Stagnation Point Flow of Nanofluid over a Moving Plate with Convective Boundary Condition and Magnetohydrodynamics

A theoretical investigation is carried out to examine the effects of volume fraction of nanoparticles, suction/injection, and convective heat and mass transfer parameters on MHD stagnation point flow of water-based nanofluids (Cu and Ag). The governing partial differential equations for the fluid flow, temperature, and concentration are reduced to a system of nonlinear ordinary differential equations. The derived similarity equations and corresponding boundary conditions are solved numerically using Runge-Kutta Fehlberg fourth-fifth order method. To exhibit the effect of the controlling parameters on the dimensionless velocity, temperature, nanoparticle volume fraction, skin friction factor, and local Nusselt and local Sherwood numbers, numerical results are presented in graphical and tabular forms. It is found that the friction factor and heat and mass transfer rates increase with magnetic field and suction/injection parameters.


Introduction
In recent years, the requirements of modern technology have encouraged interest in fluid flow studies which involve interaction of several phenomena.One such study is stagnation point flow over a permeable surface which plays an important role in many engineering problems including petroleum industries, ground water flows, extrusion of a polymer sheet from a dye, and boundary layer control.The study of a stagnation point flow towards a solid surface in moving fluid is traced back to Hiemenz in 1911.He analyzed twodimensional stagnation point flow on stationary plate using a similarity transformation.Consequently, many investigators have extended that idea to different aspects of the stagnation point flow problem.For instance, Chamkha [1] investigated the mixed convection MHD flow near the stagnation point of a vertical semi-infinite surface.Ramachandran et al. [2] studied the dual solution analysis for different range of the buoyancy parameter.Devi et al. [3] extended the work of [2] and obtained a similarity solution for unsteady case.
Grosan and Pop [4] studied axisymmetric mixed convection nanofluid flow past a vertical cylinder, Shateyi and Makinde [5] investigated MHD stagnation point flow towards a radially stretching convectively heated disk, Ibrahim et al. [6] examined MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet, and Mustafa et al. [7] reported stagnation point flow of a nanofluid towards a stretching sheet.
In modern metallurgy and metal-working process, the magnetohydrodynamic (MHD) flow of an electrically conducting fluid towards a stretching surface is significant.These processes include the fusing of metals with electrical furnace by using magnetic field and cooling the inner first wall of the nuclear reactor containment vessel where the hot plasma is isolated from the wall by applying a magnetic field.Ishak et al. [8] studied MHD mixed convection flow near the stagnation point on a vertical permeable surface.Ishak et al. [9] numerically investigated the MHD stagnation point flow over a stretching sheet by using Keller box method and observed that velocity at a point increases with strong magnetic field when the free stream velocity is greater than the stretching velocity.Furthermore, Mahapatra et al. [10] investigated MHD stagnation point flow of a non-Newtonian fluid towards a stretching sheet and observed that, for a given magnetic parameter, the skin friction increases with an increase in power law index.Some more related studies to stagnation point flow can be found in [11][12][13][14][15][16][17][18].
Choi and Eastman [19] was the first who introduced the theory of nanofluid.Since then it has been an active field of research for about two decades.Nanofluid is a single phase mixture of suspended nanometer sized solid particles and fibers in conventional base fluids.Water, ethylene glycol mixture, toluene, and so forth are commonly used as base fluids.Wang and Mujumdar [20] found that the addition of very small amount of nanoparticles to conventional heat transfer liquids enhanced the thermal conductivity of the fluid up to approximately two times.Nanofluids offer many diverse advantages in industrial applications such as microelectronics, biomedicine, transportation, fuel cell, and nuclear reactors [21].The literature is comprehensively occupied by nanofluids studies [22][23][24][25][26][27][28][29].
In this study, our main objective is to analyze the effect of volume fraction of nanoparticles on MHD stagnation point flow towards a moving surface with convective heat and mass transfer parameters.The governing boundary layer equations have been transformed to a two-point boundary value problem using similarity variables.These similarity equations were solved numerically using Runge-Kutta Fehlberg fourthfifth order method.The effects of governing parameters on the dimensionless velocity, temperature, and particle concentration as well as on local skin friction, and Nusselt and Sherwood numbers have been investigated.

Mathematical Formulation
We considered the two-dimensional, steady, and laminar boundary layer flow of water-based nanofluids containing two types of nanoparticles, Cu and Ag, over a moving flat plate.A uniform magnetic field of strength  0 was applied in a direction normal to plane y = 0. We assumed that the plate was heated by a fluid with heat transfer coefficient ℎ  and temperature   .The concentration of the fluid at the plate surface was assumed to be   while the uniform temperature and concentration far away from the plate were taken as  ∞ and  ∞ , respectively.Under these assumptions, the steady boundary layer equations governing the flow, heat, and mass transfer are The boundary conditions for the velocity components, temperature, and nanoparticle fraction are defined as → ∞: Here,  and V are the velocity components along the axes  and , respectively. ∞ is the free stream velocity,  is species diffusivity,  is electrical conductivity,  is concentration of the nanofluid, and ]  ,   ,   are the kinematic viscosity, density, and thermal diffusivity of the nanofluid, respectively, which are expressed as follows [7,29]: We introduce the following transformations: where ℎ() is the dimensionless concentration and  represents the stream function and is defined as  = /, V = −/, so that ( 1) is satisfied identically.The governing equations are reduced by using (7) as follows: 1 The transformed boundary conditions are where primes denote differentiation with respect to  and the four parameters are defined as Here,   is suction (  > 0) and injection (  < 0) parameter,  is magnetic parameter, Pr is Prandtl number, Sc is Schmidt number,   is convective heat transfer parameter, and   is convective mass transfer parameter.
The quantities of practical interest, in this study, are the local skin friction coefficient and local Nusselt and Sherwood numbers which are defined as where   is the shear stress at surface of the wall and   and   are the wall heat and mass fluxes, respectively.Using (7), we obtain where Re  =  ∞ /]  is the local Reynolds number.

Numerical Solution
Using similarity transformation, the governing equations of the problem are reduced to a system of nonlinear, coupled ordinary differential equations (8) which are solved numerically by Runge-Kutta-Fehlberg fourth-fifth order method for different values of parameters such as magnetic parameter, suction/injection parameter, nanoparticles volume fraction, and convective heat and mass transfer parameter for waterbased nanofluids.The effects of these parameters on the dimensionless velocity, temperature, and concentration as well as skin friction and the rate of heat and mass transfer are investigated.Since the physical domain of considered problem is unbounded, whereas the computational domain has to be finite, in all our computations we have used the value of  max = 10 to ensure that all numerical solutions approach the asymptotic values correctly for all values of physical quantities considered in this study.For these numerical computations, the step size and convergence criteria are chosen to be 0.001 and 10 −6 , respectively.The physical properties of the fluid, water, and the nanoparticles are given in Table 1.To validate the obtained numerical solution, comparison has been made with previously published data from the literature for skin friction in Table 2 and they are found to be in a favorable agreement.Simultaneous effects of different parameters on friction factor and Nusselt and Sherwood numbers are presented in Table 3, while other parameter values have been kept preset.From Table 3, it is clear that all physical quantities of interest, that is, skin friction and Nusselt and Sherwood numbers, are the increasing functions of  and   .

Results and Discussion
Numerical computations are carried out for several sets of values of the governing parameters using the Runge-Kutta-Fehlberg fourth-fifth order method.In order to illustrate the salient features of the model, the numerical results are presented in Figures 1-9.In this study we considered Pr = 6.2.
Figures 1(a) and 1(b) are presented to show the influence of the volume fraction of nanoparticles on the dimensionless velocity.It is observed that the momentum boundary layer thickness is smaller for Cu-water as compared to Ag-water.
In the absence of magnetic field, the dimensionless velocity, inside the boundary layer, is smaller and increases with magnetic field.It is also observed that the magnetic field reduces the boundary layer thickness in both cases.This decrease is because of Lorentz force which resists the transport phenomena.From Figures 2(a) and 2(b), it is evident that the nanofluid velocity profiles width decreases (becomes narrow) with the increasing value of nanoparticle volume fraction , and vice versa.Increasing the volume fraction of nanoparticle, on the other hand, increases the thermal conductivity of the nanofluid, and we therefore observe that the thermal boundary layer thickness becomes thicker.Figures 3(a) and 3(b) illustrate the effect of the volume fraction of nanoparticles in the absence/presence of suction parameter on the dimensionless velocity.It is noticed that the hydrodynamic boundary layer thickness is smaller for Cuwater as compared to Ag-water.In the absence of suction, the dimensionless velocity, inside the boundary layer, is smaller and increases with suction.It is also evident that the increasing value of suction reduces the boundary layer thickness in both the cases.
Figures 4(a) and 4(b) are drawn to analyze the influence of volume fraction of nanoparticles on the dimensionless temperature in both hydrodynamic and hydromagnetic flows of water-based nanofluids.It is observed that the dimensionless temperature at the wall is higher for hydrodynamic flows and lowers with an increase in the volume fraction of nanoparticles in both cases.This is due to the increase in thermal conductivity with an increase in the volume fraction of nanoparticles.It is also noticed from Figures 4(a) and 4(b) that the wall temperature decreases with an increase in the magnetic field.It is due to the fact that transverse magnetic field creates a drag (Lorentz force) which resists the flow and decreases the wall temperature.
Figures 5(a) and 5(b) are presented to show the effect of magnetic parameter and volume fraction of nanoparticles (Cu and Ag) on the dimensionless concentration.It is observed that in the absence of nanoparticles and the magnetic parameter, the dimensionless concentration is higher within the boundary layer and decreases with an increase in the magnetic field.However, no appreciable effect of both parameters on the dimensionless concentration could be observed.
The variation of skin friction with magnetic parameter, suction/injection parameter, and volume fraction of nanoparticles is presented in Figures 6 and 7 for both nanofluids.As expected, for conventional fluid ( = 0), there is no change in the skin friction of both nanofluids.However, as the volume fraction of nanoparticles increases, the skin friction also increases.This is due to increase in density of nanofluids with the volume fraction of nanoparticles.Due to higher density of Cu, the skin friction is found to be higher for Cu-water nanofluids.A monotonic increase is noticed in skin friction with suction/injection parameter.It is also important to note that the magnetic field increases the skin friction due to an additional force which opposes the velocity of nanofluids.
Figure 8 presents the variation of Nusselt number with nanoparticle volume fraction and magnetic and convection parameters.It is evident that, for both nanofluids, the local Nusselt number or the local heat transfer coefficient increases as the solid volume fraction increases.This increase in local heat transfer coefficient is the result of the increase in thermal conductivity of nanofluid caused by increase in the population of high thermal conductivity nanoparticles.Since the magnetic field reduces the surface temperature (see Figure 4), the local Nusselt number increases with magnetic field.Also the convective parameter helps in enhancing the heat transfer from the surface.This can be observed in Figure 8.It is important to note that when   → ∞, the convective boundary condition is reduced to an isothermal boundary condition.
Finally, Figure 9 shows the effects of magnetic parameter, convective mass transfer parameter, and Schmidt number for both Cu and Ag nanofluids on Sherwood number.It is seen from the figure that the mass transfer rate increases with an increase in magnetic parameter and Schmidt number.Further, it is found that the mass transfer rate is enhanced with convective mass transfer parameter.

Conclusion
The effect of volume fraction of nanoparticles, suction/ injection, convective heat, and mass transfer for MHD stagnation point of water-based nanofluids over a moving     plate is investigated numerically.The governing equations are converted into ordinary differential equations by using appropriate similarity transformations.The similarity equations are then solved by Runge-Kutta Fehlberg fourth-fifth order method.The obtained results are displayed graphically to illustrate the effect of the different physical parameters on the dimensionless velocity, temperature, and concentration as well as the local skin friction and the local Nusselt and Sherwood numbers.The main findings of the study are as follows: (i) The hydrodynamic boundary layer decreases with magnetic and suction parameters.
(ii) The skin friction factor monotonically increases with suction parameter.
(iii) The local Nusselt and Sherwood numbers are the increasing functions of magnetic and suction parameters.

Figure 7 :
Figure 7: Variation of skin friction with volume fraction of nanoparticles.

Figure 8 :
Figure 8: Variation of dimensionless heat transfer rate with   , , and .

Figure 9 :
Figure 9: Variation of dimensionless mass transfer rate with   , , and .

Table 2 :
Comparison of skin friction for different values of  at  =   = 0.

Table 3 :
Values related to the reduced skin friction coefficient and reduced Nusselt and Sherwood numbers for different values of the governing parameters (for Cu-water).