MHD Mixed Convective Boundary Layer Flow of a Nanofluid through a Porous Medium due to an Exponentially Stretching Sheet

Magnetohydrodynamic MHD boundary layer flow of a nanofluid over an exponentially stretching sheet was studied. The governing boundary layer equations are reduced into ordinary differential equations by a similarity transformation. The transformed equations are solved numerically using the Nactsheim-Swigert shooting technique together with Runge-Kutta sixorder iteration schemes. The effects of the governing parameters on the flow field and heat transfer characteristics were obtained and discussed. The numerical solutions for the wall skin friction coefficient, the heat and mass transfer coefficient, and the velocity, temperature, and concentration profiles are computed, analyzed, and discussed graphically. Comparison with previously published work is performed and excellent agreement is observed.


Introduction
The study of Magnetohydrodynamics MHD boundary layer flow on a continuous stretching sheet has attracted considerable attention during the last few decades due to its numerous applications in industrial manufacturing processes such as the aerodynamic extrusion of plastic sheets, liquid film, hot rolling, wire drawing, glass fiber and paper production, drawing of plastic films, metal, and polymer extrusion, and metal spinning. Crane 1 was the first to consider the boundary layer flow caused by a stretching sheet which moves with a velocity varying linearly with the distance from a fixed point. The heat transfer aspect of this problem was investigated by Carragher and Crane 2 under the conditions when the temperature difference between the surface and the ambient fluid is proportional 2 Mathematical Problems in Engineering to a power of the distance from a fixed point. After this pioneering work, the flow field over an exponentially stretching surface has drawn considerable attention and a good amount of literature has been generated on this problem. Magyari and Keller 3 analyzed the steady boundary layers on an exponentially stretching continuous surface with an exponential temperature distribution. Partha et al. 4 investigated the effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface. Sajid and Hayat 5 studied the influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet by solving the problem analytically via homotopy analysis method HAM . The study of magnetohydrodynamic MHD has important applications, and may be used to deal with problems such as cooling of nuclear reactors by liquid sodium and induction flow meter, which depends on the potential difference in the fluid in the direction perpendicular to the motion and to the magnetic field 6 . At high operating temperature, radiation effect can be quite significant. Many processes in engineering areas occur at high temperatures and knowledge of radiation heat transfer becomes very important for the design of pertinent equipment 7 . Elbashbeshy 8 added new dimension to the study on exponentially continuous stretching surface. Khan 9 and Sanjayanand and Khan 10 studied the viscous-elastic boundary layer flow and heat transfer due to an exponentially stretching sheet. The numerical simulation of boundary layer flow over an exponentially stretching sheet with thermal radiation was given by Bidin and Nazar 11 . Wang 12 studied the free convection on a vertical stretching surface, also Reddy Gorla and Sidawi 13 investigated the free convection on a vertical stretching surface with suction and blowing.
Dissipation is the process of converting mechanical energy of downward-flowing water into thermal and acoustical energy. Vajravelu and Hadjinicolaou 14 analyzed the heat transfer characteristics over a stretching surface with viscous dissipation in the presence of internal heat generation or absorption. Convective boundary layer flow has wide applications in engineering as postaccidental heat removal in nuclear reactors, solar collectors, drying processes, heat exchangers, geothermal and oil recovery, building construction, and so forth. Cheng and Minkowycz 15 also studied free convection from a vertical flat plate with applications to heat transfer from a dick. Gorla et al. 16,17 solved the nonsimilar problem of free convective heat transfer from a vertical plate embedded in a saturated porous medium with an arbitrary varying surface temperature.
The term nanofluid refers to these kinds of fluids by suspending nanometer sized metallic particles in common fluids and reported that their highly enhanced thermal properties has been introduced by Choi 18 . Kang et al. 19 investigated the estimation of thermal conductivity of nanofluid using experimental effective particle volume. Abu-Nada et al. 20, 21 have showed application of nanofluids for heat transfer and also analyzed effects of inclination angle on natural convection in enclosures filled with Cu-water nanofluid. Wang and Mujumdar 22, 23 studied heat transfer characteristics of nanofluids including theoretical and numerical investigations. Ghasemi and Aminossa dati 24 have showed the periodic natural convection in a nanofluid filled enclosure with oscillating heat flux.
The study of convective instability and heat transfer characteristics of the nanofluids was considered by Kim et al. 25 . Jang and Choi 26 obtained nanofluids thermal conductivity and various parameters effect this study. The natural convective boundary layer flows of a nanofluid past a vertical plate have been described by Neild and Kuznestov 27 and Kuznestov and Neild 28 . In this model, Brownian motion and Thermophoresis are accounted with the simplest possible boundary conditions. They also studied Cheng-Minkowycz problem for natural convective boundary-layer flow in a porous medium saturated by a nanofluid. Owing to applications in science and engineering, Brownian motion The present paper studies the problem of MHD mixed convective boundary layer of a nanofluid flow over an exponentially stretching sheet. The governing equations are transformed into nonlinear coupled ordinary differential equations which depends on the combined porous and magnetic parameter R , thermal convective parameter λ T , Mass convective parameter λ M , viscosity ratio parameter ∧ , dimensionless inertia parameter ∇ , Prandtl number P r , Eckert number E c , Lewis number L e , Brownian motion parameter N b , and Thermophoresis parameter N t . The obtained nonlinear coupled ordinary differential equations are solved numerically using Nactsheim-Swigert 44 shooting iteration technique together with Runge-Kutta six-order iteration schemes. The velocity, temperature, and concentration distributions are discussed and presented graphically, and also the skin-friction coefficient, the surface heat, and mass transfer rate at the sheet are investigated.

Mathematical Formulation
Consider a steady two-dimensional flow of an incompressible viscous and electrically conducting nanofluid caused by a stretching sheet, which is placed in a quiescent ambient fluid of uniform temperature of the plate and species concentration are raised to T w > T ∞ and C w > C ∞ , respectively, which are thereafter maintained constant, where T w , C w are temperature and species concentration at the wall and T ∞ , C ∞ are temperature and species concentration far away from the plate, respectively. The x-axis is taken along the stretching sheet in the direction of the motion and y-axis is perpendicular to it. Consider that a variable magnetic field B x is applied normal to the sheet and that the induced magnetic field is neglected, which is justified for MHD flow at small magnetic Reynolds number. The sketch of the physical configuration and coordinate system are shown in Figure 1.
Under the above assumptions and usual boundary layer approximation, the MHD free convective nanofluid flow and heat and mass transfer are governed by the following see Nield and Kuznetsov 27 and Kuznetsov and Nield 28 : where u and v are the velocities in the x-and y-directions, respectively, ρ is the fluid density, υ the kinematic viscosity, υ the reference kinematic viscosity, K the variable thermal conductivity, c p the specific heat at constant pressure, T and C the fluid temperature and concentration in the boundary layer, c * ε 2 is the inertia parameter, α is the thermal diffusivity, D B is the Brownian diffusion coefficient, and D T is the thermophoresis diffusion coefficient. The boundary condition for the model is where u • is the reference velocity, T • , C • the reference temperature and concentration, respectively, and L is the reference length. To obtain similarity solutions, it is assumed that the magnetic field B x is of the form where B • is the constant magnetic field. Also the variable thermal conductivity can be taken as the form Mathematical Problems in Engineering 5 In order to attains a similarity solution to 2 with the boundary conditions 2.2 , the following dimensionless variables are used:

2.5
From the above transformations, the nondimensional, nonlinear, and coupled ordinary differential equations are obtained as where the notation primes denote differentiation with respect to η and the parameters are defined as combined porous and magnetic parameter , Lewis number .

2.7
The transformed boundary conditions

2.8
The physical quantities of the skin-friction coefficient, the reduced Nusselt number, and reduced Sherwood number are calculated, respectively, by the following: where R e x xu • e x/L /ν is the local Reynolds number.

Numerical Technique
The system of nondimensional, nonlinear, and coupled ordinary differential equations 2.6 with boundary condition 2.8 are solved numerically using standard initially value solver the shooting method. For the purpose of this method, the Nactsheim-Swigert shooting iteration technique 44 together with Runge-Kutta six order iteration scheme is taken and determines the temperature and concentration as a function of the coordinate η. Extension of the iteration shell to above equation system of differential 2.8 is straightforward; there are three asymptotic boundary condition and hence three unknown surface conditions f 0 , θ 0 and ϕ 0 .   shows a favorable agreement, as presented in Table 1.Therefore, the present results are very accurate. The physical representation of the present study is shown in Figures 2-21.         Figure 11 shows the dimensionless temperature distribution θ η for different values of E c where λ T 5.0, λ M 2.0, R 4.0, ∇ 1.0, ∧ 1.5, P r 1.0, L e 5.0, N t 0.1, and N b 0.1. It shows that the thermal boundary layer increases gradually as E c is increased. Figure 12 shows   Figure 6: Effect of E c on velocity profiles.       Figure 12: Effect of ∧ on concentration profiles.    0.1. It shows that the concentration boundary layer decreases gradually as E c is increased.

Results and Discussions
Since the physical interest of the problem, the skin-friction coefficient −f , the Nusselt number −θ at the sheet and the Sherwood number −ϕ at the sheet are plotted against Brownian motion parameter N b and illustrated in Figures 17-21. Figure 17 illustrates

Conclusions
The effect of radiation on steady MHD mixed convection boundary layer flow of a nanofluid over an exponentially stretching sheet was investigated. The results are presented for the effect of various parameters. The velocity, temperature, and concentration effects as well as the skin-friction coefficient, Nusselt number, and Sherwood number effects along the exponential sheet are studied and shown graphically. However, the comparison with previously published work 11 is performed and excellent agreement is observed. From the present study the important findings are listed below. Nomenclature B 0 : Magnetic induction C: Nanoparticle concentration C • : Reference concentration C w : Nanoparticle concentration at stretching surface C ∞ : Ambient nanoparticle concentration as y tends to infinity C f : Skin-friction coefficient c p : Specific heat capacity D B : Brownian diffusion coefficient D T : Thermophoresis diffusion coefficient g: Acceleration due to gravity G r : Grashof number G m : Modified Grashof number L e : Lewis number L: Reference length N u : Nusselt number N b : Brownian motion parameter N t : Thermophoresis parameter P : Fluid pressure P r : Prandtl number R: Combined porous and magnetic parameter R e : Local Reynolds number S h : Sherwood number T : Fluid temperature T • : Reference temperature T w : Temperature at the stretching surface T ∞ : Ambient temperature as y tends to infinity u, v: Velocity components along x-and y-axes, respectively u • : Reference velocity x, y: Cartesian coordinates measured along stretching surface.