Magnetohydrodynamic Boundary Layer Flow of Nanofluid over an Exponentially Stretching Permeable Sheet

A mathematical model of the steady boundary layer flow of nanofluid due to an exponentially permeable stretching sheet with external magnetic field is presented. In the model, the effects of Brownian motion and thermophoresis on heat transfer and nanoparticle volume friction are considered. Using shooting technique with fourth-order Runge-Kutta method the transformed equations are solved. The study reveals that the governing parameters, namely, the magnetic parameter, the wall mass transfer parameter, the Prandtl number, the Lewis number, Brownianmotion parameter, and thermophoresis parameter, have major effects on the flowfield, the heat transfer, and the nanoparticle volume fraction.Themagnetic fieldmakes enhancement in temperature and nanoparticle volume fraction, whereas the wall mass transfer through the porous sheet causes reduction of both. For the Brownian motion, the temperature increases and the nanoparticle volume fraction decreases. Heat transfer rate becomes low with increase of Lewis number. For thermophoresis effect, the thermal boundary layer thickness becomes larger.


Introduction
The term "nanofluid" was proposed by Choi [1], referring to dispersions of nanoparticles in the base fluids such as water, ethylene glycol, and propylene glycol.The thermal conductivity enhancement characteristic of nanofluid was observed by Masuda et al. [2].Buongiorno [3] discussed the reasons behind the enhancement in heat transfer for nanofluid and he found that Brownian diffusion and thermophoresis are the main causes.Later, Nield and Kuznetsov [4] and Kuznetsov and Nield [5] investigated the natural convective boundary layer flow of a nanofluid employing Buongiorno model.
The study of boundary layer flow and heat transfer due to stretching surface has numerous applications in industry and technology, such as in polymer extrusion, drawing of copper wires, artificial fibers, paper production, hot rolling, wire drawing, glass fiber, metal extrusion and metal spinning, and continuous stretching of plastic films.Crane [6] first studied the boundary layer flow due to linearly stretching sheet.Many researchers [7][8][9][10][11][12][13][14][15][16][17] extended the work of Crane, whereas Magyari and Keller [18] considered the boundary layer flow and heat transfer due to an exponentially stretching sheet.
The flow and heat transfer over an exponentially stretching surface were investigated by Elbashbeshy [19] taking wall mass suction.Khan and Sanjayanand [20] presented the boundary layer flow of viscoelastic fluid and heat transfer over an exponentially stretching sheet with viscous dissipation effect and Partha et al. [21] reported a similarity solution for mixed convection flow past an exponentially stretching surface.Ishak [22] studied the magnetohydrodynamic (MHD) boundary layer flow over an exponentially shrinking sheet in presence of thermal radiation.Bhattacharyya [23] discussed the boundary layer flow and heat transfer caused due to an exponentially shrinking sheet and Bhattacharyya and Pop [24] showed the effect of external magnetic field on the flow over an exponentially shrinking sheet.Recently, Bhattacharyya and Vajravelu [25] described the stagnationpoint boundary layer flow due to exponentially shrinking sheet for Newtonian fluid and Bachok et al. [26] investigated the same problem for nanofluid.
The boundary layer flow of nanofluid past a linearly stretching sheet was first studied by Khan and Pop [27] introducing the model of Nield and Kuznetsov [4].The boundary layer flow induced in a nanofluid due to a linearly stretching 2 Physics Research International sheet with convective boundary condition was described by Makinde and Aziz [28].Kandasamy et al. [29] investigated the MHD boundary layer flow of a nanofluid past a vertical stretching permeable surface with suction/injection.Mustafa et al. [30] reported the flow of a nanofluid near a stagnationpoint towards a stretching surface.Rana and Bhargava [31] illustrated the steady, laminar boundary layer flow due to the nonlinear stretching of a flat surface in a nanofluid.Later, Hady et al. [32] analysed the boundary layer flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation and variable wall temperature.Makinde et al. [33] discussed the combined effects of buoyancy force and magnetic field on stagnation-point flow and heat transfer in a nanofluid flow towards a stretching sheet.A theoretical study of unsteady boundary layer flow of a nanofluid over a permeable stretching/shrinking sheet was reported by Bachok et al. [34].Recently, Nadeem and Lee [35] obtained analytic solutions of boundary layer flow of nanofluid over an exponentially stretching surface using homotopy analysis method (HAM).
In the present paper, the MHD boundary layer nanofluid flow dynamics due to an exponentially shrinking sheet is investigated.The obtained ordinary differential equations by similarity transformations are solved numerically using shooting technique with fourth-order Runge-Kutta method.Then computed results are plotted in graphs and discussed in detail.

Mathematical Formulation
Consider the steady boundary layer flow of nanofluid over an exponentially stretching sheet in presence of a transverse magnetic field.The governing equations of motion and the energy equation may be written in usual notation as [27,35] where  and V are the velocity components in and directions, respectively,  is the kinematic viscosity,   is the density of the base fluid,  is the temperature,  ∞ is constant temperature of the fluid in the inviscid free stream,  is the thermal conductivity, ()  is the effective heat capacity of nanoparticles, ()  is heat capacity of the base fluid,  is nanoparticle volume fraction,   is the Brownian diffusion coefficient, and   is the thermophoretic diffusion coefficient.Here, the variable magnetic field () is taken in the form [22,24] where  0 is a constant.The boundary conditions are given by where   is the variable temperature at the sheet with  0 being a constant which measures the rate of temperature increase along the sheet,   is the variable wall nanoparticle volume fraction with  0 being a constant, and  ∞ is constant nanoparticle volume fraction in free stream.The stretching velocity   is given by where  > 0 is stretching constant.A physical model with the coordinate system of the problem is sketched in Figure 1.
Here V  is the variable wall mass transfer velocity and is given by where V 0 is a constant with V 0 < 0 for mass suction and V 0 > 0 for mass injection.Now, we introduce the similarity transformations: where Ψ is the stream function with  = / and V = −/ and  is the similarity variable.
In view of relations in (6) we finally obtain the following self-similar equations: where  = 2 2 0 / is the magnetic parameter, Pr = / is the Prandtl number, and Le = /  is the Lewis number.The dimensionless parameters Nb (Brownian motion parameter) and Nt (thermophoresis parameter) are defined as The boundary conditions (3) reduce to the following forms: where  = −V 0 /√/2 is the wall mass transfer parameter. > 0 (V 0 < 0) corresponds to mass suction and  < 0 (V 0 > 0) corresponds to mass injection.The quantities of physical interest for this problem are the local skin friction coefficient   , the local Nusselt number Nu  , and the local Sherwood number Sh  , which are, respectively, defined as [35] that is, where Re  =   / is the local Reynolds number.

Numerical Method for Solution
The highly nonlinear coupled ODEs (7) along with the boundary conditions ( 9) form a two-point boundary value problem (BVP) and those are solved using shooting method [36][37][38][39][40][41].The following first-order system is set: with the boundary conditions The set of nonlinear first-order ordinary differential equations (12) with boundary conditions (13) have been solved by shooting method using the fourth-order Runge-Kutta algorithm with a systematic guessing of (0), that is,   (0), (0), that is,   (0), and (0), that is,   (0).The step size is taken as Δ = 0.01 and the suitable finite value of  → ∞,  ∞ , is taken as 20 in all cases.The guess values  (0),   (0), and   (0) are adjusted using "secant method" to give better approximation for the solution.An asymptotic convergence criterion of 10 −5 level for the boundary conditions  ( ∞ ) = 0, ( ∞ ) = 0, and ( ∞ ) = 0 is taken in the computation.

Results and Discussion
The numerical solutions are obtained using the above numerical scheme for some values of the governing parameters, namely, the magnetic parameter (), the wall mass transfer parameter (), the Prandtl number (Pr), the Lewis number (Le), Brownian motion parameter (Nb), and thermophoresis parameter (Nt).Effects of , , Pr, Le, Nb, and Nt on the steady boundary layer flow, heat transfer, and nanoparticle volume fraction over exponentially stretching sheet in nanofluid are discussed in detail.
To ensure the numerical accuracy, the values   (0) and (∞) are compared with the results of Magyari and Keller [18] in Table 1 without magnetic field ( = 0) and with nonporous stretching sheet ( = 0) and those are found in excellent agreement.Thus, we are very much confident that the present results are accurate.

Magyari and Keller
1.281808 1.28180838 (∞) 0.905639 0.90564328 of magnetic parameter are shown in Figure 2. The velocity reduces with the increase of magnetic parameter.The magnetic field opposes the transport process.Actually, the increase of  leads to the increase of the Lorentz force arising because of interaction of magnetic and electric fields for the motion of an electrically conducting fluid, and the stronger Lorentz force produces much more resistance to the transport phenomena.On the other hand, the temperature and the nanoparticle volume fraction increase with .The Lorentz force has the tendency to increase the temperature and nanoparticle volume fraction in nanofluid motion.Consequently, the thermal boundary layer thickness and nanoparticle volume fraction boundary layer thickness become thicker for stronger magnetic field.
In Figure 3, the velocity, temperature, and nanoparticle volume fraction are presented for variation in wall mass transfer parameter .With increasing values of the mass suction parameter ( > 0), the velocity  (), temperature (), and nanoparticle volume fraction () in the boundary layer region decrease, whereas, due to the increase of mass injection ( < 0), all those increase.Due to mass suction, the fluid is brought closer to the sheet and it thins velocity boundary layer thickness as well as the thermal and nanoparticle volume boundary layer thicknesses.Opposite effect is found for mass injection case; that is, the fluid is taken away from the sheet.Consequently, the velocity, thermal, and nanoparticle volume boundary layer thicknesses become broader.
The influences of the Prandtl number Pr and the Lewis number Le on the temperature and nanoparticle volume fraction are depicted in Figures 4 and 5, respectively.The increment of Prandtl number results in major effects on temperature as well as on nanoparticle volume fraction.The thermal boundary layer thickness reduces with Prandtl number and it happens due to decrease of thermal diffusivity for the increment of Prandtl number.The nanoparticle volume fraction exhibits overshoot near the sheet for higher values of Pr, though the nanoparticle volume boundary layer thickness reduces.Hence, with uniform thermophoretic particle deposition, for larger values of Prandtl number the nanoparticle volume fraction () is higher in the fluid adjacent to the sheet than the value at the wall.Very minor variation (initially increasing near the sheet and then decreasing away from the sheet) is observed in the temperature with the increase in the Lewis number.For large values of Le, the nanoparticle volume fraction significantly decreases and also the nanoparticle volume boundary layer thickness reduces.But, for smaller value of Le the overshoot is found in () near the sheet (Figure 5(b)).The Brownian diffusion effect becomes nominal for larger values of Lewis number and for which the nanoparticle volume boundary layer thickness decreases.
The effect of Brownian motion parameter Nb on the dimensionless temperature and the dimensionless nanoparticle volume fraction is plotted in Figure 6 (Figures 6(a   increment in the thermophoresis force which tends to move nanoparticles from hot to cold areas and consequently it increases the magnitude of temperature profiles and nanoparticle volume fraction profiles.Ultimately, the thickness of nanoparticle volume boundary layer becomes significantly large for slightly increased value of thermophoresis parameter. We now discuss the variations of the physical quantities of engineering importance, that is, the local skin friction coefficient   , the local Nusselt number Nu  , and the local Physics Research International Sherwood number Sh  for different values of , , Pr, Le, Nb, and Nt.And for which the quantities −  (0), −  (0), and −  (0) related to local skin friction coefficient, the local Nusselt number and the local Sherwood number, respectively, are plotted in Figures 8,9,10,11,12,13,14,15,and 16 for various values of parameters.The values of −  (0), −  (0), and −  (0) versus the mass transfer parameter  are plotted in Figure 8 for different values magnetic parameter.For stronger magnetic field the value of −  (0), that is, the local skin friction coefficient, increases, whereas the values of −  (0), that is, local Nusselt number, and −  (0), that is, local Sherwood number, decreases.Also, the values of −  (0) and −  (0) are depicted against Brownian motion parameter Nb and thermophoresis parameter Nt for various  in Figures 9 and 10.Similar results are also observed in those graphs.In physical viewpoint, it can be noticed that the powerful Lorentz force that arose in flow field for larger magnetic field reduces the values of local Nusselt number and local Sherwood number.On the other hand, the skin friction coefficient increases (decreases) with mass suction (injection) (Figure 8(a)).−  (0) and −  (0) are displayed against Nb and Nt for several  in Figures 11 and 12. From all graphical results it should be noted that, similar to that of skin friction, the local Nusselt number and the local Sherwood number also decrease with injection and increase with suction.But for greater values of thermophoresis parameter Nt, the contrary effect of mass transfer parameter on the local Sherwood number is obtained, namely, the value of−  (0) increases with mass injection and decreases with mass suction (Figure 12 The heat transfer rate is strengthened when the Prandtl number increases, whereas it is worth noting that the Lewis number affects the Nusselt number (reduction) and the Sherwood number (increment) in completely reverse manner compared to that of Prandtl number.For small values of Lewis number, the Brownian diffusion effect is large and accordingly increased heat transfer rate is found (greater Nusselt number, Figure 15(a)).From Figures 9 to 16, the variations in local Nusselt number and the local Sherwood number for Brownian motion parameter Nb and thermophoresis parameter Nt can be understood.For high rate of Brownian motion, that is, for increasing Nb, the values of −  (0) reduce and those of −  (0) increase.But the increase in thermophoresis parameter Nt causes reduction in both, the local Nusselt number and the local Sherwood number.

Concluding Remarks
The laminar MHD boundary layer flow of nanofluid past an exponentially permeable stretching sheet has been studied.The spatial focus is offered to the effects of Brownian motion and thermophoresis on the heat transfer in boundary layer flow of nanofluid due to an exponentially stretching sheet in presence of magnetic field.The findings of the analysis can be summarized as follows.

Figure 1 :
Figure 1: Physical model and coordinate system.
(b)).The effects of Prandtl number Pr and Lewis number Le on the local Nusselt number Nu  and the local Sherwood number Sh  are illustrated in Figures 13-16.Due to higher values of Prandtl number the local Nusselt number increases and local Sherwood number reduces.

Figure 14 :Figure 15 :
Figure 14: Variation of −  (0) with (a) Brownian motion parameter Nb and (b) thermophoresis parameter Nt, for different values of Prandtl number Pr.