Numerical Study of Radiative Magnetohydrodynamics Viscous Nanofluid Due to Convective Stretching Sheet with the Chemical Reaction Effect

A numerical investigation was performed for the radiative magnetohydrodynamic (MHD) viscous nanofluid due to convective stretching sheet. Heat and mass transfer were investigated in terms of viscous dissipations, thermal radiation and chemical reaction. The governing Partial Differential Equations (PDEs) were transformed into an arrangement of non-linear Ordinary Differential Equations (ODEs) by using the similarity transformation. The resulting system of ODEs is solved numerically by using shooting method along with Adams-Moulton Method of order four with the help of the computational software FORTAN. Furthermore, we compared our results with the existing results for especial cases. which are in an excellent agreement. The numerical solution obtained the velocity, temperature and concentration profiles. The figures showed differences among the parameters. Moreover, the numerical values of Nusselt and Sherwood numbers were presented and analyzed through tables.


INTRODUCTION
The transfer of heat by the movement of fluids from one place to another is called convective heat transfer, which is a combination of heat diffusion and bulk fluid flow, that are called conduction and advection, respectively. In engineering problems, convective heat transfer has wide applications. A large number of investigations on nanofluids (i.e. the mixture of fluid and nanoparticles) showed that this approach can improve thermal conductivity in fluids. Nanofluid is a fluid containing nanometersized particles called nanoparticles, which are made of metals, oxides, carbides, etc. Nanofluids have properties that make them potentially useful in many heat transfer applications. They exhibit enhanced thermal conductivity and convective heat transfer coefficient. Choi [1] studied the enhancement of thermal conductivity of fluids with nanoparticles. Eastman et al. [2] reviewed the detailed work performed on convective transport in nanofluids. The different theories of heat transfer in nanofluids were discussed by Boungiorno [3]. Kuznetsov and Nield [4] studied the convective nanofluid in a vertical plate, and later they extended their work for porous medium as well [5].
The impact of thermal rays is essential in space technology and high temperature processes. At the point when the temperature variation is very high, the linear thermal radiation causes a noticeable error. To overcome such errors, nonlinear thermal radiation is taken into account. The impact of the chemical reaction and thermal radiation on the flow over a stretching surface with an outer heat source was explained by Krishna et al. [6]. Researchers conducted a series of research work to highlight the importance of thermal radiation [7][8][9].

ISSN: 0067-2904
Science, 2020, Vol. 61, No. 7, pp: 1733-1744 Iraqi The chemical reaction can further be classified by considering the heterogeneous and homogenous processes. In the case of a strong compound system, the reaction is heterogeneous. In most of the cases of chemical reaction processes, the concentration rate depends upon the species itself, as discussed by Magyari et al. [10]. Devi and Kandasmy [11] analyzed the impact of the homogenous chemical reaction with heat and mass transfer laminar flow along with semi-infinite horizontal plate. Chamkha and Rashad [12] described the impact of the chemical reaction on MHD flow in the presence of heat generation or absorption of uniform vertical permeable surface. In addition, many significant features of the MHD flow past a stretching sheet were presented and elaborated in the literature [13][14][15].
In this article, we present a review study of Gbadeyan et al. [16] and then extend the flow analysis with variable thermophysical properties. Also, we present a comparison of the obtained numerical results with those from a previously published study [16]. The results showed that both approaches are in excellent agreement.

MATHEMATICAL FORMULATION
Let us consider the numerical investigation of MHD boundary layer flow of an incompressible nanofluid. The flow is two-dimensional that passes a stretching surface with a porous medium. The plate is stretched with a velocity of w u ax  , where a is a constant along x  direction. In addition, the fluid is flowing in the presence of a magnetic field which is supposed to be applied along the y  direction. The temperature at surface is w T , whereas w C represents the nanoparticle concentration at surface. The uniform temperature and the nanofluid volume fraction far from the surface of the sheet are T  and C  respectively, as y tends to  .
The following system of equations is incorporated for the mathematical model [16]: Where u and v are the components of velocity in the x and y directions, respectively, (see Nomenclature table). The associated boundary conditions for the above system are The radiative heat flux r q is given as [13] *4 This can be expanded with the help of Taylor series along with [15], as follows: The following process was implemented to convert the PDEs to ODEs [16]: The final dimensionless form of the governing model is [16]: The BCs have the form: The equations for the dimensional form of skin-friction coefficient, Nusselt number, and Sherwood number, respectively, are as follows:

NUMERICAL ANALYSIS
The set of non-linear coupled differential equations (13) -(15) with the conditions (16) is solved numerically. Firstly, it is noticed that the heurist infinity for the independent variable is selected as  is iteratively found using Newton's method, After finding   f  we solved equations (14) and ((15) with the initial condition (16).
Thereby, the following notations were considered.
By incorporating the above notations, a system of first order ODEs is achieved, that is stated below.

RESULTS AND DISCUSSION
Gbadeyan [16] used a six order Runge-Kutta method for the numerical solution of the discussed model. In the present survey, the shooting technique along with Adams-Moulton Method of fourth order were opted for reproducing the solution of [16]. In Tables-1 and 2, comparisons of Nusselt number and Sherwood number for different values of and are displayed. We compared the results with those of the published work of Gbadeyan [16], with a good agreement of numerical results being achieved.
As a test of the accuracy of our numerical computations in   Figure-1 shows the impact of the magnetic parameter on temperature profile. The resistive force generated due to the interaction of the conducting fluid and the magnetic field reduces the temperature at all points with a transverse compression of profiles reducing the thermal boundary layer thickness due to transverse magnetic field. The influence of the thermal radiation parameter on the profile of temperature distribution is displayed in Figure- The impact of the Prandtl number for the variation of temperature distribution is visualized in Figure-3. This figure provides the evidence that the fluid with increasing values of the Prandtl number represents a weak energy diffusion. Thus, higher values of the Prandtl number result in a strong reduction in temperature profile. Figures-(4 and 5) explain the effects of the Biot number (Bi) on temperature and concentration distribution. Increasing the value of dimensionless parameter increases temperature and concentration distribution. Physically, the Biot number is basically the ratio between resistance of the heat transfer in the body and resistance at body surface. This is because of the fact that the convective heat exchange along the surface will enhance the momentum boundary layer. Figures 6 and 7 are presented to visualize the effects of the thermophoresis parameter and Brownian motion= on temperature and concentration profiles, respectively. In these figures, it is observed that temperature and concentration profiles show an increment in response to growing values of the dimensionless parameters ( and . Physically, heated particles leave away from high temperature to low temperature so that the temperature of the fluid increases.
The results of the concentration vs. Lewis number are illustrated in Figure 8. Increasing Le corresponds to the increase in concentration. As a result, the concentration on the surface increases initially, but after a while, it starts to decrease at a short distance away from the surface. Figure 9 shows the influence of Eckert number ( on the dimensionless temperature profile. Eckert number is the ratio of the kinetic energy dissipated in the flow to the thermal energy conducted into or away from the fluid. As shown in the figure, temperature profile increases with the increase in the values of .
In Figure-10, we can see the impact of reaction rate on concentration profile. When the chemical reaction parameter increases, the boundary layer thickness decreases. This is due to the fact that the chemical reaction in this system results in consumption of the chemical and results in decrease of concentration profile.

CONCLUSION
After a thorough investigation, we have reached the following conclusions: Higher values of magmatic parameter and radiation parameter enhance the temperature profile.