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Newtonian steady state flow of fluids with electrical conduction properties was examined adjacent to a moving heated vertical plate subjected to a magnetic field and a heat source/sink. The impact of magnetic parameter, Prandtl number, permeability coefficient, heat source/sink volumetric rate and temperature difference between heated plate and ambient temperature. A reduced system of ODEs was created via group similarity method. The solution led to some important results. Increasing permeability coefficient of the plate material resulted in a significant increase in flow velocity and a slight increase in heat flux but the magnitude of shear stress and temperature distribution decreased. Moreover, increasing the magnetic parameter, M, led to a significant decrease in velocity and a decrease in heat flux, whereas shear stress and temperature distribution increased. Furthermore, increasing Prandtl number, Pr, reduced the velocity significantly and the heat flux slightly. On the other hand, the magnitude of shear stress and temperature distribution increased. In case of using heat source, the increase in its energy rate decreased the heat flux with no significant effect on shear stress. Finally, the increment of temperature difference led to noticeable increase in velocity and a slight increase in heat flux, whereas the shear stress decreased.
electrically conducting fluids, group method, magnetic parameter, Prandtl number
For many decades, Newtonian flows attracts many researchers to investigate and study their behaviors. Steady and unsteady fluid dynamics were studied for different cases of operations to model and simulate many engineering applications. Power generators, cooling systems of nuclear reactors and liquid metal flow control are few examples of such applications. Electrically conducting fluids subjected to magnetic field, which are also known as magnetohydrodynamic (MHD) fluids, are important fluids models. Many researchers have studied different cases of MHD fluids using numerous methods. Kataria and Patel [1] studied the effects of heat generation on MHD fluid flow through porous medium past an oscillating vertical plate. Huang and Liu [2], analyzed the features of laminar MHD fluid in a pipe. Liu and Guo [3] examined the fractional Maxwell MHD fluid. Ahmad et al. [4] used a periodically accelerated plate to find a new analytical technique for MHD fluid flow. Ajam et al. [5] used Buongiorno’s model and found a new analytical approximation of MHD resulting from a stretching permeable surface. Chen et al. [6] obtained a solution for fractional viscoelastic MHD fluid using Lie group similarity over a stretching sheet. Khan et al. [7] attained a numerical solution of MHD flow with homogenous heterogeneous reactions. Prasad et al. [8] studied the thermal properties of MHD Casson fluid. Umavathi et al. [9] studied the effect of temperature on MHD flow in a vertical channel. Ahmed et al. [10] investigated the nonNewtonian Maxwell fluid with variable thermal conductivity. Rehman et al. [11] studied MHD flow of Casson fluid in stretching cylinder. Different mathematical methods were exploited to investigate and analyze numerous cases of fluid dynamics. Lie Infinitesimal and group methods [1214], homotopy method [1517], finite element [1820] and finite volume [2123] are examples for such common methods. Several modeling of MHD fluids have been studied [24, 25].
Inspired by all these researches, the present work provides analytical and numerical solutions for heated moving vertical plate submerged in MHD fluid. The objectives of the recent study are to combine many parameters to the considered flow and investigate their effect on the velocity profile, shear stress, heat distribution and heat flux inside the boundary layer. The considered parameters are magnetic parameter, permeability coefficient, Prandtl number, temperature difference and volumetric heat rate of a heat source/sink.
Consider a moving vertical porous plate immersed in MHD fluid with temperature adjacent to plate of T_{w} while temperature outside boundary layer is $T_{\infty}$. The flow undergoes a constant pressure and subjected to a constant magnetic field of density B_{0} in ydirection that results in Lorentz force in xdirection $\left(\sigma B_{0}^{2} u\right)$. The momentum in ydirection has been neglected while the heat diffusion is more significant in ydirection. Based on the previous assumptions, the physical model is depicted in Figure 1 while the governing equations are described as:
Figure 1. Physical model of the problem
$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$ (1)
$u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}=v \frac{\partial^{2} u}{\partial y^{2}}+g \beta\left(TT_{\infty}\right)\frac{\sigma}{\rho} B_{0}^{2} u\frac{v}{k} u$ (2)
$u \frac{\partial T}{\partial x}+v \frac{\partial T}{\partial y}=\alpha \frac{\partial^{2} T}{\partial y^{2}}+\frac{16 \sigma^{*}}{3 \rho c_{p} \chi} T_{\infty}^{3} \frac{\partial^{2} T}{\partial y^{2}}+\frac{Q}{\rho c_{p}}\left(TT_{\infty}\right)$ (3)
where, v is the kinematic viscosity, T is the temperature, k stands for permeability while $\alpha$ is the thermal diffusivity $\alpha=\frac{v}{P r}$ (Pr is Prandtl number).
The flow is Subjected to the following boundary conditions:
$u(x, 0)=u_{w}(x), u(x, \infty)=0, v(x, 0)=0,\\T(x, 0)=T_{\infty}$ (4)
The equations, Eq. (1)  Eq. (3), are normalized using the following substitutions:
$U(x, y)=\frac{u(x, y)}{u_{w}(x, y)}$ (5)
$\theta=\frac{TT_{\infty}}{T_{w}T_{\infty}}$ (6)
The equations, Eq. (1)  Eq. (3), are transformed to:
$U \frac{d u_{w}}{d x}+u_{w} \frac{\partial U}{\partial x}+\frac{\partial V}{\partial y}=0$ (7)
$U u_{w}\left(U \frac{d u_{w}}{d x}+u_{w} \frac{\partial U}{\partial x}\right)+v u_{w} \frac{\partial U}{\partial y}$ $v u_{w} \frac{\partial^{2} U}{\partial y^{2}}g \beta \theta \Delta T\frac{\sigma}{\rho} B_{0}^{2} U u_{w}+\frac{v}{k} U u_{w}=0$ (8)
$U u_{w} \frac{\partial \theta}{\partial x}+v \frac{\partial \theta}{\partial y}\alpha \frac{\partial^{2} \theta}{\partial y^{2}}\frac{16 \sigma^{*} T_{\infty}^{3}}{3 \rho c_{p} \chi} \frac{\partial^{2} \theta}{\partial y^{2}}\frac{q}{\rho c_{p}} \theta=0$ (9)
The boundary conditions are:
$U(x, 0)=1, v(x, 0)=0, \theta(x, 0)=1$ $U(x, \infty)=0, \theta(x, \infty)=0$ (10)
A group transformation of one parameter, a, was used to reduce the PDE system into an ODE system in one similarity variable, $\eta$.
3.1 Formulation of the problem using group method
The group structure is assumed to be on the form [12, 13]:
$G: \bar{S}=K^{s}(a) S+Q^{s}(a)$ (11)
where, S stands for the system variables. The differential coefficients function, $K^{S}$ and $Q^{s}$, are real constants. The partial derivatives are defined as:
$\left. \begin{matrix} {{{\bar{S}}}_{{\bar{\iota }}}}=(\frac{{{k}^{s}}}{{{k}^{i}}}){{S}_{i}} \\ {{{\bar{S}}}_{\overline{\iota J}}}=(\frac{{{k}^{s}}}{{{k}^{i}}{{k}^{j}}}){{S}_{i}}_{j} \\\end{matrix} \right\}i=x,y\text{ and j=x,y}$ (12)
3.2 Analysis of the problem
Eqns. (7)  (9) are transformed to:
$\bar{U} \frac{d \bar{u}_{w}}{d \bar{x}}+\bar{u}_{w} \frac{\partial \bar{U}}{\partial \bar{x}}+\frac{\partial \bar{v}}{\partial \bar{y}}=H_{1}(a)\left[U \frac{d u_{w}}{d x}+u_{w} \frac{\partial U}{\partial x}+\frac{\partial v}{\partial y}\right]$ (13)
$\bar{U} \bar{u}_{w}\left(\bar{U} \frac{d \bar{u}_{w}}{d \bar{x}}+\bar{u}_{w} \frac{\partial \bar{U}}{\partial \bar{x}}\right)+\bar{v} \bar{u}_{w} \frac{\partial \bar{v}}{\partial \bar{y}}v \bar{u}_{w} \frac{\partial^{2} \bar{U}}{\partial \bar{y}^{2}}$$g \beta \bar{\theta} \Delta T\frac{\sigma}{\rho} B_{0}^{2} \bar{U} \bar{u}_{w}+\frac{v}{K} \bar{U} \bar{u}_{w}$$=H_{2}(a)\left[U u_{w}\left(U \frac{d u_{w}}{d x}+u_{w} \frac{\partial U}{\partial x}\right)+v u_{w} \frac{\partial U}{\partial y}\right.$$\left.v u_{w} \frac{\partial^{2} U}{\partial y^{2}}g \beta \theta \Delta T\frac{\sigma}{\rho} B_{0}^{2} U u_{w}+\frac{v}{k} U u_{w}\right]$ (14)
$\bar{U} \bar{u}_{w} \frac{\partial \bar{\theta}}{\partial \bar{x}}+\bar{v} \frac{\partial \bar{\theta}}{\partial \bar{y}}\alpha \frac{\partial^{2} \bar{\theta}}{\partial \bar{y}^{2}}\frac{16 \sigma^{*} T_{\infty}^{3}}{3 \rho c_{p} \chi} \frac{\partial^{2} \bar{\theta}}{\partial \bar{y}^{2}}\frac{q}{\rho c_{p}} \bar{\theta}$$=H_{3}(a)\left[U u_{w} \frac{\partial \theta}{\partial x}+v \frac{\partial \theta}{\partial y}\alpha \frac{\partial^{2} \theta}{\partial y^{2}}\frac{16 \sigma^{*} T_{\infty}^{3}}{3 \rho c_{p} \chi} \frac{\partial^{2} \theta}{\partial y^{2}}\right.$$\left.\frac{q}{\rho c_{p}} \theta\right]$ (15)
Invariance condition [12, 13] for Eq. (13)Eq. (15) leads to:
$K^{y}=K^{v}=1, K^{x}=K^{U} K^{u_{w}}, K^{\theta}=K^{x}$ (16)
$Q^{U}=Q^{u_{w}}=Q^{v}=Q^{\theta}=0$ (17)
Finally, the full group structure can be described as:
$G:\left\{ \begin{matrix} {{G}_{1}}\left\{ \begin{matrix} \overline{x}={{K}^{x}}x+{{Q}^{x}} \\ \overline{y}=y+{{Q}^{y}} \\\end{matrix} \right. \\ {{G}_{2}}\left\{ \begin{matrix} \overline{U}={{K}^{U}}U \\ \begin{align} & \overline{{{u}}}_{w}=\frac{{{k}^{x}}}{{{k}^{{{u}_{w}}}}}{{u}_{w}} \\ & \overline{v}=V \\ & \overline{\theta}{ ={{K}^{x}}\theta } \\\end{align} \\\end{matrix} \right. \\\end{matrix} \right.$ (18)
3.3 The complete transform of MHD system
Recalling Morgan's theorem for group method of one parameter, the independent variables can be reduced into one similarity variable. Moreover, the dependent variables, $U, u_{w}, v$ and $\theta$ are transformed into new invariant variables. Morgan’s theorem states:
$\sum_{i=1}^{6}\left(\gamma_{i} S_{i}+\delta_{i}\right) \frac{\partial q_{i}}{\partial s_{i}}=0$ (19)
where, $S_{i}$ refers to the original system variables ( $x, y, U, u_{w}, V, \theta$) and $q_{i}$ refers to the transformed variables while the $\gamma_{i}$ and $\delta_{i}$ are defined as:
$\left\{\begin{array}{l}\gamma_{i}=\frac{\partial K^{S} i(a)}{\partial a} \\ \delta_{i}=\frac{\partial Q^{S_{i}(a)}}{\partial a}\end{array}\right.$ （20）
3.3.1 Transformation of the independent variables
Applying Eq. (19) to the original independent variables, helps in attaining a new similarity variable in the form:
$\eta(x, y)=y \Gamma(x)$ (21)
whereas the dependent variables are transformed to new invariant variables in the form:
$\left\{\begin{array}{l}U=\omega(x) F(\eta) \\ u_{w}=u_{w}(x) \\ v=V(\eta) \\ \theta=\pi(x) E(x)\end{array}\right.$ (22)
where, $\Gamma(x), \omega(x), u_{w}(x)$ and $\pi(x)$ are arbitrary functions which will be evaluated during the reduction process of the system. The system, Eq. (7)  Eq. (9), will be reduced to the following system where dashes indicate derivatives with respect to $\eta$:
$V^{\prime}+\left(v^{2}+\frac{av^{3}}{v}\right) F=0$ （23）
$v^{2} F^{2}+\left(\frac{av^{3}}{v}\right) F^{2}+v F^{\prime}$
$=F^{\prime \prime}+g \beta \Delta T E\left(\frac{M}{v}+\frac{1}{K}\right) v^{2} F$ (24)
$a E F+v V E^{\prime}=\left(\frac{v}{P r}+H\right) E^{\prime \prime}+\frac{q v^{2}}{\rho c_{p}} E$ (25)
Such that the arbitrary functions were evaluated to be:
$\Gamma(x)=\frac{1}{v}, \eta(x, y)=\frac{y}{v}, \pi(x)=\frac{c_{0}}{v^{3}} x+c_{1}$,$\omega(x)=c_{2}\left(\frac{c_{0}}{v^{3}} x+c_{1}\right)^{\frac{v^{3}}{a}}, u_{w}=\frac{v}{c_{2}}\left(\frac{c_{0}}{v^{3}} x+c_{1}\right)^{\frac{c_{0}v^{3}}{c_{0}}}$ (26)
where, $M=\frac{\sigma}{\rho} B_{0}^{2}, H=\frac{16 \sigma^{*}}{3 \rho c_{p} \chi} T_{\infty}^{3}$. Moreover, $c_{0}, c_{1}$ and $c_{2}$ are arbitrary constants.
The obtained ODE system is subjected to the following conditions:
$F(0)=1, V(0)=0, E(0)=1$,$F(\infty)=0, E(\infty)=0$ (27)
Eqns. (23)  (25) are numerically solved using shooting method. The effects of Prandtl number, Pr, magnetic parameter, M, permeability coefficient of the plate, k, the volumetric heat rate of the source/sink, q, and temperature difference, $\Delta T$, were investigated for MHD flow.
4.1 Effect of permeability coefficient, k
The results showed that, increasing permeability coefficient of the plate material led to a corresponding significant increase in the fluid velocity and a slight increase in the heat flux inside the boundary layer. On the other hand, the magnitude of shear stress and temperature distribution decreased with increasing permeability coefficient. This can be explained by the fact that the pores in porous plate facilitated the flow diffusion and heat flux as shown in Figures 25.
4.2 Effect of magnetic parameter, M
The results showed that, increasing magnetic parameter, M, or increasing the magnetic flux density led to a corresponding significant decrease in the fluid velocity and a slight decrease in the heat flux inside the boundary layer. This is due to the retarding effect of Lorentz magnetic forces on the fluid diffusion. On the other hand, the magnitude of shear stress and temperature distribution increased with increasing M values as illustrated in Figures 69.
4.3 Effect of Prandtl number, Pr
The results showed that, increasing Prandtl number, Pr, or increasing the viscosity of the fluid led to a corresponding significant decrease in flow velocity and a slight decrease in the heat flux inside the boundary layer. This is due to the effect of increasing fluid viscosity on retarding the fluid flow. On the other hand, the magnitude of shear stress and temperature distribution increase with increasing Pr values as illustrated in Figures 1013.
Figure 2. Effect of permeability coefficient on velocity
Figure 3. Effect of permeability coefficient on shear stress
Figure 4. Effect of permeability coefficient on temperature distribution
Figure 5. Effect of permeability coefficient on heat flux
Figure 6. Effect of magnetic parameter on velocity
Figure 7. Effect of magnetic parameter on shear stress
Figure 8. Effect of magnetic parameter on temperature
Figure 9. Effect of magnetic parameter on heat flux
Figure 10. Effect of Prandtl number on velocity
Figure 11. Effect of Prandtl number on shear stress
Figure 12. Effect of Prandtl number on temperature
Figure 13. Effect of Prandtl number on heat flux
4.4 Effect of source/sink volumetric heat rate, q
The results showed that, increasing source energy from 0.2 to 1 slightly increased the fluid velocity and slightly decreased the heat flux, whereas it had no significant effect on shear stress. Moreover, increasing sink energy from 2 to 3 had a slight effect on the velocity and heat flux while it had no significant effect on shear stress as shown in Figures 1416.
4.5 Effect of temperature difference, $\left(\Delta T=T_{w}T_{\infty}\right)$
Also, the results showed that, increasing $\Delta T$ led to a corresponding significant increase in the fluid velocity and a slight increase in the heat flux inside the boundary layer. This is due to the energy gained by the fluid which activated the fluid particles. On the other hand, the magnitude of shear stress decreased with increasing $\Delta T$ values as illustrated in Figures 1719.
Figure 14. Effect of source/sink heat rate energy on velocity
Figure 15. Effect of source/sink heat rate energy on shear stress
Figure 16. Effect of source/sink heat rate energy on heat flux
Figure 17. Effect of temperature difference between the plate and ambient temperature on velocity
Figure 18. Effect of temperature difference between the plate and ambient temperature on shear stress
Figure 19. Effect of temperature difference between the plate and ambient temperature on heat flux profile
Investigation and analysis of electrically conducting fluid containing a moving vertical plate has been executed in the presence of external magnetic field and heat source/sink. The following results have been obtained
(1) Increasing the permeability coefficient, k, of the plate material increases the fluid velocity and heat flux, significantly, whereas it decreases the shear stress and temperature distribution.
(2) If a heat source exists, the increment in its volumetric heat rate, q, slightly increases the fluid velocity, but slightly decreases heat flux with no significant effect on shear stress.
(3) If a heat sink exists, the increment in its volumetric heat rate, q, has a slight effect on the fluid velocity and heat flux, but it has no noticeable effect on shear stress.
(4) The temperature difference between the heated plate and ambient temperature outside the boundary layer increases the fluid velocity and the heat flux. On the contrary, it decreases the shear stress and temperature distribution.
(5) Increasing the magnetic parameter, M, and Prandtl number, Pr, decrease velocity and heat flux due to the Lorentz forces and fluid viscosity increment, respectively, which obstruct the fluid particles. The opposite behavior occurs with shear stress and temperature distribution.
The authors would like to express their gratitude to the reviewers and the editor for their valuable comments and suggestions that enrich this paper to get out to the light.
a 
Group parameter 
B_{0} 
magnetic flux density 
$c_{p}$ 
specific heat at constant pressure 
g 
gravity acceleration 
k 
permeability coefficient 
K, Q 
Group coefficient function 
Pr 
Prandtl number 
q 
volumetric rate of heat generation or absorption 
T 
temperature of the fluid 
$T_{\infty}$ 
temperature outside the boundary layer 
u 
velocity component in xdirection 
v 
velocity component in ydirection 
x 
vertical distance 
y 
horizontal distance from the plate 
Greek symbols


$\alpha$ 
thermal diffusivity $\frac{v}{P r}$ 
$\beta$ 
volumetric coefficient of expansion 
$\eta$ 
similarity independent variable 
v 
kinematic viscosity of the fluid 
$\rho$ 
fluid density 
$\sigma$ 
fluid conductivity 
$\sigma^{*}$ 
StefanBoltzmann constant 
x 
the mean absorption coefficient 
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