MHD Slip Flow of Newtonian Fluid past a Stretching Sheet with Thermal Convective Boundary Condition , Radiation , and Chemical Reaction

An analysis is carried out to study the problem of heat and mass transfer flow over a moving permeable flat stretching sheet in the presence of convective boundary condition, slip, radiation, heat generation/absorption, and first-order chemical reaction. The viscosity of fluid is assumed to vary linearly with temperature. Also the diffusivity is assumed to vary linearly with concentration. The governing partial differential equations have been reduced to the coupled nonlinear ordinary differential equations by using Lie group point of transformations. The system of transformed nonlinear ordinary differential equations is solved numerically using shooting techniques with fourth-order Runge-Kutta integration scheme. Comparison between the existing literature and the present study was carried out and found to be in excellent agreement. The effects of the various interesting parameters on the flow, heat, and mass transfer are analyzed and discussed through graphs in detail. The values of the local Nusselt number, the local skin friction, and the local Sherwood number for different physical parameters are also tabulated.


Introduction
Investigations of laminar boundary flow of an electrically conducting fluid over a moving continuous stretching surface are important in many manufacturing processes, such as materials manufactured by polymer extrusion, continuous stretching of plastic films, artificial fibers, hot rolling, wire drawing, glass fiber, metal extrusion and metal spinning, cooling of metallic sheets, or electronic chips.The first and foremost work regarding the boundary-layer behavior in moving surfaces in quiescent fluid was considered by Sakiadis [1].Heat and mass transfers on stretched surface with suction and injection was introduced by Fox et al. [2].P. S. Gupta and A. S. Gupta [3] studied the same problem for linearly stretching sheet.Heat transfer past a moving continuous plate with variable temperature was studied by Soundalgekar and Murty [4] and Grubka and Bobba [5].Subsequently, many researchers [6][7][8][9][10][11][12] worked on the problem of moving or stretching plates under different situations.
MHD viscous flow over a stretching sheet in the presence of slip velocity was studied by many authors, such as Martin and Boyd [13], Fang and Lee [14], Ariel et al. [15], Andersson [16], Wang [17,18], Mukhopadhyay and Anderson [19], Fang [20], and Hayat et al. [21].Fang et al. [22] studied MHD viscous flow over a permeable shrinking sheet.They observed that the velocity at the wall increased with slip parameter.Mahmoud and Waheed [23] included the effects of slip and heat generation/absorption on MHD mixed convection flow of a micropolar fluid over a heated stretching surface.Recently, Hamad et al. [24] investigated heat and mass transfer over a moving porous plate with hydrodynamic slip and thermal convective boundary conditions.They found that the wall slip velocity and wall shear stress are a decreasing function of the slip parameter.
Thermal radiation effects on an electrically conducting fluid arise in many practical applications such as electrical power generation, solar power technology, nuclear reactors, and nuclear waste disposal (see Mahmoud [25] and Chamkha [26]).Radiation effects on boundary layer flow with and without applying a magnetic field have been investigated by many authors [27][28][29][30][31][32][33] under certain conditions.
Heat and mass transfer problems with a chemical reaction have received a considerable amount of attention due to their importance in many chemical engineering processes, for example, food processing, manufacturing of ceramics, and polymer production.On the other hand, the study of heat generation or absorption in moving fluids is important in problems dealing with chemical reactions and those concerned with dissociating fluids.Possible heat generation effects may alter the temperature distribution and, consequently, the particle deposition rate in nuclear reactors, electronic chip, and semiconductor wafers.Many investigators [34][35][36][37][38][39][40] displayed the effects of heat generation and chemical reactions on heat transfer over a stretching sheet under various conditions.Recently, thermal convective surface boundary conditions were used by several authors such as Ishak [41], Makinde [42], Yao et al. [43], Hamad et al. [24], and Ferdows et al. [44].
Lie group point of transformations can be used to generate similarity transformations.It reduces the number of independent variables of the partial differential equations and keeps the system and associated initial and boundary conditions invariant.Reviews for the mathematical theory and applications of Lie group point of transformations to differential equations may be found in the texts by Olver [45], Bluman and Kumei [46], Hill [47], and Ibragimov and Kovalev [48].This technique has been applied by many researchers to solve different flow phenomena over different geometries (see [49][50][51][52]).
Motivated by the previous works, the aim of the present work is to study the problem of heat and mass of Newtonian fluid with thermal convective boundary condition, slip velocity, radiation, heat generation/absorption, and firstorder chemical reaction taking into account the viscosity dependent on the temperature and the thermal diffusivity dependent on the concentration.This problem has not been studied before despite its applications in many engineering processes, such as nuclear waste, polymer production, dispersion of chemical pollutants through water-saturated soil, geothermal energy extractions, plasma studies, liquid metal fluids, and power generation systems.

Mathematical Formulation of the Problem
Consider laminar boundary layer flow heat and mass transfer of viscous incompressible Newtonian fluid over moving stretching sheet as shown in Figure 1.A magnetic field of nonuniform () is applied parallel to the  axis.The bottom surface of the plate is heated by convection from a hot fluid of temperature   , and this generates a heat transfer coefficient ℎ  (), and  ∞ is the ambient fluid temperature.The mass concentration of the species at the plate is also maintained uniform   which is also higher than the ambient fluid concentration  ∞ .Figure 1 shows the coordinates and flow model.Under these assumptions and boundary layer approximations, the governing equations are subject to the boundary conditions where  and V are the velocities in  (along the sheet) and  (normal to the sheet) directions, respectively,   () is the mass transfer velocity,  is the coefficient of dynamic viscosity, ] = (/) is the kinematic coefficient of viscosity, () is the variable diffusivity,  0 is the electric conductivity,  is the density of the fluid,   is the specific heat at constant pressure,  is the thermal conductivity,  0 () is the velocity slip factor having dimension (velocity) −1 ,   () is the velocity of moving plate,   () is the suction or injection velocity through the surface,  is the temperature,  is the species concentration, () is the rate of chemical reaction, and  0 () is the heat generation/absorption rate.The subscripts denote wall conditions and free stream conditions, respectively.
Using Rosseland approximation for radiation [53], we can write   = −(4 1 /3 * )( 4 /), where  1 is the Stefan-Boltzmann constant and  * is the mean absorption coefficient.Assuming that the temperature difference is within the flow such that  4 may be expanded in a Taylor series and expanding  4 about  ∞ and neglecting higher orders, we get ∞ .Therefore, (3) becomes We assume that the temperature dependent viscosity is varying linearly and is given by (see [54]) where  ∞ is the coefficient of viscosity far away from the plate and  0 and  0 are constants depend on the nature of the fluid.We choose  0 = 1.Also here, we assume that the concentration diffusivity varies linearly as (see [55]) where   is the constant concentration diffusivity and  is a constant.
We introduce the nondimensional parameters with  being characteristic length, Re =  0 /] is the Reynolds number, and  0 is a reference velocity.Also, we introduce stream function  defined as The continuity equation ( 1) is satisfied identically and ( 2), ( 4), and ( 6) yield The boundary conditions ( 5) now become where , and  * = / 0 .The quantities Pr, , , , and Sc are the Prandtl number, the viscosity parameter, the concentration diffusivity parameter, the radiation parameter, and the Schmidt number, respectively.

Symmetries of the Problem
In this section, we apply the Lie group of transformations which leaves the equations ( 11)-( 13) invariant.Firstly, we consider Lie group of transformations with independent variables  and  and dependent variables , Θ, and Φ for the problem  * =  * (, , , , ; ) ,  * =  * (, , , , ; ) ,  * =  * (, , , , ; ) , Θ * = Θ * (, , , , ; ) , Φ * = Φ * (, , , , ; ) , (16) where  is the group parameter.The infinitesimal generator of the group ( 16) can be expressed in the following vector form: in which , , Ψ, Θ, and Φ are infinitesimal functions of the group variables.Then, the corresponding one-parameter Lie group of transformations is given by The action of the infinitesimal generator Γ is extended to all derivatives appearing in equations ( 11)-( 13) through the third prolongation where Ψ [] , Ψ [] , Θ [] , . . ., Ψ [] are the infinitesimal functions corresponding the derivatives.The infinitesimal generator Γ is a point symmetry of ( 11)- (12) if Since the coefficients of Γ do not involve derivatives, we can separate (20) with respect to derivatives and solve the result over a determined system of linear homogeneous partial differential equations known as the determining equations.After straightforward calculations for the determining equations, we find the solutions , , Ψ, Θ, and Φ of the determining equations in the form where   ( = 1, 2, 3, 4) are arbitrary constants and () is the infinite parameter Lie group transformation, and the functions satisfy the following ordinary differential equations: which directly give the form for the functions (), (), and (), where  1 ,  2 , and  3 are arbitrary constants and  1 ̸ =  2 .Imposing the restrictions from boundaries and from the boundary conditions on the infinitesimals, we obtain () =  4 = 0.

Physical Quantities of Interest.
The parameters in which we are interested for our problem are the local skin friction coefficient   , the local Nusselt number Nu  , and the local Sherwood number Sh  , which are defined, respectively, as where Re  = (/]) indicates the local Reynolds number.

Numerical Solution and Discussion
The transformed system of nonlinear ordinary differential equations (31) with the boundary conditions (32) has been solved numerically by using a fourth-order Runge-Kutta integration along with shooting techniques.In order to verify the accuracy of our results, we have compared the computed values of −  (0) with the previously published works by Hamad et al. [24], and Hayat et al. [21] with   = 0 and  → ∞ at different values of  as shown in Table 1.The results of −  (0), −  (0) and −  (0) for Pr = 0.72, Sc = 0.22,  = 0.2,  = 0.5, and   = −0.5 for various values of  are compared with those obtained by Hamad et al. [24] and are listed in Table 2.Moreover, Table 3 represents the comparison of −  (0) between our results and the results obtained by Hamad et al. [24] for Pr = 0.72,  = 0.2,  = 0.5,  −  (0) −  (0) −  (0) Hamad et al. [24] P r e s e n tw o r k H a m a de ta l .[ 2, 3, and 4 show the dimensionless velocity profiles   (), the dimensionless temperature profiles and the dimensionless mass profiles in the boundary layer for various values of the viscosity parameter .These figures depict that increase in the value of  the dimensionless velocity decreases near the surface but increases as larger distances, while the dimensionless temperature and the dimensionless mass in the boundary layer region decrease as the viscosity parameter increases.
Figures 5-7 exhibit the effect of the suction or injection parameter   on the dimensionless velocity profiles, the dimensionless temperature, and the dimensionless mass profiles.It is shown from Figure 5 that the suction parameter decreases the velocity indicating the usual fact that suction stabilizes the boundary layer growth, while injection increases the velocity in the boundary layer region indicating that injection helps the flow penetrate more into the fluid.In Figure 6, it is found that the temperature decreases as the suction parameter increases.This means that larger suction leads to faster cooling of the plate, while the temperature increases as the injection parameter increases; this is because of the heat transfer from fluid to surface.Figure 7 illustrates that the mass decreases as the suction parameter increases and increases as the injection parameter increases.
The dimensionless velocity profiles for different values of the Boit number  are described in Figure 8. From this figure, we observe that an increase in  leads to increase in   near the wall but it decreases with larger distances, while Figure 9 shows that the dimensionless temperature decreases as the Boit number  increases.
Effect of the chemical reaction parameter  0 on the dimensionless concentration profiles is shown in Figure 10.It is seen that the dimensionless concentration of the fluid decreases with the increase of  0 .
Figures 11, 12, and 13 illustrate the influence of slip velocity parameter  on the velocity, temperature, and concentration profiles, respectively.It is obvious that slip parameter decreases the velocity   and increases both dimensionless temperature  and the dimensionless concentration .
Figures 14, 15, and 16 show the velocity, temperature, and concentration distributions for various values of the magnetic field .We can notice that increasing  decreases the velocity   and increases the temperature  and concentration; this result qualitatively agrees with the expectations; this is because the application of a transverse magnetic field to an electrically conducting fluid gives rise to a resistive-type force called the Lorentz force.This force has the tendency to slow down the motion of the fluid in the boundary layer and to increase its temperature and concentration.
Figure 17 illustrates the effect of the concentration diffusivity parameter  on the concentration profiles; one can observe that the concentration profiles increase as  increases.
The effects of the Prandtl number Pr on the dimensionless velocity and temperature dimensionless velocity are Table 5: Values of (−1/2)  (Re  ) 1/2 , −Nu  (Re  ) −1/2 , and −Sh  (Re  ) −1/2 for   = 0.5, Pr = 0.72, Sc = 0.6,  = 0.5, and  = 1.illustrated in Figures 18 and 19, respectively.From Figure 18, it can be seen that as the Prandtl number Pr increases, the dimensionless velocity decreases near the surface but increases with larger distances.Figure 19 illustrates that the dimensionless temperature decreases with increasing Pr; this is because, when the Prandtl number increases the thickness of the thermal boundary layer decreases and, hence, the temperature decreases.

𝜁 𝐾
The effects of the presence of heat source ( > 0) or heat sink ( < 0) in the boundary layer on the velocity and temperature profiles are presented in Figures 20 and 21, respectively.From Figure 20, one sees that the dimensionless velocity decreases with the heat source ( > 0) increasing, while it increases with the heat sink ( < 0) increasing.The presence of heat source in the boundary layer generates energy which causes the temperature of the fluid to increase.On the other hand, the presence of heat sink in the boundary layer absorbs energy which causes the temperature of the fluid to decrease, and this corresponds with the observation in Figure 21, in which we note that the dimensionless temperature increases with the heat source ( > 0) increasing, while it decreases with the heat sink ( < 0) increasing.
For different values of radiation parameter , the dimensionless velocity profiles are plotted in Figure 22.It is obvious that velocity decreases with the increase in .The effects on the temperature profiles are presented in Figure 23.From this figure, we observed that, as the value of  increases, the temperature profiles increase, with an increase in the thermal boundary layer thickness.

Conclusions
In this study, the symmetries of the problem of heat and mass transfer flow over a moving permeable flat stretching sheet in the presence of thermal radiation, variable viscosity, a uniform transverse magnetic field, chemical reaction, heat generation/absorption, suction/injection, convective boundary condition, and slip velocity are obtained using Lie group analysis.Numerical solutions of the resulting system of nonlinear ordinary differential equations are obtained by using the shooting method coupled with Runge-Kutta scheme.The effects of various parameters on the dimensionless velocity, temperature, and concentration profiles have been studied graphically.The influence of some parameters on the local skin friction, local Nusselt number, and local Sherwood number is tabulated.It was found that with the increase of the convective parameter , injection parameter   < 0, and the slip velocity , the absolute concentration increases, while it decreases with the increase in the magnetic parameter , viscosity parameter , suction parameter   > 0, and chemical reaction parameter  0 .

Figure 2 :
Figure 2: Velocity distribution for various values of .

6 =Figure 3 :
Figure 3: Temperature distribution for various values of .

Figure 18 : 22 Figure 19 :
Figure 18: Velocity distribution for various values of Pr.

Figure 22 :
Figure 22: Velocity distribution for various values of .

Table 4 .
In all comparisons, it is found that our results are in excellent agreement with the previous published results.Further, the values of the local skin friction, the local Nusselt number, and the local Sherwood number for different values of ,  0 , , , , and  are listed in Table5.It is noted from this table that the local skin friction increases with increasing , heat absorption ( < 0), and , while it decreases by increasing , , and heat generation ( > 0).The local Nusselt number decreases with the increase of ,  and heat generation ( > 0), while the local Nusselt number increases with the increase of , heat absorption ( < 0), and .The local Sherwood number increases with the increase of , heat absorption ( < 0), and  0 , while it decreases with the increasing of , , , and heat generation ( > 0).