Heat transfer on magnetohydrodynamic (MHD) stagnation - point flow in nanofluid past a porous shrinking/stretching sheet with variable stream condition in the presence of blowing at the surface

This paper deals with the steady two-dimensional magnetohydrodynamic (MHD) stagnation-point flow of nanofluid and heat transfer with thermal radiation past a stretching/shrinking sheet in the presence of injection are investigated numerically and simulated with Maple 18 software. Brownian motion and thermophoresis are included in employed model of nanofluid simultaneously. The system of nonlinear ordinary differential equations governing the flow is converted into a system of coupled higher order nonlinear ordinary differential equation by using similarity transformations. The results show that the flow field velocity, temperature and concentration profiles are influenced appreciably by the applied magnetic field, Brownian motion, heat radiation, porous parameter and thermophoresis particle deposition. 
 
   
 
 Key words: Magnetohydrodynamics, injection, stagnation point flow, heat transfer, Brownian motion, thermophoresis.


INTRODUCTION
In past few years, an innovative way of improving the heat transfer and thermal conductivities using an electrically conducting fluid has attracted reasonable attention in many fields of science and technology because of its wide applications.Nanofluids are the suspension of metallic, non-metallic or polymeric nanosized powders in base liquid which are employed to enhance the heat transfer rate in various applications.Nanofluid was first introduced by Choi (1995), refers to the fluids with suspended nanoparticles.Most of the conventional heat transfer fluids such as water, ethylene glycol and mineral oils have low thermal conductivity and thus are inadequate to attend the requirements of today's cooling rate.Nanofluid are emerged its applications in medical science, biomedical industry, power generation in nuclear reactors, hybrid-powered engines, engine cooling, vehicle thermal management, domestic refrigerator, heat exchanger, nuclear reactor coolant (Ahmadreza, 2013).
Stagnation region is a point of conduct where the flow J. Mech.Eng.Res.
regarding a body and the fluid particles has zero velocity with respect to the body.The fluid motion near the stagnation region is determined by stagnation flow.The Bernoulli equation presents that the static pressure is highest when the velocity is zero and therefore static pressure is at its maximum value at stagnation points.
Along with static pressure, stagnation region encounters the strongest heat transfer and rate of mass decomposition.Chiam (2001) investigated flow around the stagnation point over a stretching sheet.Mahapatra and Gupta (2001) studied the stagnation-point flow towards a stretching sheet taking various stretching and stagnation flow velocities.They concluded that two different types of boundary layer are formed near the stretching sheet.Layek et al. (2007) analyzed the heat and mass transfer for boundary layer stagnation point flow towards a stretching sheet subjected to heat absorption/ generation.Nadeem et al. (2010) defined solutions for boundary layer flow in the region of the stagnation point towards a stretching sheet.Nowadays large numbers of studies are going on boundary layer stagnation flow over a stretching sheet.
Compared to stretching sheet, only small amount of works has been done on the flow over a shrinking sheet.There are two conditions that the flow towards a shrinking sheet is likely to exist, whether an adequate suction/injection on the boundary is imposed or a stagnation flow is considered (Miklavčič and Wang, 2006).Wang (2008) investigated that the steady flow over a shrinking sheet is not occurred because the vorticity generated due to the shrinking sheet is not confined inside the boundary layer.Recently, Faraz et al. (2011) analyzed that the two-dimensional viscous flow over a shrinking sheet by employing an analytical approach while Bhattacharyya and Layek (2011) considered that the boundary layer stagnation point flow towards a shrinking sheet with thermal radiation and it is predicted that the solutions exist for a certain range of the shrinking parameter.
All the studies refer to the stagnation point flow towards stretching/shrinking sheet in a viscous and Newtonian fluid.To the author's knowledge, a few studies have been communicated and compared stretching/shrinking sheets with regard to magnetohydrodynamic boundary layer stagnation point flow past a stretching/shrinking sheet.The objective of the current study is therefore compared with the effects of magnetic field, Brownian motion, thermophoresis and thermal radiation on the relevant nanofluid flow past a stretching/shrinking sheet in the presence of injection.

FLOW ANALYSIS
Consider the steady two-dimensional MHD stagnation-point flow of an incompressible viscous electrically conducting nanofluid impinging normally on porous stretching/shrinking surface.The fluid is subjected to a uniform transverse magnetic field of strength B0. Figure 1 illustrates the physical model and the coordinate system, where the x and y axes are measured along the surface of the sheet and normal to it, respectively.
It is assumed that the velocity of the stretching/shrinking surface is cx x u w  ) ( and the velocity outside boundary layer is ) ( , a and c are constants with 0  a .We note that 0  c and 0  c correspond to stretching and shrinking surface, (Khan and Pop, 2010) respectively.The governing boundary layer equations for steady two-dimensional laminar nanofluid flow past a stretching/shrinking sheet and subjected thermal radiation can be written as: In Equation ( 2), we have neglected the induced magnetic field since the magnetic Reynolds number for the flow is considered to be very small.This assumption is justified for flow of electrically conducting fluids such as liquid metals e.g., mercury, liquid sodium etc.Let u and v are the velocity components along the x and y directions, By Rosseland approximation (Samir Kumar Nandy, Ioan Pop, 2014), the radiative heat transfer as:  is the Stefan-Boltzmann constant and 1 k is the mean absorption T can be expanded in a Taylor series about  T and neglecting higher order terms, we get . Hence: The stream function and the dimensionless variables are defined as: Therefore the equations of ( 1) -( 4) become: -modified Grashof number.
Equation ( 10) presents that the temperature actually does not depend on (Pr) and ) (R independently (Fang and Zhang, 2010), but depend only on a combination of them which is the effective Prandtl number . Equuatin (10) can be written as:  is the ratio of the rates of stretching/shrinking velocity and the free stream velocity.
The physical quantities of interest are the skin friction, the local Nusselt number and the local Sherwood number which are defined as: (14) w  is the shear stress along the stretching / shrinking surface, and m q are the wall heat and mass fluxes, respectively.Hence: is the local Reynolds number.

RESULTS AND DISCUSSION
The system that coupled Equations ( 9) to (11) are highly nonlinear and it cannot be solved analytically and numerical solutions subject to the boundary conditions (Fang and Zhang, 2010) are obtained using the computer algebra software Maple 18.This software utilizes a fourth-fifth order Runge-Kutta-Fehlberg method as default to solve boundary value problems numerically using the desolve command and the Maple worksheet is listed in Appendix A. The numerical results are illustrated in the form of dimensionless velocity, temperature and concentration.Numerical computations are carried out for the governing parameters, namely, magnetic parameter ( ), thermal radiation parameter ( ), effective Prandtl number ( ), Brownian motion parameter ( ), thermophoresis parameter ( ), Lewis number ( ),  Wang (2008).The results depict a good agreement with their results since the errors are found to be very less.This may be due to the fact that we have utilized Runge-Kutta-Fehlberg method (Maple 18 software) which has fifth-order accuracy.Thus, the present results are more accurate than their results.
It is also observed from Figure 2 The effect of porosity and magnetic strength on velocity profiles in the presence of stretching and shrinking surface with uniform Brownian motion and injection are shown in Figures 3 and 4. Due to the combined effect of permeability of the porous medium and kinematic viscosity of the nanofluid, it is shown that the velocity of the nanofluid increases with increase of and M for both stretching/shrinking sheet.The lorentz force acts in the opposite direction to the flow of nanofluid, opposing the motion of the nanofluid but injection provides an additional effect to the flow.In the presence of stretching/shrinking surface, there is peak formation of velocity profiles at / for and / for respectively.It is interesting to note that the momentum boundary layer for M is higher than that of due to the combined effect of density and kinematics viscosity of the nanofluid.
The effects of Grashof number with injection on the velocity profile for selected values of parameters are shown in Figure 5.As the parameter increase, the velocity of nanofluid firstly increases for both stretching/shrinking surface /( ) and then decreases ( )/ , respectively.In the presence of stretching/shrinking surface, there is a peak formation of velocity profiles occurs at / .It is interesting to ` Balachandar et al. 27   note that the momentum boundary layer firstly thick and then thin with increase of Grashof number due to effect of buoyancy to viscous force acting on a nanofluid.Figure 6 presents the influence of Lewis number on nanoparticle volume fraction profiles in the presence of injection of the surface.In the presence of shrinking sheet, it is observed that the nanoparticle volume fraction profile firstly increases and then decreases ( ) with increase of Lewis number because of thermal diffusivity to mass diffusivity of the nanofluid.In the case of stretching sheet, it is seen that the nanoparticle volume fraction decreases with increase of Lewis number.
The effect of Brownian motion on the temperature and nanoparticle volume fraction profiles with and S are shown in Figure 7(a)-(b).In the presence of stretching and shrinking surface, it is predicted that temperature of the nanofluid increases whereas the nanoparticle volume fraction firstly increases /( ) and then decreases ( )/ with the increase of N b .Both stretching and shrinking sheet, it is also observed that the thermal boundary layer thickness for shrinking sheet is stronger than that of stretching sheet with the increase of N b .The physical phenomenon behind this, is that the increased Brownian motion increases the thickness of thermal and diffusion boundary layer and then decreases the thickness of the diffusion boundary layer, which ultimately enhances the temperature and nanoparticle volume fraction firstly and then decelerates the nanoparticle volume fraction.Nanoparticle volume fraction profiles goes up and down because of the injection at the surface provides an alternative force in the flow region and its affect on surrounding liquids, plays a dominant role on heat and mass transfer.
Figure 8(a) and (b) depict the characteristic temperature and nanoparticle volume fraction profiles for different values of the of thermophoresis parameter ( ).It is predicted that the temperature decreases for stretching sheet and increases for shrinking sheet whereas the thermal boundary layer thickness for shrinking sheet is stronger than that of stretching sheet with increase of It is also observed that the nanoparticle volume fraction increases for stretching sheet while it first decreases and then increases ( with increase of It is interesting to note that the nanoparticle volume fraction profiles goes up and down because of the injection at the shrinking surface.This means that the nanoparticle volume fraction near the surface is stronger than that of the nanoparticle volume fraction at the surface and consequently, the nanoparticles are  expected to transfer to the surface due to the size of the nanoparticles and the steepness of the temperature gradient, the heat conductivity and heat absorption of the nanoparticles play a dominant role on fluid.firstly increases and then decreases ( ) for shrinking surface with increase of R .It is important to note that the nanoparticle volume fraction profiles goes up and down because of blowing at the shrinking surface.It is interesting to note that the increase of thermal radiation for shrinking sheet plays a dominant role on temperature and nanoparticle volume fraction profiles compared to that of stretching sheet.Due to the effect of injection, shrinking surface exerts a force in flow region over nanoparticle volume fraction which results in decrease of diffusion boundary layer ( ) as temperature increases.Temperature and nanoparticle volume fraction is dependable on injection.
Figure 10(a) and (c) observe the influence of the injection S on velocity, temperature and nanoparticle volume fraction profiles in the boundary layer with uniform Brownian motion of the nanofluid.It is shown that the velocity of the nanofluid firstly decreases for both stretching/shrinking surface /( ) and then increases ( )/ with decrease of injection parameter.The temperature and nanoparticle volume fraction increase with decrease of injection strength .The physical explanation for such behavior is interesting to note that the heated fluid is blown towards the stretching/shrinking surface where the buoyancy forces can act to retard the fluid due to high influence of the viscosity of the nanofluid.It is interesting to observe that the nanoparticle volume fraction profiles moves up and down with decrease of injection at the shrinking sheet due to the joined effects of kinematic viscosity and injection at the shrinking surface.

Conclusions
The effects of various governing parameters viz.magnetic, Brownian motion, thermophoresis, thermal radiation parameters, Grashof and Lewis numbers on flow field and heat transfer characteristics of the stagnation point flow of a nanofluid towards a porous stretching and shrinking sheet in the presence of injection is investigated.The effects of these parameters on the dimensionless velocity, temperature and nanoparticle volume fraction can be summarized as follows: i.It is interesting to note that the velocity of the nanofluid increases with increase of the magnetic and porosity strength at the stretching/shrinking sheet and the momentum boundary layer for magnetic field is stronger than that of porosity due to the combined effect of density and kinematics viscosity of the nanofluid.ii.Momentum boundary layer firstly thick and then thin with increase of Grashof number due to effect of buoyancy to viscous force acting on a nanofluid.iii.Nanoparticle volume fraction profile firstly increases and then decreases with increase of Lewis number because of thermal diffusivity to mass diffusivity of the nanofluid.iv.Thermal and diffusion boundary layer thickness for shrinking sheet is stronger than that of stretching sheet J. Mech.Eng.Res. with increase of Brownian motion and thermophoresis particle deposition.v. Nanoparticle volume fraction profiles goes up and down with increase of and because the injection at the surface provides an alternative force in the flow region and its affect on surrounding fluids, plays a dominant role on heat transfer and nanoparticle volume fraction of the nanofluid.vi.Increase of thermal radiation plays a dominant role on temperature and nanoparticle volume fraction for shrinking sheet compared to that of stretching sheet.
It is concluded that the flow and heat transfer properties of MHD convection boundary layer flow in the stagnationpoint region can be controlled by Brownian motion and thermophoresis of the nanofluid in the presence of injection at the surface.
stream velocity, k is the permeability of the porous medium, T is the fluid temperature and C is the nanoparticle volume fraction,  is the electrical conductivity of the fluid,  is the kinematic viscosity, m  is the thermal diffusivity, f  is the density of the base fluid, B D is the Brownian diffusion coefficient, T D is the thermophoresis diffusion coefficient, p c is the specific heat at constant pressure, r q is the radiative heat flux and  is the ratio of the effective heat capacity of the nanoparticle material to the heat capacity of the ordinary fluid.It is considered that the temperature w T and the nanoparticle volume fraction w C are constant at the surface and the ambient temperature  T and the nanoparticle volume fraction  C attain to constant values.The boundary conditions (for stretching/shrinking surface):

Figure 1 .
Figure 1.Physical model and the coordinate system.
(a) -(b) that the agreement with the numerical solution of and profiles for different values of Brownian motion ( ) exactly correlates with the first solution of Samir Kumar Nandy and Pop (2014).

Figure 3 .
Figure 3. Impact of porosity parameter on velocity profiles.

Figure 5 .
Figure 5. Role of Grashof number on velocity profiles.

Figure 6 .
Figure 6.Lewis number on nanoparticle volume fraction.
Figure 9(a) and (b) present the temperature and nanoparticle volume fraction profiles for different values of the thermal radiation parameter R in the presence of blowing at the stretching/shrinking surface.The temperature of stretching/shrinking surface firstly decreases /( ) and then increases ( )/ with increase of R whereas the nanoparticle volume fraction firstly decreases and then increases ( ) for stretching surface while ` 30 J. Mech.Eng.Res.

Table 1 .
Comparison of the present results for