Numerical Analysis of Three‐Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary Conditions through a Porous Medium Numerical Analysis of Three-Dimensional MHD Nanofluid Flow over a Stretching Sheet with Convective Boundary Conditions through a Porous Medium

Numerical analysis has been carried out on the problem of three-dimensional magneto- hydrodynamic boundary layer flow of a nanofluid over a stretching sheet with convictive boundary conditions through a porous medium. Suitable similarity trans- formations were used to transform the governing partial differential equations into a system of ordinary differential equations. We then solved the resultant ordinary differ- ential equation by using the spectral relaxation method. Effects of the dimensionless parameters on velocity, temperature and concentration profiles together with the friction coefficients, Nusselt and Sherwood numbers were discussed with the assistance of graphs and tables. The velocity was found to decrease with increasing values of the magnetic, stretching and permeability parameters. The local temperature was observed to rise as the Brownian motion, thermophoresis and Biot numbers increased. The con- centration profiles diminish with increasing values of the Lewis number and chemical reaction parameter.


Introduction
Many researchers have over the past few years paid significant attention to the study of boundary layer flow heat and mass transfer over a stretching sheet due to its industrial and engineering applications. These applications include cooling of papers, glass-fibre production, plastic sheets and polymer extrusion, hot rolling wire drawing, metal spinning, stretching of rubber sheets and crystal growing. The quality and final product formation in these processes are dependent on the rate of stretching and cooling.
Since the pioneering study by Crane [1] who presented an exact analytical solution for the steady two-dimensional flow due to a stretching surface in a quiescent fluid many studies on stretched surfaces have been done [1][2][3][4][5].
Thermal conductivity of nanoparticles has been shown in recent research on nanofluid to change the fluid characteristics. The thermal conductivity of the base liquid with the enhanced conductivity of nanofluid and the turbulence induced by their motion contribute to a remarkable improvement in the convective heat transfer coefficient. This feature of nanofluid makes them attractive to a wide variety of industries, ranging from transportation to energy production and supply to electronics. They can be used in welding equipment, high heat flux and to cool car engines, among other applications. Many researchers [6][7][8][9][10] have studied the boundary layer flow of a nanofluid caused by a stretching surface.
Shateyi and Prakash [11] carried out a numerical analysis on the problem of magneto hydrodynamic boundary layer flow of a nanofluid over a moving surface in the presence of thermal radiation. Kuznetsov and Nield [12] examined the influence of nanoparticles on natural convection boundary layer flow past a vertical plate, using a model in which Brownian motion and thermophoresis are accounted for. Aziz and Khan [13] investigated using a similarity analysis of the transport equations by their numerical computations to the natural convective flow of a nanofluid over a convectively heated vertical plate.
Makinde and Aziz [14] numerically studied the boundary layer flow induced in a nanofluid due to a linearly stretching sheet. Hayat et al. [15] addressed the MHD flow of second grade nanofluid over a nonlinear stretching sheet. Zhao et al. [16] studied the three-dimensional nanofluid bio-convection near a stagnation attachment. Sheikholeslami and Ganji [17] studied two-dimensional laminar-forced convection nanofluids over a stretching surface in a porous medium. The study used different models of nanofluid based on different formulas for thermal conductivity and dynamic viscosity. Nayak et al. [18] did a numerical study on the mixed convection of copper-water nanofluid inside a differentially heated skew enclosure. Recently, Mabood and Das [19] analysed MHD flow and melting heat transfer of a nanofluid over a stretching surface. Naramgari and Sulochana [20] analysed the momentum and heat transfer of MHD nanofluid embedded with conducting dust particles past a stretching surface in the presence of volume fraction of dust particles. Sandeep et al. [21] analysed the unsteady MHD radiative flow and heat transfer characteristics of a dusty nanofluid over an exponentially permeable stretching in the presence of volume fraction of dust and nanoparticles. Sheikholeslami et al. [22] computationally investigated nanofluid flow and heat transfer in a square heated rectangular body. Sheikholeslami and Ganji [23] provided a review of researches on nanofluid flow and heat transfer via semi-analytical and numerical methods. Lastly, Naramgari and Sulochana [20] analysed the three-dimensional MHD Newtonian and non-Newtonian fluid flow over a stretching surface in the presence of thermophoresis and Brownian motion.
The main objective of this chapter is to numerically analyse the influence of convective boundary conditions on the model of three-dimensional magnetohydrodynamic, nanofluid flow over a stretching sheet through a porous medium in the presence of thermophoresis and Brownian motion as well as thermal radiation. The governing partial differential equations use suitable similarity transformations. The transformed governing equations are solved numerically using the spectral relaxation method (SRM). The effects of dimensionless parameters on velocity components, temperature and concentration profiles together with the skin friction coefficients, local Nusselt and Sherwood numbers are discussed with the aid of tables and graphs.

Mathematical formulation
We consider a three-dimensional steady incompressible MHD nanofluid flow, heat and mass transfer over a linearly stretching sheet through a porous medium. The sheet is assumed to be stretched along the xy -plane while the fluid is placed along the z-axis. A uniform magnetic field B 0 is applied normally to the stretched sheet and the induced magnetic field is neglected by assuming very small Reynolds number. We assume that the sheet is stretched with linear velocities u ¼ ax and v ¼ by along the xy-plane, respectively, with constants a and b. Under the above assumptions and the boundary approximation, the governing equations for the current study are given by: where u, v and w are the velocity components in the x, y and z-directions, respectively, T is the fluid temperature, C is the fluid concentration, k 1 is the permeability, v is kinematic viscosity, ρ is the fluid density, τ is the ratio of the heat capacitances, D B and D T are the Brownian motion and thermopheric diffusion coefficients and c p is the specific heat capacity.
The corresponding boundary conditions for the flow model are: We have h f as the convective heat transfer coefficient, h s is the convective mass transfer coefficient and T f and C f are the convective fluid temperature and concentration below the moving sheet.

Similarity transformation
In order to non-dimensionalise the governing equations, we introduce the following similarity equations [24]. These transformations also transform the partial differential equations into a system of ordinary differential equations which is then solved using the spectral relaxation method: Upon substituting the similarity variables into Eqs.
(2)-(5), we obtain the following system of ordinary equations The corresponding boundary conditions are f 0 ð∞Þ ! 0, g 0 ð∞Þ ! 0, θð∞Þ ! 0, ∅ð∞Þ ! 0: Primes denote differentiation with respect to η and parameters appearing in Eqs (9)-(14) are defined as: are the Biot numbers and c ¼ b=a is the stretching parameter. The quantities of engineering interest are the skin-friction coefficient C f along the x-and y-direction (C f x and C f y ), the Nusselt number and Sherwood number. These quantities are defined as follows: where τ wx , τ wy are the wall shear along x-and y-directions, respectively, and q w and q m are the heat flux and mass flux at the surface, respectively.Upon using the similarity variables into the above expressions, we obtain the following:

Method
To solve the set of ordinary differential Eqs. (9)-(12) together with the boundary conditions (13) and (14), we employ the Chebyshev pseudo-spectral method known as spectral relaxation method. This is a recently developed method, and the details of the method are found in Motsa et al. [25]. This method transforms sets of non-linear ordinary differential into sets of linear ordinary differential equations. The entire computational procedure is implemented using a program written in MATLAB computer language. The nanofluid velocity, temperature, the local skin-friction coefficient and the local Nusselt and Sherwood numbers are determined from these numerical computations.
To apply the SRM to the non-linear ordinary differential equations, we first set f 0 ðηÞ ¼ pðηÞ and g 0 ðηÞ ¼ qðηÞ. We then write the equations as follows: The boundary conditions become In view of the SRM, we then obtain the following iterative scheme: The above equations form a system of linear decoupled equations which can be solved iteratively for r ¼ 1, 2, …. Starting from initial guesses p 0 ðηÞ, q 0 ðηÞ, θðηÞ, ∅ðηÞ . Applying the Chebyshev pseudo-spectral method to the above equations, we obtain where I is the identity matrix of size ðN þ 1Þ ðN þ 1Þ. The initial guesses are obtained as:

Results and discussion
The system of ordinary differential Eqs. (9)-(12) subject to the boundary conditions (13) and (14) is numerically solved by applying the spectral relaxation method. The SRM results presented in this chapter were obtained using N ¼ 40 collocation points, and also the convergence was achieved after as few as six iterations. We also use these default values for the parameters Pr ¼ 0:71,  with increasing values of the thermophoresis parameter. Lastly, Table 2 shows the influence of the Biot numbers on the heat transfer and mass transfer rates and they both increase with increasing values of the thermal Biot number. But we noticed opposite effects when the solutal Biot number increases. Figures 1 and 2 display the effect of permeability parameter on the velocity profiles. We observe that the tangential velocity profiles decrease as the values of the permeability parameter. Also, the transverse velocity (f 0 ðηÞ) is reduced by the increasing values of the permeability parameter as more nanofluid is taken away from the boundary layer. This explains the thinning of the velocity boundary layers as the values of K increases (Figure 1). Figures 3 and 4 depict the effect of the magnetic field parameter on the velocity profiles. As expected, we observe that both velocity components are greatly reduced as the values of the magnetic parameter increase. This is because physically increasing the values of magnetic field strength produces a drag-like force known as the Lorentz force. This force acts against the flow when the magnetic field is applied in the normal direction, as in this chapter. Figures 5 and 6 display the influence of the stretching parameter on the velocity fields. It is seen from Figure 5              the nanofluid increases with increases values of the permeability parameter. From Figure 14, we observe that the temperature profiles increase with the increasing values of the magnetic field parameter. Figure 15 displays the effect of thermal radiation parameter R on the     temperature profiles. We observe in this figure that increasing the values of the thermal radiation produces a significant reduction in the thermal condition of the fluid flow.
Lastly, the effect of the Lewis number on the concentration profiles is depicted on Figure 16.
Large values of the Lewis number implies increased values of the Schmidt number which results in the thinning of the solutal boundary layer.

Conclusion
A three-dimensional magnetohydrodynamic nanofluid, heat and mass transfer over a stretching surface with convective boundary conditions through a porous medium. The transformed governing equations are solved numerically using the spectral relaxation method. The accuracy of the SRM was validated against the MATLAB in-built bvp4c routine for solving boundary value problems. The following conclusions are driven from this study: • The effect of increasing the magnetic field parameter is to reduce the momentum boundary layer there and to increase the thermal and solutal boundary layer thickness. The same effect on the flow characteristics is also experienced by increasing values of the stretching parameter (c).
• We observed that the local temperature rises as the Brownian motion, thermophoresis, permeability parameter and Biot numbers intensify. But opposite influences are observed when the values of the thermal radiation and stretching parameters increase. • Increasing values of the Lewis number (Le) dimishes the concentration of the nanoparticles.
• The rise in the stretching ratio parameter increases the Nusselt and Sherwood number.
• Lastly, the Nusselt number decreases, while the Sherwood number increases as the Brownian motion and thermophoresis effects increase.

Pr
Prandtl number q w , q m heat and mass fluxes at the surface