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

Authors: M. Ferdows; Md. Shakhaoath Khan; Md. Mahmud Alam; Shuyu Sun
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

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 six-order 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.


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## Body

## 1. 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 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 was reported by [29–36]. Bachok et al. [37] have showed the steady boundary layer flow of a nanofluid past a moving semiinfinite flat plate in a uniform free stream. It was assumed that the plate is moving in the same or opposite directions to the free stream to define resulting system of nonlinear ordinary differential equations.Khan and Pop [38, 39] formulated the problem of laminar boundary layer flow of a nanofluid past a stretching sheet. They also expressed free convection boundary layer nanofluid flow past a horizontal flat plate. Hamad and Pop [40] discussed the boundary layer flow near the stagnation-point flow on a permeable stretching sheet in a porous medium saturated with a nanofluid. Hamad et al. [41] investigates free convection flow of a nanofluid past a semiinfinite vertical flat plate with the influence of magnetic field. Very recently Shakhaoath et al. [42, 43] investigate the effects of thermal radiation and magnetic field on the boundary layer flow of a nanofluid over a stretching surface.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 (λT), Mass convective parameter  (λM), viscosity ratio parameter (∧), dimensionless inertia parameter (∇), Prandtl number (Pr), Eckert number (Ec), Lewis number (Le), Brownian motion parameter (Nb), and Thermophoresis parameter (Nt). 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.


## 2. 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 toTw(>T∞) and Cw(>C∞), respectively, which are thereafter maintained constant, where Tw, Cw 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.Figure 1
Physical model and coordinate system.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]):
(2)∂u∂x+∂v∂y=0u∂u∂x+v∂u∂y=υ~∂2u∂y2-νKu-c*ε2u2-σB2ρu+gβ(T-T∞)+gβ*(C-C∞)u∂T∂x+v∂T∂y=α∂2T∂y2+υ~cp(∂u∂y)2+τ{DB(∂T∂y·∂C∂y)+DTT∞(∂T∂y)2}u∂C∂x+v∂C∂y=DB∂2C∂y2+DTT∞∂2T∂y2,
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, cp the specific heat at constant pressure, T and C the fluid temperature and concentration in the boundary layer, c*ε2 is the inertia parameter, α is the thermal diffusivity, DB is the Brownian diffusion coefficient, and DT is the thermophoresis diffusion coefficient.The boundary condition for the model is(2.2)u=uw=u∘ex/L,v=0,T=Tw=T∞+T∘ex/2L,C=Cw=C∞+C∘ex/2L,aty=0u=0,T→T∞,C→C∞,asy→∞,
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
(2.3)B=B∘ex/2L,
where B∘ is the constant magnetic field. Also the variable thermal conductivity can be taken as the form
(2.4)K=k∘ex/2L.In order to attains a similarity solution to (2) with the boundary conditions (2.2), the following dimensionless variables are used:
(2.5)η=yu∘2υLex/2L,ψ=2υu∘Lex/2Lf(η),θ=θ(η)=T-T∞Tw-T∞,φ=φ(η)=C-C∞Cw-C∞,u=∂ψ∂y=u∘ex/Lf′,v=-∂ψ∂x=-υu∘2Lex/2L(f+ηf′).From the above transformations, the nondimensional, nonlinear, and coupled ordinary differential equations are obtained as(2.6)∧f′′′+ff′′-2(∇+1)f′2-2Rf′+2λTθ+2λMφ=0θ′′-Prf′θ+Prfθ′+∧PrEcf′′2+PrNbθ′φ′+PrNtθ′2=0φ′′+Lefφ′-Lef′φ+(NtNb)θ′′=0,
where the notation primes denote differentiation with respect to η and the parameters are defined as
(2.7)R=(νLk∘u∘+σB∘2Lρu∘)(combinedporousandmagneticparameter),∧=υ~ν(viscosityratioparameter),∇=c*ε2L(dimensionlessinertiaparameter),λT=GrRex2=gβ(Tw-T∞)uw2(thermalconvectiveparameter),λM=GmRex2=gβ(Cw-C∞)uw2(massconvectiveparameter),pr=υα(Prandtlnumber),Ec=u∘2T∘cp(Eckertnumber),NB=(ρc)pDB(Cw-C∞)υ(ρc)f(Brownianmotionparameter),Nt=(ρc)pDT(Tw-T∞)υT∞(ρc)f(Thermophoresisparameter),Le=υDB(Lewisnumber).The transformed boundary conditions(2.8)f=0,f′=1,θ=1,φ=1,atη=0f′=0,θ=0,φ=0, asη→∞.The physical quantities of the skin-friction coefficient, the reduced Nusselt number, and reduced Sherwood number are calculated, respectively, by the following:(2.9)Cf(Rex)-1/2=-f′′(0),Nu(Rex)-1/2=-θ′(0),Sh(Rex)-1/2=-φ′(0),
where R_(e_x)=(xu∘ex/L)/ν is the local Reynolds number.


## 3. 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).


## 4. Results and Discussions

The system of governing equation (2.6) are coupled both in heat and mass transfer; it is clear that analytical solution is not possible. Thus, computations have been carried out based on numerical technique. The values of the governing parameters are chosen arbitrary. However, the numerical results are presented for some representative values of these governing parameters. In order to see the physical insight, the numerical values of velocity (f′), temperature (θ), and concentration (φ) with the boundary layer have been computed for different parameters as the combined porous and magnetic parameter (R), Thermal convective parameter (λT), Mass convective parameter  (λM), viscosity ratio parameter (∧), dimensionless inertia parameter (∇), Prandtl number (Pr), Eckert number (Ec), Lewis number (Le), Brownian motion parameter (Nb), and Thermophoresis parameter (Nt). In order to assess the accuracy of the numerical results as the results for the reduced Nusselt number -θ′(0) for different values of Prandtl number (Pr) and Eckert number (Ec), the present results compared with Bidin and Nazar [11]. Comparison with the existing results 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.Table 1
Comparison for the reduced Nusselt number-θ′(0) for the values of λT=λM=R=∇=Le=Nb=Nt=0.0.
P
r
E
c
=
0.0
E
c
=
0.0
E
c
=
0.0
Bidin and Nazar [11] (K=0.0)
Presentresults
Bidin and Nazar [11] (K=0.0)
Presentresults
Bidin and Nazar [11] (K=0.0)
Presentresults
1
0.9547
0.9550
0.8622
0.8629
0.5385
0.5392
2
1.4714
1.4719
1.3055
1.3062
0.7248
0.7250
3
1.8691
1.8701
1.6882
1.6890
0.8301
0.8309Figure 2
Effect of∧ on velocity profiles.Figure 3
Effect ofR on velocity profiles.Figure 4
Effect ofλT on velocity profiles.Figure 5
Effect ofλM on velocity profiles.Figure 6
Effect ofEc on velocity profiles.Figure 7
Effect of∧ on temperature profiles.Figure 8
Effect ofR on temperature profiles.Figure 9
Effect ofλT on temperature profiles.Figure 10
Effect ofλM on temperature profiles.Figure 11
Effect ofEc on temperature profiles.Figure 12
Effect of∧ on concentration profiles.Figure 13
Effect ofR on concentration profiles.Figure 14
Effect ofλT on concentration profiles.Figure 15
Effect ofλM on concentration profiles.Figure 16
Effect ofEc on concentration profiles.Figure 17
Effect of∧ on Skin-friction coefficient.Figure 18
Effect ofR on skin-friction coefficient.Figure 19
Effect of∧ on heat transfer rate.Figure 20
Effect ofR on heat transfer rate.Figure 21
Effect of∧ on mass transfer rate.Figure2 displays the dimensionless velocity distribution f′(η) for different values of ∧ where λT=5.0,λM=2.0,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1Then for above case it is observed that velocity profiles increase as ∧ increases.Figure3 exhibits the dimensionless velocity distribution f′(η) for different values of R where λT=5.0,λM=2.0,∧=1.5,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the velocity reduces steeply near the exponential stretching sheet as R is increased.Figure4 represents the dimensionless velocity distribution f′(η) for different values of λT where ∧=1.5,λM=2.0,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1It shows that the velocity rises as λT is increased.Figure5 depicts the dimensionless velocity distribution f′(η)  for different values of λM where λT=5.0,∧=1.5,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the velocity reduces as  λM is increased.Figure6 depicts the dimensionless velocity distribution f′(η) for different values of Ec where λT=5.0,λM=2.0,R=4.0,∇=1.0,∧=1.5,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the velocity rises as Ec is increased.Figure7 represents the dimensionless temperature distribution θ(η) for different values of ∧ where λT=5.0,λM=2.0,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the thermal boundary layer rises as ∧ is increased causing the fluid temperature to reduce at every point other than the wall.Figure8 shows the dimensionless temperature distribution θ(η) for different values of R where λT=5.0,λM=2.0,∧=1.5,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1andNb=0.1 It shows that the thermal boundary layer increases as R is increased.Figure9 exhibits the dimensionless temperature distribution θ(η) for different values of λT where ∧=1.5,λM=2.0,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the thermal boundary layer reduces as λT is increased.Figure10 portrays the dimensionless temperature distribution θ(η) for different values of λM where λT=5.0,∧=1.5,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the thermal boundary layer rises as λM is increased.Figure11 shows the dimensionless temperature distribution θ(η) for different values of Ec where λT=5.0,λM=2.0,R=4.0,∇=1.0,∧=1.5,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the thermal boundary layer increases gradually as Ec is increased.Figure12 shows the dimensionless concentration distribution φ(η) for different values of ∧ where λT=5.0,λM=2.0,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the concentration boundary layer rises as ∧ is increases.Figure13 exhibits the dimensionless concentration distribution φ(η) for different values of R where λT=5.0,λM=2.0,∧=1.5,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the concentration boundary layer rises as R is increased.Figure14 depicts the dimensionless concentration distribution φ(η) for different values of λT where ,∧=1.5,λM=2.0,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It is observed that the increasing effect of λT is to reduce concentration distribution as concentration species is dispersed away.Figure15 exhibits the dimensionless concentration distribution φ(η) for different values of λM where λT=5.0,∧=1.5,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the concentration boundary layer reduces as λM is increased.Figure16 shows the dimensionless concentration distribution φ(η) for different values of Ec where λT=5.0,λM=2.0,R=4.0,∇=1.0,∧=1.5,Pr=1.0,Le=5.0,Nt=0.1,andNb=0.1. It shows that the concentration boundary layer decreases gradually as Ec is increased.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 (Nb) and illustrated in Figures 17–21.Figure17 illustrates the skin-friction coefficient (-f′′) plotted for the different values of ∧ where λT=5.0,λM=2.0,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,andNb=0.1. It is verified that there is a decrease in skin-friction coefficient as ∧ increases.Figure18 displays the skin-friction coefficient (-f′′) plotted for the different values of R where λT=5.0,λM=2.0,∧=1.5,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,andNb=0.1. It is verified that there is a decrease in skin-friction coefficient as R increases.Figure19 depicts the Nusselt number (-θ′) plotted for the different values of ∧ where λT=5.0,λM=2.0,R=4.0,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,andNb=0.1. It is verified that there is decrease in Nusselt number as ∧ increases.Figure20 displays the Nusselt number (-θ′) plotted for the different values of R where λT=5.0,λM=2.0,∧=1.5,∇=1.0,Ec=0.2,Pr=1.0,Le=5.0,andNb=0.1. It is verified that there is a decrease in Nusselt number as R increases.Figure21 shows the Sherwood number (-φ′) plotted for the different values of ∧ where λ1=4.0,α=1.0,Fw=2.0,Q=1.0,Nt=0.5,M=1.0,Pr=0.71,andLe=5.0. It is verified that there is a decrease in Sherwood number as ∧ increases.


## 5. 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.(1)
Momentum, thermal, and concentration boundary layer thickness increase as the viscous ratio parameter increases. Whereas skin-friction coefficient, surface heat, and mass transfer rate decreases.(2)
Momentum boundary layer thickness reduces; on the other hand, thermal and concentration boundary layer increases steeply near the exponential stretching sheet as the combined porous and Magnetic parameter is increased. Whereas skin-friction coefficient and surface heat transfer rate decreases.(3)
Momentum and thermal boundary layer increase whereas the concentration boundary layer decreases gradually as the Eckert number is increased.



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