frac{\text{d}f\left(\theta \right)}{\text{d}\theta };\\ \frac{\partial \tilde{T}}{\partial X}=\frac{\partial \theta }{\partial X}\frac{\text{d}\tilde{T}\left(\theta \right)}{\text{d}\theta }=-\frac{Y}{2X\sqrt{X}}\frac{\text{d}\tilde{T}\left(\theta \right)}{\text{d}\theta };\\ \frac{\partial \tilde{C}}{\partial X}=\frac{\partial \theta }{\partial X}\frac{\text{d}\tilde{C}\left(\theta \right)}{\text{d}\theta }=-\frac{Y}{2X\sqrt{X}}\frac{\text{d}\tilde{C}\left(\theta \right)}{\text{d}\theta };\end{array}$(30)

$\{\begin{array}{l}\frac{\partial U}{\partial Y}=\frac{\partial f\left(\theta \right)}{\partial Y}=\frac{\partial \theta }{\partial Y}\frac{\text{d}f\left(\theta \right)}{\text{d}\theta }=\frac{1}{\sqrt{X}}\frac{\text{d}f\left(\theta \right)}{\text{d}\theta };\\ \frac{\partial \tilde{T}}{\partial Y}=\frac{\partial \theta }{\partial Y}\frac{\text{d}\tilde{T}\left(\theta \right)}{\text{d}\theta }=\frac{1}{\sqrt{X}}\frac{\text{d}\tilde{T}\left(\theta \right)}{\text{d}\theta };\\ \frac{\partial \tilde{C}}{\partial Y}=\frac{\partial \theta }{\partial Y}\frac{\text{d}\tilde{C}\left(\theta \right)}{\text{d}\theta }=\frac{1}{\sqrt{X}}\frac{\text{d}\tilde{C}\left(\theta \right)}{\text{d}\theta };\end{array}$(31)

$\{\begin{array}{l}\frac{{\partial }^{2}U}{\partial Y^2}=\frac{1}{X}\frac{{\text{d}}^{2}f\left(\theta \right)}{\text{d}{\theta }^2};\\ \frac{{\partial }^2\tilde{T}}{\partial Y^2}=\frac{1}{X}\frac{{\text{d}}^2\tilde{T}\left(\theta \right)}{\text{d}{\theta }^2};\\ \frac{{\partial }^2\tilde{C}}{\partial Y^2}=\frac{1}{X}\frac{{\text{d}}^2\tilde{C}\left(\theta \right)}{\text{d}{\theta }^2}.\end{array}$(32)

Thus, the continuity Equation (25) becomes:

$\frac{\partial V}{\partial Y}=\frac{Y}{2X\sqrt{X}}\frac{\text{d}f\left(\theta \right)}{\text{d}\theta }$

$\iff$ $\text{d}V=\frac{Y}{2X\sqrt{X}}\frac{\text{d}f\left(\theta \right)}{\text{d}\theta }\text{d}Y.$

As $Y=\theta \sqrt{X}$ and $\text{d}Y=\sqrt{X}\text{d}\theta$ ,

then:

$\text{d}V=\frac{X\theta }{2X\sqrt{X}}\frac{\text{d}f\left(\theta \right)}{\text{d}\theta }\text{d}\theta$

$\iff$ $\text{d}V=\frac{\theta }{2\sqrt{X}}\text{d}f\left(\theta \right)$

$\iff$ $V\left(X,\theta \right)=\frac{1}{2\sqrt{X}}{\displaystyle {\int }_0^{\theta }\epsilon \text{d}f\left(\epsilon \right)}$ .

Integration by parts gives us:

$V\left(X,\theta \right)=\frac{1}{2\sqrt{X}}\left[\theta f-{\displaystyle {\int }_0^{\theta }f\left(\epsilon \right)\text{d}\epsilon }\right].$ (33)

If we introduce the function $F\left(\theta \right)={\displaystyle {\int }_0^{\theta }f\left(\epsilon \right)\text{d}\epsilon }$ , Equation (33) becomes:

$V\left(X,\theta \right)=\frac{1}{2\sqrt{X}}\left[\theta F^{\prime }-F\right],$ (34)

with

$U=f=F^{\prime }.$

From the relations or systems of Equations (30)-(32) and (34), Equations (26)-(28) become respectively:

$F^{‴}+\frac{1}{2}{F}^{″}F-\left(M+k_p\right)F^{\prime }+G_r\tilde{T}+G_m\tilde{C}=0,$ (35)

${\tilde{T}}^{″}+\frac{1}{2}{P}_{r}F{\tilde{T}}^{\prime }+E_c{P}_r{F^{″}}^2+P_{r}S\tilde{T}+P_r{D}_u{\tilde{C}}^{″}=0,$ (36)

${\tilde{C}}^{″}+\frac{1}{2}{S}_{c}F{\tilde{C}}^{\prime }-S_c\gamma \tilde{C}+S_c{S}_r{\tilde{T}}^{″}=0,$ (37)

with: $\frac{\text{d}f}{\text{d}\theta }=f^{\prime }=F^{″}$ ; $\frac{{\text{d}}^{2}f}{\text{d}{\theta }^2}=f^{″}=F^{‴}$ ; $\frac{\text{d}\tilde{T}}{\text{d}\theta }={\tilde{T}}^{\prime }$; $\frac{{\text{d}}^2\tilde{T}}{\text{d}{\theta }^2}={\tilde{T}}^{″}$; $\frac{\text{d}\tilde{C}}{\text{d}\theta }={\tilde{C}}^{\prime }$; $\frac{{\text{d}}^2\tilde{C}}{\text{d}{\theta }^2}={\tilde{C}}^{″}$.

The boundary conditions give:

$\{\begin{array}{l}U=1,V=-\alpha F_w,F=F_w,\tilde{T}=1,\tilde{C}=1\text{at}\theta =0\\ U=0,\tilde{T}\to 0,\tilde{C}\to 0\text{at}\theta \to \infty ,\end{array}$(38)

with $\alpha >0$ .

The following non-dimensional quantities represent the skin friction coefficient ($C_f$ ), the Nusselt number ($N_u$ ) and the Sherwood number ($S_h$ ) respectively:

$C_f=-\frac{1}{2\sqrt{2}}{\left(G_r\right)}^{\frac{3}{4}}{\left(\frac{\text{d}f}{\text{d}\theta }\right)}_{{}_{\theta =0}}$

$N_u=\frac{1}{\sqrt{2}}{\left(G_r\right)}^{\frac{3}{4}}{\left(\frac{\text{d}\tilde{T}}{\text{d}\theta }\right)}_{{}_{\theta =0}}$

$S_h=\frac{1}{\sqrt{2}}{\left(G_r\right)}^{\frac{3}{4}}{\left(\frac{\text{d}\tilde{C}}{\text{d}\theta }\right)}_{{}_{\theta =0}}$

3. Numerical Solution Method

For the numerical solution, we used the bvp4c technique, a solver integrated into MATLAB (Shampine et al. [23]). Thus, the highly coupled nonlinear ordinary differential Equations (35)-(37) with boundary conditions (38) are solved by defining:

$A=1-P_r{D}_u{S}_c{S}_r;$ $A_1=P_r/A;$

$A_2=P_{r}S/A;$ $A_3=P_r{E}_c/A;$

$A_4=S_c{D}_u{P}_r/A;$ $A_5=\gamma A_4;$

$A_6=S_c/A;$ $A_7=\gamma A_6;$

$A_8=S_c{S}_r{P}_r/A;$ $A_9=S_c{S}_r{P}_{r}S/A;$

$A_{10}=S_c{S}_r{E}_c{P}_r/A;$

$F=y_1;$ $F^{\prime }=y_2;$ $F^{″}=y_3;$

$\tilde{T}=y_4;$ ${\tilde{T}}^{\prime }=y_5;$

$\tilde{C}=y_6;$ ${\tilde{C}}^{\prime }=y_7;$

$F^{‴}={y^{\prime }}_3=-0.5y_3{y}_1+\left(M+kp\right)y_2-G_r{y}_4-G_m{y}_6;$

${\tilde{T}}^{″}={y^{\prime }}_5=-0.5A_1{y}_1{y}_{5\text{'}}-A_2{y}_4-A_3{y}_3^2+0.5A_4{y}_1{y}_7-A_5{y}_6;$

${\tilde{C}}^{″}={y^{\prime }}_7=-0.5A_6{y}_1{y}_7+A_7{y}_6+0.5A_8{y}_1{y}_5+A_9{y}_4+A_{10}{y}_3^2.$

For $\theta =0$ , $y_1=0$ ; $y_2=1$ ; $y_4=1$ and $y_6=1$ .

For $\theta \to \infty$ , $y_2=0$ ; $y_4=0$ and $y_6=0$ .

4. Results and Discussions

The numerical results of the present study were obtained for various parameter values affecting velocity, temperature, concentration, wall friction coefficient, Nusselt number, and Sherwood number across a moving vertical porous permeable plate, using the bvp4c method in MATLAB.

These results were computed for different values of the porous medium permeability parameter ($k_p$ ), Eckert number ($E_c$ ), Soret number ($S_r$ ), magnetic parameter ($M$ ), Schmidt number ($S_c$ ), Prandtl number ($P_r$ ), heat source parameter ($S$ ), chemical reaction parameter ($\gamma$ ), Grashof number ($G_r$ ), modified Grashof number ($G_m$ ), Dufour number ($D_u$ ), injection/suction rate, and the plate movement rate, and are presented in Figures 2-16.

In the numerical solution process, to obtain the results shown in the figures and tables, the following default parameter values were defined: $M=1.0$ , $G_r=10$ ; $G_m=5$ ; $\gamma =0.50$ ; $k_p=1.0$ ; $S_r=1.0$ ; $S=0.10$ ; $P_r=0.63$ ; $E_c=0.01$ ; $D_u=0.5$ ; $S_c=0.22$ ; $F_w=0$ . In each case, $\theta$ is along the horizontal axis.

Figure 2. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $M$ against $\theta$ .

Figure 3. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $k_p$ against $\theta$ .

Figure 4. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $\gamma$ against $\theta$ .

Figure 5. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $P_r$ against $\theta$ .

Figure 6. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $S_c$ against $\theta$ .

Figure 7. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $G_r$ against $\theta$ .

Figure 8. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $G_m$ against $\theta$ .

Figure 9. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $Sr$ against $\theta$ .

Figure 10. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $E_c$ against $\theta$ .

Figure 11. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $D_u$ against $\theta$ .

Figure 12. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of $S$ against $\theta$ .

Figure 13. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of blowing velocity against $\theta$ .

Figure 14. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different values of suction velocity against $\theta$ .

Figure 15. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different positive values of $U_0$ against $\theta$ .

Figure 16. (a) Velocity profiles, (b) temperature profiles and (c) concentration profiles for different negative values of $U_0$ against $\theta$ .

Figure 2 illustrates the variations in velocity, temperature, and concentration profiles under the influence of different values of the magnetic parameter ($M$ ). An increase in the magnetic parameter $M$ results in a gradual decrease in the fluid velocity (Figure 2(a)). This behavior is attributed to the generation of a resistive Lorentz force within the fluid, which acts to oppose the motion and thereby reduces the velocity profile. Conversely, increasing the magnetic parameter enhances both the temperature and concentration profiles (Figure 2(b) and Figure 2(c)), due to the additional energy dissipation and reduced convective transport.

Similarly, an increase in the permeability parameter ($k_p$ ) leads to a decrease in the fluid velocity profiles (Figure 3(a)), while increasing the temperature (Figure 3(b)) and concentration profiles (Figure 3(c)). This behavior can be attributed to the presence of the porous medium, which introduces an additional resistive force in the fluid flow. This resistance reduces the velocity distribution and, in turn, enhances the temperature and concentration distributions.

Figure 4 illustrates the effect of the chemical reaction parameter ($\gamma$ ) on the velocity, temperature, and concentration profiles. As shown, the velocity (Figure 4(a)) and concentration (Figure 4(c)) decrease with an increase in the chemical reaction parameter ($\gamma$ ). In contrast, the temperature (Figure 4(b)) increases as $\gamma$ increases. Physically, this behavior can be attributed to the fact that positive values of the chemical reaction parameter ($\gamma >0$ ) correspond to a destructive chemical reaction, which reduces the concentration of the reactive species and, consequently, the flow velocity.

As for the Prandtl number ($P_r$ ) shown in Figure 5, an increase in $P_r$ significantly decreases the velocity (Figure 5(a)) and temperature (Figure 5(b)), while increasing the concentration of fluid particles (Figure 5(c)). This behavior is expected, as fluids with higher Prandtl numbers possess greater viscosity, which reduces the flow velocity and decreases the thickness of the thermal boundary layer. Consequently, a thinner thermal boundary layer leads to reduced heat transfer.

In Figure 6, the velocity, temperature, and concentration profiles are influenced by the Schmidt number ($S_c$ ). As $S_c$ increases, the concentration of nanoparticles (Figure 6(c)) decreases, which in turn leads to a reduction in the velocity profiles (Figure 6(a)), while the temperature profiles exhibit a slight increase (Figure 6(b)). Physically, the Schmidt number ($S_c$ ) is a dimensionless parameter defined as the ratio of momentum diffusivity (viscous diffusivity) to mass diffusivity. Therefore, an increase in $S_c$ implies lower mass diffusivity, which restricts mass transport and results in decreased concentration and velocity profiles.

Figure 7 and Figure 8 illustrate the effects of the Grashof number ($G_r$ ) and the modified Grashof number ($G_m$ ) on the velocity, temperature, and concentration profiles, respectively. It is observed that increasing either $G_r$ (Figure 7) or $G_m$ (Figure 8) significantly enhances the velocity profiles (Figure 7(a) and Figure 8(a)). However, this increase in buoyancy-driven flow leads to a reduction in both temperatures (Figure 7(b) and Figure 8(b)) and concentration profiles (Figure 7(c) and Figure 8(c)). Physically, the Grashof number represents the ratio of buoyancy to viscous forces within the boundary layer. A higher Grashof number indicates stronger thermal buoyancy effects, which enhance fluid motion due to gravitational forces acting on density variations. As a result, the flow accelerates, leading to increased velocities, while the enhanced mixing and thinning of the thermal and concentration boundary layers contribute to the observed decreases in temperature and concentration profiles.

The effects of the Soret number ($S_r$ ) are illustrated in Figure 9. An increase in $S_r$ leads to higher velocity and concentration profiles, as shown in Figure 9(a) and Figure 9(c), respectively. This enhancement is due to the Soret effect, which describes mass flux induced by temperature gradients. As $S_r$ increases, it promotes the diffusion of species from regions of higher temperature to lower temperature, thereby enhancing both fluid motion and species concentration. Consequently, the dynamic and concentration boundary layers become thicker. Conversely, the temperature profile (Figure 9(b)) decreases with increasing $S_r$ , indicating a reduction in the thermal boundary layer thickness.

The effects of the Eckert number ($E_c$ ) on the velocity, temperature, and concentration profiles are presented in Figure 10. As observed in Figure 10(a) and Figure 10(b), both the fluid velocity and temperature increase with rising $E_c$ . This behavior is attributed to the conversion of kinetic energy into internal energy due to viscous dissipation, which enhances thermal energy within the fluid. As a result, the temperature rises and the flow accelerates. However, an increase in the Eckert number slightly reduces the concentration profile (Figure 10(c)), likely due to the dominance of thermal effects over mass diffusion in the flow field.

In Figure 11, an increase in the Dufour number $D_u$ leads to higher velocity (Figure 11(a)) and temperature (Figure 11(b)) profiles. Conversely, a slight decrease is observed in the concentration profile (Figure 11(c)).

From Figure 12, it is observed that an increase in the heat source parameter ($S$ ) slightly enhances the velocity (Figure 12(a)) and temperature (Figure 12(b)) profiles. However, this increase in $S$ leads to a slight reduction in the concentration of fluid particles (Figure 12(c)).

The effects of injection and suction velocities are illustrated in Figure 13 and Figure 14, respectively. When the plate is subjected to an injection velocity, an increase in this velocity leads to higher flow velocity, temperature, and concentration profiles of the fluid particles. In other words, increasing the initial negative values of the stream function ($F_w$ ) enhances the velocity, temperature, and concentration distributions (Figure 13), thereby thickening the corresponding boundary layers. Conversely, when the plate is subjected to a suction velocity, an increase in this velocity results in a reduction in the flow velocity, temperature, and concentration profiles. That is, increasing the initial positive values of the stream function ($F_w$ ) diminishes these profiles (Figure 14), effectively thinning the boundary layers.

Figure 15 illustrates that increasing the plate’s motion speed in the direction of the fluid flow enhances the initial flow velocity $U_0$ , thereby increasing the flow velocity profiles near the wall (Figure 15(a)). This increase in plate speed leads to a slight reduction in both the temperature (Figure 15(b)) and concentration (Figure 15(c)) profiles. In contrast, when the plate moves in the opposite direction of the fluid flow (Figure 16), increasing its speed gradually decreases the initial flow velocity $U_0$ and, consequently, reduces the flow velocity profiles near the wall (Figure 16(a)). This reverse motion results in a slight increase in the temperature (Figure 16(b)) and concentration (Figure 16(c)) profiles.

Tables 1-3 illustrate the effects of various parameters on the skin friction coefficient, Nusselt number, and Sherwood number for $\theta =0.6867$ . An increase in $M$ or $k_p$ (Table 1), or $S_c$ or $\gamma$ (Table 2), leads to a decrease in both the skin friction coefficient and the Nusselt number, while the Sherwood number increases. Conversely, increasing $E_c$ (Table 1), $D_u$ or $S$ (Table 2), or $U_0^{-}$ or $F_w^{-}$ (Table 3) results in higher skin friction and Sherwood numbers but a lower Nusselt number. An increase in $P_r$ (Table 1), or $U_0^{+}$ or $F_w^{+}$ (Table 3) reduces the skin friction and Sherwood numbers, while enhancing the Nusselt number. Increasing $G_r$ (Table 1) raises the skin friction coefficient, but reduces both the Nusselt and Sherwood numbers. Similarly, an increase in $S_r$ (Table 2) enhances the skin friction coefficient and Nusselt number, but decreases the Sherwood number. It is important to note that $F_w^{+}$ denotes the suction velocity, $F_w^{-}$ the injection velocity, $U_0^{+}$ the plate velocity in the direction of the fluid flow, and $U_0^{-}$ the plate velocity opposite to the fluid flow direction. Table 4 shows the results of R. Biswas et al. [24], and Table 5 shows the results of our work. After comparing these two tables, we notice that our work is in good agreement and represents the complement of the work of R. Biswas et al. [24].

Table 1. Effects of different parameters ($M$ , $k_p$ , $G_r$ , $P_r$ , $E_c$ ) on skin friction coefficient, Nusselt number and Sherwood number.

$M$	$k_p$	$G_r$	$P_r$	$E_c$	$C_f$	$N_u$	$S_h$
0.50					0.036612	0.308713	0.315315
0.70					0.030986	0.308679	0.315318
0.90					0.025428	0.308644	0.315320
	1.0				0.036612	0.308713	0.315315
	1.5				0.022674	0.308627	0.315322
	2.0				0.009143	0.308540	0.315330
		5			0.036612	0.308713	0.315315
		10			0.089843	0.183843	0.187506
		15			0.116425	0.135713	0.138411
			0.63		0.036612	0.308713	0.315315
			0.71		0.036581	0.308832	0.315264
			1.00		0.036460	0.309322	0.315050
				0.01	0.036612	0.308713	0.315315
				0.02	0.036681	0.307826	0.315704
				0.03	0.036750	0.306940	0.316092

Table 2. Effects of different parameters ($D_u$ , $S_c$ , $\gamma$ , $S_r$ , $S$ ) on skin friction coefficient, Nusselt number and Sherwood number.

$D_u$	$S_c$	$\gamma$	$S_r$	$S$	$C_f$	$N_u$	$S_h$
0.5					0.036612	0.308713	0.315315
0.0					0.036740	0.305505	0.316722
0.5					0.036928	0.300783	0.318793
	0.22				0.036612	0.308713	0.315315
	0.60				0.035641	0.302353	0.335455
	0.78				0.034893	0.297348	0.351308
		0.5			0.036612	0.308713	0.315315
		1.5			0.036007	0.304871	0.327477
		2.5			0.035417	0.301090	0.339450
			2		0.036612	0.308713	0.315315
			3		0.036679	0.308782	0.315107
			4		0.036758	0.308865	0.314856
				0.01	0.036612	0.308713	0.315315
				0.02	0.036756	0.305173	0.316868
				0.03	0.036902	0.301617	0.318427

Table 3. Effects of different parameters ($U_0^{-}$ , $U_0^{+}$ , $F_w^{+}$ , $F_w^{-}$ ) on skin friction coefficient, Nusselt number and Sherwood number.

$U_0^{-}$	$U_0^{+}$	$F_w^{+}$	$F_w^{-}$	$C_f$	$N_u$	$S_h$
−0.4				0.306284	0.300423	0.316289
−0.8				0.380453	0.297138	0.316959
−1.0				0.417056	0.295350	0.317356
	0.4			0.154100	0.305790	0.315463
	0.8			0.076092	0.307846	0.315318
	1.0			0.036612	0.308713	0.315315
		0		0.036612	0.308713	0.315315
		0.4		0.028334	0.322734	0.313796
		0.8		0.019527	0.337212	0.312062
			0	0.036612	0.308713	0.315315
			−0.4	0.044362	0.295148	0.316620
			−0.8	0.051593	0.282039	0.317709

Table 4. Represents the previous results by R. Biswas et al. [24].

Parameters	$U$	$\tilde{T}$	$\tilde{C}$	$C_f$	$N_u$	$S_h$
$M$	$\downarrow$			$\downarrow$
$k_p$	$\downarrow$			$\downarrow$
$G_r$	$\uparrow$			$\uparrow$	$\downarrow$	$\downarrow$
$P_r$		$\downarrow$		$\downarrow$	$\uparrow$
$E_c$		$\uparrow$
$S_c$			$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\gamma$	$\downarrow$		$\downarrow$		$\downarrow$	$\uparrow$
$S_r$	$\uparrow$			$\uparrow$
$S$	$\uparrow$	$\uparrow$

Table 5. Represents the present results.

Parameters	$U$	$\tilde{T}$	$\tilde{C}$	$C_f$	$N_u$	$S_h$
$M$	$\downarrow$	$\uparrow$	$\uparrow$	$\downarrow$
$k_p$	$\downarrow$	$\uparrow$	$\uparrow$	$\downarrow$
$G_r$	$\uparrow$	$\downarrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$
$P_r$	$\downarrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\uparrow$
$E_c$	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\uparrow$
$S_c$	$\downarrow$	$\uparrow$	$\downarrow$	$\downarrow$	$\downarrow$	$\uparrow$
$\gamma$	$\downarrow$	$\uparrow$	$\downarrow$		$\downarrow$	$\uparrow$
$S_r$	$\uparrow$	$\downarrow$	$\uparrow$	$\uparrow$
$S$	$\uparrow$	$\uparrow$	$\downarrow$	$\uparrow$	$\downarrow$	$\uparrow$

5. Conclusions

From the above study, it is noted that:

1) Increasing the magnetic parameter $M$ or the permeability parameter $k_p$ improves the temperature and concentration profiles as well as the Sherwood number, but reduces the velocity profiles and deteriorates the skin friction coefficient and the Nusselt number.

2) Increasing the Schmidt number $S_c$ or the chemical reaction parameter $\gamma$ decreases the skin friction coefficient and the Nusselt number, as well as the velocity and concentration profiles, but improves the Sherwood number and the temperature profiles.

3) Increasing the Eckert number or Dufour number or the heat source parameter improves the velocity and temperature profiles as well as the skin friction coefficient and Sherwood number, but decreases the concentration profiles and the Nusselt number.

4) The movement of the plate in the direction of fluid flow increases the fluid velocity and the Nusselt number and reduces the temperature and concentration profiles, as well as the skin friction coefficient and the Sherwood number. On the other hand, the movement of the plate in the opposite direction of fluid flow has an opposite effect on these profiles.

5) Injection speed increases the velocity, temperature, and concentration profiles, as well as the skin friction coefficient and Sherwood number, but worsens the Nusselt number. In contrast, suction speed has the opposite effect on these profiles.

6) The Grashof number or modified Grashof number improves the velocity profiles and the skin friction coefficient, but decreases the temperature and concentration profiles, as well as the Nusselt number and the Sherwood number.

7) The Soret number $S_r$ increases the particle velocity and concentration as well as the skin friction coefficient and the Nusselt number, but decreases the temperature and the Sherwood number.

8) The Prandlt number decreases the velocity and temperature profiles as well as the skin friction coefficient and the Sherwood number, but increases the concentration and the Nusselt number.

Nomenclature

$B_0$	Magnetic component (Wb∙m−2)
$C$	Concentration of fluid (-)
$\tilde{C}$	Dimensionless fluid concentration (-)
$C_f$	Skin friction (-)
$C_s$	concentration susceptibility (J/kg∙K)
$C_p$	Specific heat at constant pressure (J/m3∙K)
$C_w$	concentration at the plate surface (mol/l)
$C_{\infty }$	concentration at far away from the plate (mol/l)
$D_m$	Molecular diffusivity of the concentration (-)
$D_u$	Dufour Number (-)
$E$	The electric field.
$E_c$	Eckert number (-)
$F$	The non-dimensional stream function
$f$ or $U$	The mean velocity nondimensional profile.
$f^{\prime }$	The first derivative of velocity.
$f^{″}$	The second derivative of velocity.
$F_w$	Initial value of F
$F_w^{+}$	Initial positive value of F
$F_w^{-}$	Initial negative value of F
$g$	Acceleration due to gravity (m∙s−2)
$G_r$	Grashof number (-)
$G_m$	Modified Grashof number (-)
$J$	The current density vector
$k$	Thermal conductivity (W/m∙K)
$k_1$	Reaction rate constant (-)
$k_p$	Permeability of porous medium (-)
$k^{\text{*}}$	Darcy permeability (-)
$k_T$	Thermal diffusion ratio (-)
$L$	Plate length or maximum value of x
$M$	Magnetic parameter
$N_u$	Nusselt number (-)
$P_r$	Prandlt number (-)
$Q$	Heat absorption quantity (-)
$S$	Heat source parameter (-)
$S_c$	Schmidt number (-)
$S_h$	Sherwood number (-)
$S_r$	Soret number (-)
$T$	Temperature of fluid (K)
$T_w$	Temperature at the plate surface (K)
$T_{\infty }$	Temperature at far away from the plate (K)
$T_m$	Mean fluid temperature (K)
$\tilde{T}$	Dimensionless fluid temperature (-)
$U$	Dimensionless primary velocity (m/s)
$U_0$	Uniform velocity (m/s)
$V_0$	Injection/suction velocity in the y direction
$u$ , $v$	Velocity components (m/s)
$U$ , $V$	Dimensionless velocity components
$x$ , $y$	Dimensional cartesian coordinates (m)
$X$ , $Y$	Dimensionless cartesian coordinates

Greek Symbols

${\beta }_T$	Thermal expansion coefficient (-)
${\beta }_c$	Concentration expansion coefficient (-)
$\rho$	Fluid density (kg∙m−3)
$\nabla$	Nabla operator
$\varphi =\overline{v}$	The amplitude function
$\delta$ or ${\delta }_{BL}$	Boundary layer thickness
$\mu$	Dynamic viscosity (Pa∙s)
${\mu }_e$	The magnetic permeability
$\nu$	Kinematic viscosity (m2∙s−1)
$\theta$	The nondimensional coordinate
$\gamma$	Chemical reaction parameter (-)
${\epsilon }_e$	Absolute permittivity of the fluid
$\sigma$	The fluid electrical conductivity
$\uparrow$	Increase
$\downarrow$	Decrease

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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