MHD boundary layer flow and heat transfer of nanofluidsover a nonlinear stretching sheet: A numerical study

F. Mabood a,n, W.A. Khan b, A.I.M. Ismail a

a School of Mathematical Sciences, Universiti Sains Malaysia, Penang 11800, Malaysiab Department of Mechanical Engineering, University of Waterloo, Waterloo, ON, Canada N2L 3G1

a r t i c l e i n f o

Article history:Received 13 June 2014Received in revised form19 August 2014Available online 16 September 2014

Keywords:MHDNanofluidStretching sheetViscous dissipationLewis number

a b s t r a c t

The MHD laminar boundary layer flow with heat and mass transfer of an electrically conducting water-based nanofluid over a nonlinear stretching sheet with viscous dissipation effect is investigatednumerically. This is the extension of the previous study on flow and heat transfer of a nanofluid overnonlinear stretching sheet (Rana and Bhargava, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) 212–226). The governing equations are reduced to nonlinear ordinary differential equations using suitablesimilarity transformation. The effects of the governing parameters on dimensionless quantities likevelocity, temperature, nanoparticle concentration, friction factor, local Nusselt, and Sherwood numbersare explored. It is found that the dimensionless velocity decreases and temperature increases withmagnetic parameter, and the thermal boundary layer thickness increases with Brownian motion andthermophoresis parameters.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Nanofluids are used to enhance the thermal conductivity ofbase fluids like water, ethylene glycol, propylene glycol etc. Theyhave several engineering and biomedical applications in cooling,cancer therapy and process industries. The enhancement ofthermal conductivity of conventional heat transfer fluids throughsuspensions of solid particles is a relatively recent development inengineering technology. The resulting effect of these suspensionsis to increase the coefficient of heat transfer. The suspendedparticles are able to increase the thermal conductivity and heattransfer performance since the thermal conductivity of solidmetals is higher than base fluids. Major advantages of nanofluidsare that they are more stable, have sufficient viscosity and betterwetting, spreading and dispersion properties on solid surface evenfor modest nanoparticle concentrations [1].

The nanoparticles used in nanofluids are typically made ofmetals (Al, Cu), oxides (Al2O3), carbides (SiC), nitrides (AlN, SiN) ornonmetals (graphite, carbon nanotubes) and the base fluid isusually a conductive fluid, such as water (as in this study) orethylene glycol. Nanoparticles range in diameter between 1 and100 nm. Experimental studies have shown that nanofluids

commonly need only contain up to a 5% volume fraction ofnanoparticles to ensure effective heat transfer enhancements [2].Nanofluids offer many diverse advantages in application such asmicroelectronics, fuel cell, nuclear reactors, biomedicine andtransportation [3]. Important work on the boundary layer flow ofa nanofluid over a stretching sheet has been reported by Khan andPop [4] using Buongiorno’s model [5]. Rana and Bhargava [6]conducted similar research for a nonlinear stretching sheet usingfinite element and finite difference methods. The effect of con-vective surface boundary condition on the boundary layer flow ofnanofluid over a stretching sheet was discussed by Makinde andAziz [7] whilst Mustafa et al. [8] investigated boundary layer flowfor an exponential stretching sheet by using homotopy analysismethod for the computation of analytical solutions. Abel et al. [9]investigated the steady buoyancy-driven dissipative magneto-convective flow from a vertical nonlinear stretching sheet. Severalother studies have addressed various aspects of regular/nanofluids (including comparison) with stretching sheet [10–19].

Electrically-conducting nanofluid flows, which respond to theimposition of magnetic fields, have received relatively significantconsiderations. The solution of boundary layer equation for apower law fluid in magneto-hydrodynamics is obtained by Helmy[20], whereas Chiam [21] investigated hydromagnetic flow over asurface stretching with a power-law velocity using shootingmethod. Ishak et al. [22] investigated the hydromagnetic flowand heat transfer adjacent to a stretching vertical sheet. Nourazaret al. [23] investigated MHD forced-convective flow of nanofluid

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journal homepage: www.elsevier.com/locate/jmmm

Journal of Magnetism and Magnetic Materials

http://dx.doi.org/10.1016/j.jmmm.2014.09.0130304-8853/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author.E-mail addresses: mabood1971@yahoo.com (F. Mabood),

wkhan_2000@yahoo.com (W.A. Khan), izani@cs.usm.my (A.I.M. Ismail).

Journal of Magnetism and Magnetic Materials 374 (2015) 569–576

over a horizontal stretching flat plate with variable magnetic fieldincluding the viscous dissipation. Zeeshan et al. [24] examined theMHD flow of third grade nanofluid between coaxial porouscylinders whilst Chamkha and Aly [25] have considered MHD freeconvective boundary-layer flow of a nanofluid along a permeableisothermal vertical plate in the presence of heat source or sink.MHD mixed convective flow of nanofluid over a stretching sheetwas very recently investigated by Matin et al. [26]. Some otherrelated studies can be found in [27–34].

The above literature review reveals that no study has beenreported for MHD boundary layer flow and heat transfer of ananofluid over a nonlinearly stretching sheet with viscous dis-sipation effects. The objective of the present paper is therefore toextend the work of Rana and Bhargava [6] by considering MHDboundary layer flow with viscous dissipation effect. The metho-dology adopted will be to reduce the governing partial differentialequations into a system of ordinary differential equations andsolving the resulting system using the Runge–Kutta–Fehlbergfourth fifth order method.

2. Flow analysis and mathematical formulation

Consider a two dimensional, steady and incompressible viscousflow of a water-based nanofluid past over a nonlinear stretchingsurface. The sheet is extended with velocity uw xð Þ ¼ axn with fixedorigin location, where n is a nonlinear stretching parameter, a is aconstant and x is the coordinate measured along the stretchingsurface. The nanofluid flows at y¼0, where y is the coordinatenormal to the surface. The fluid is electrically conducted due to anapplied magnetic field B xð Þ normal to the stretching sheet. Themagnetic Reynolds number is assumed small and so, the inducedmagnetic field can be considered to be negligible. The walltemperature Tw and the nanoparticle fraction Cw are assumedconstant at the stretching surface. When y tends to infinity, theambient values of temperature and nanoparticle fraction aredenoted by T1 and C1, respectively. The considered physicalsystem is of importance in modern nano-technological fabricationand thermal materials processing. It is important to note that theconstant temperature and nanoparticle fraction of the stretchingsurface Tw and Cw are assumed to be greater than the ambienttemperature and nanoparticle fraction T1, C1, respectively. Thecoordinate system and the flow model are shown in Fig. 1. Thegoverning equations of momentum, thermal energy and nanopar-ticles equations can be written as, Khan and Pop [4]

∂u∂x

þ∂v∂y

¼ 0; ð1Þ

u∂u∂x

þv∂u∂y

¼ ν∂2u∂y2

�σB2ðxÞρf

u; ð2Þ

u∂T∂x

þv∂T∂y

¼ α∂2T∂y2

þτ DB∂C∂y

∂T∂y

þDT

T1� ∂T

∂y

� �2( )

þ ν

cp

∂u∂y

� �2

; ð3Þ

u∂C∂x

þv∂C∂y

¼DB∂2C∂y2

þ DT

T1

� �∂2T∂y2

; ð4Þ

The boundary conditions for the velocity, temperature andnanoparticle fraction are defined as [6]:

y¼ 0 : uw ¼ axn; v¼ 0; T ¼ Tw; C ¼ Cw; ð5Þ

y¼1 : u¼ 0; v¼ 0; T ¼ T1; C ¼ C1: ð6Þ

Here, u and v are the velocity components along the x and y-axes, respectively. α¼ k= ρcð Þf is the thermal diffusivity, σ iselectrical conductivity, ν is the kinematic viscosity, ρf is the densityof the base fluid, DB is the Brownian diffusion coefficient and DT isthe thermophoresis diffusion coefficient. τ¼ ρcð Þp= ρcð Þf is the ratiobetween the effective heat capacity of the nanoparticle materialand heat capacity of the fluid, c is the volumetric volumecoefficient, ρp is the density of the particles, and C is rescalednanoparticle volume fraction. We assume that the variable mag-netic field B xð Þ is of the form B xð Þ ¼ B0x n�1ð Þ=2. This form of B xð Þ hasalso been considered by many authors including [20–22].

Rana and Bhargava [6] introduced the following transforma-tions

η¼ yffiffiffiffiffiffiffiffiffiffiffiffiaðnþ1Þ

2ν

qxðn�1Þ=2; u¼ axnf 0 ηð Þ; v¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiaν nþ1ð Þ

2

qx n�1ð Þ=2 f ηð Þþn�1

nþ1ηf0 ηð Þ

� �;

θðηÞ ¼ T�Tð Þ1= Tw�T1ð Þ; ϕðηÞ ¼ C�C1ð Þ= Cw�C1ð Þ; ð7Þwhere ψ represent the stream function and is defined as u¼ ∂ψ=∂y,v¼ �∂ψ=∂x, so that Eq. (1) is satisfied identically. The governingEqs. (2)–(4) are reduced by using Eq. (7) as follows:

f‴þ f f ″� 2nnþ1

� �f 02�Mf 0 ¼ 0; ð8Þ

1Pr

θ″þ f θ0 þNbϕ0θ0 þNtθ02þEcf ″2 ¼ 0; ð9Þ

ϕ″þLefϕ0 þNtNb

θ″¼ 0: ð10Þ

The transformed boundary conditions

f 0ð Þ ¼ 0; f 0 0ð Þ ¼ 1; θ 0ð Þ ¼ 1; ϕ 0ð Þ ¼ 1;

f 0 1ð Þ ¼ 0; θ 1ð Þ ¼ 0 ϕ 1ð Þ ¼ 0: ð11Þwhere primes denote differentiation with respect to η, Theinvolved physical parameters are defined as:

Pr¼ να ; Le¼ ν

DB; Nb¼ ðρcÞpDBðCw �C1Þ

ðρcÞf ν ; Nt ¼ ðρcÞpDT ðTw �T1ÞðρcÞf T1ν ;

M¼ 2σB20

aρf ðnþ1Þ ; Ec¼ uw2

cpðTW �T1Þ : ð12Þ

Here, Pr, Le, Nb, Nt, M and Ec denote the Prandtl number, theLewis number, the Brownian motion parameter, the thermophor-esis parameter, magnetic parameter and Eckert number, respec-tively. This boundary value problem is reduced to the classicalproblem of flow and heat and mass transfer due to a stretchingsurface in a viscous fluid when n¼ 1 and Nb¼Nt ¼ 0 inEqs. (9) and (10).

The quantities of practical interest, in this study, are the localskin friction Cf x, Nusselt number Nux and the Sherwood numberShx which are defined as

Cf x ¼μfρu2

w

∂u∂y

� �y ¼ 0

; Nux ¼xqw

kðTw�T1Þ ; Shx ¼xqm

DBðCw�C1Þ ; ð13Þ

Nanofluid

,w wT C

Stretching force

,y v

B(x)u w = axn

,T C

,x uO

∞ ∞

Fig. 1. Flow configuration and coordinate system.

F. Mabood et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 569–576570

where k is the thermal conductivity of the nanofluid, and qw, qmare the heat and mass fluxes at the surface, respectively, given by

qw ¼ � ∂T∂y

� �y ¼ 0

; qm ¼ �DB∂C∂y

� �y ¼ 0

; ð14Þ

Substituting Eq. (7) into Eqs. (13)–(14), we obtain

Re1=2x Cf x ¼ffiffiffiffiffiffiffiffiffiffiffinþ12

rf ″ð0Þ; Re�1=2

x Nux ¼ �ffiffiffiffiffiffiffiffiffiffiffinþ12

rθ0ð0Þ;

Re�1=2x Shx ¼ �

ffiffiffiffiffiffiffiffiffiffiffinþ12

rϕ0ð0Þ; ð15Þ

where Rex ¼ uwx=ν is the local Reynolds number.

3. Results and discussion

The reduced Eqs. (8)–(11) are nonlinear and coupled, and thustheir exact analytical solutions are not possible. They can be solvednumerically using Runge–Kutta–Fehlberg fourth fifth ordermethod for different values of parameters such as magneticparameter, Prandtl, Eckert and the Lewis numbers, the Brownianmotion parameter and the thermophoresis parameter. The effectsof the emerging parameters on the dimensionless velocity, tem-perature, skin friction, the rates of heat and mass transfer areinvestigated. The step size and convergence criteria were chosento be 0.001 and 10�6, respectively. The asymptotic boundaryconditions in Eq. (11) were approximated by using a value of 10for ηmax as follows:

ηmax ¼ 10; f 0 10ð Þ ¼ θ 10ð Þ ¼ ϕ 10ð Þ ¼ 0; ð16ÞThis ensures that all numerical solutions approached the

asymptotic values correctly.To validate the present solution, comparisons have been made

with previously published data in the literature for �θ0 0ð Þ and�ϕ0 0ð Þ in Tables 1–4, and they are found to be in an excellentagreement. Effects of magnetic and viscous dissipation parameterson friction factor, Nusselt and Sherwood numbers are presented inTable 5, while keeping other parameter values preset. FromTables 1–5, it is clear that the Nusselt number is a decreasingfunction of M, Ec, n Nt and Le, whereas Sherwood number is found

to be a decreasing function of M, n and Nt, and an increasingfunction of Pr, Ec and Le. An increase in the Lewis number Lemeans that the fluid becomes more viscous, it, therefore, causes anincrease in the rate of mass transfer. As expected, increasing theLewis number reduces the rate of heat transfer. The values of theskin friction coefficient can be observed in an increasing mannerfor various values of M in Table 5.

Fig. 2 exhibits the effect of magnetic and nonlinear stretchingparameters on the dimensionless velocity. It is observed that thevelocity profile of the nanofluid is insignificantly reduced withincreasing values ofM and n. An increase in magnetic parameterMresults in a strong reduction in dimensionless velocity f 0. This isdue to the fact that magnetic field introduces a retarding bodyforce which acts transverse to the direction of the appliedmagnetic field. This body force, known as the Lorentz force,decelerates the boundary layer flow and thickens the momentumboundary layer, and hence induces an increase in the absolutevalue of the velocity gradient at the surface as shown in Table 5.

Figs. 3 and 4 are prepared to show the influence of magneticparameter M and nonlinear stretching parameter n on the dimen-sionless temperature and nanoparticle concentration. The dimen-sionless temperature and concentration profiles are clearlyobserved to be significantly enhanced with increasing magneticparameter. As the Lorentz force is a resistive force which opposesthe fluid motion, so heat is produced and as a result, the thermalboundary layer thickness and nanoparticle volume fraction bound-ary layer thickness become thicker for stronger magnetic field.From the same Figs. 3 and 4, it is also observed that the variationin the dimensionless temperature and nanoparticle concentrationdue to nonlinear stretching parameter is slightly negligible forboth positive and negative values of n.

The influence of Brownian motion parameter Nb and thermo-phoresis parameter Nt on dimensionless temperature and dimen-sionless nanoparticle concentration for the fixed values of otherparameters are shown in Figs. 5 and 6, respectively. It is noticedthat the dimensionless temperature increases and the nanoparti-cle volume fraction decreases with increasing values of Brownianmotion parameter Nb, further we also observed that both dimen-sionless temperature and nanoparticle volume fraction increasesdue to thermophoretic parameter Nt. This is due to the fact thatthe thermophoretic force generated by the temperature gradientcreates a fast flow away from the stretching surface. In this waymore heated fluid is moved away from the surface, and conse-quently, as Nt increases, the temperature within the boundarylayer increases, the fast flow from the stretching sheet carries withit thermophoretic force leading to an increase in the concentrationboundary layer thickness as shown in Figs. 5 and 6. As expected,an increase in the Brownian motion parameter Nb thickens thethermal boundary layer. On the other side, an increase in theBrownian motion parameter Nb increases the diffusion of nano-particles due to the Brownian effect and consequently decreases

Table 1Comparison of �θ0ð0Þ for Pr and n values when Nb¼Nt ¼M ¼ 0.

Pr n Ec Rana and Bhargava[6]

Cortell[16]

Zaimi et al.[17]

Presentresults

1 0.2 0.0 0.6113 0.610262 0.61131 0.611310.5 0.5967 0.395277 0.59668 0.596681.5 0.5768 0.574537 0.57686 0.576862.0 – – 0.57245 0.572453.0 0.5672 0.564472 0.56719 0.567194.0 – – 0.56415 0.56415

10.0 0.5578 0.554960 0.55783 0.557835 0.2 1.5910 1.607175 1.60757 1.60757

0.5 1.5839 1.586744 1.58658 1.586581.0 – – 1.56787 1.567871.5 1.5496 1.557463 1.55751 1.557512.0 – – 1.55093 1.550933.0 1.5372 1.542337 1.54271 1.54271

10.0 1.5260 1.528573 1.52877 1.528771 0.2 0.1 – 0.574985 – 0.57528

0.5 – 0.556623 – 0.556791.5 – 0.530966 – 0.531003.0 – 0.517977 – 0.51808

10.0 – 0.505121 – 0.505405 0.2 – 1.474764 – 1.47502

0.5 – 1.436789 – 1.437021.5 – 1.381861 – 1.382073.0 – 1.352768 – 1.35375

10.0 – 1.324772 – 1.32538

Table 2Comparison of Nusselt and Sherwood numbers when Pr¼ Le¼ 2 and M¼ Ec¼ 0.

n Nt Nb Rana and Bhargava [6] Present result

�θ0ð0Þ �ϕ0ð0Þ �θ0ð0Þ �ϕ0ð0Þ

0.2 0.1 0.5 0.5160 0.9012 0.5148 0.90140.3 0.4533 0.8395 0.4520 0.84020.5 0.3999 0.8048 0.3987 0.8059

3.0 0.1 0.4864 0.8445 0.4852 0.84470.3 0.4282 0.7785 0.4271 0.77910.5 0.3786 0.7379 0.3775 0.7390

10.0 0.1 0.4799 0.8323 0.4788 0.83250.3 0.4227 0.7654 0.4216 0.76600.5 0.3739 0.7238 0.3728 0.7248

F. Mabood et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 569–576 571

the reduced Nusselt number. An increase in the Brownian motionparameter Nb of the fluid leads to a decrease in the concentrationinside the boundary layer.

Figs. 7 and 8 show the effects of Ec and Le on the dimensionlesstemperature and dimensionless nanoparticle concentration for thefixed values of other parameters. In Fig. 7, it is observed that thedimensionless temperature slightly decreases and increases withLe and Ec, respectively. In fact, Lewis number Le expresses therelative contribution of thermal diffusion rate to species diffusionrate in the boundary layer regime. An increase in Le values willreduce thermal boundary layer thickness and will be accompaniedwith a decrease in temperature. Larger Le values will also suppressconcentration values i.e. inhibit nanoparticle species diffusion, asobserved in Fig. 8. There will be a much greater reduction inconcentration boundary layer thickness than thermal boundary

layer thickness, over an increment in Lewis number, as it is evidentby Figs. 7 and 8. On the other hand, the Eckert number Ec controlsthe fluid motion and a value of zero corresponds to the case of noviscous dissipation. The presence of viscous dissipation in theenergy equation acts as an internal heat source due to the action ofviscous stresses, so that the dimensionless temperature overshootsat Ec¼ 0:3 as compared with the caseEc¼ 0. It is well known thatsituations in which viscous dissipation is significant are thoseinvolving flows with large velocities and high viscosity. Asympto-tic convergence of all profiles is observed with greater transversecoordinate, confirming to the accurate imposition of infinityboundary conditions.

Now we focus on the variations of quantities of the physicalinterest from engineering point of view. That is, local skin frictionCf x, the local Nusselt number Nux and the local Sherwood numberShx. Fig. 9 depict that for larger values of magnetic parameter M,the skin friction coefficient shows the increasing behavior corre-sponding to the increasing values of nonlinear stretching para-meter n. This means that fluid motion on the wall of the sheet isaccelerated when we strengthen the effects of parameters.

In Figs. 10–13, the local Nusselt and Sherwood numbers forvarious values of M, Ec, Le, n, Nb and Nt for fixed value of Pr¼ 6:2are plotted. The values of the local Nusselt number versus Lewisnumber, magnetic and nonlinear stretching parameters are

Table 3Comparison of Nusselt and Sherwood numbers when Nb¼Nt ¼ 0:5 and M ¼ Ec¼ 0. The values reported by Rana and Bhargava [6] are given in brackets.

n ↓Pr; Le- �θ0ð0Þ �ϕ0ð0Þ

2 10 2 10

0.2 0.7 0.3295 [0.3299] 0.3035 [0.3042] 0.8134 [0.8132] 2.3206 [2.3198]0.3 0.3262 [0.3216] 0.3003 [0.2965] 0.8067 [0.7965] 2.3110 [2.2959]3.0 0.3050 [0.3053] 0.2806 [0.2812] 0.7633 [0.7630] 2.2471 [2.2464]

10.0 0.2999 [0.3002] 0.2758 [0.2765] 0.7527 [0.7524] 2.2310 [2.2303]20.0 0.2986 [0.2825] 0.2747 [0.2753] 0.7500 [1.4548] 2.2268 [2.2261]0.2 2.0 0.3987 [0.3999] 0.2816 [0.2835] 0.8060 [0.8048] 2.4227 [2.4207]0.3 0.3959 [0.3930] 0.2793 [0.2778] 0.7971 [0.7826] 2.4112 [0.2778]3.0 0.3775 [0.3786] 0.2644 [0.2661] 0.7390 [0.7379] 2.3342 [2.3324]

10.0 0.3728 [0.3739] 0.2607 [0.2624] 0.7248 [0.7238] 2.3148 [2.3130]20.0 0.3716 [0.3726] 0.2597 [0.2614] 0.7212 [0.7201] 2.3097 [2.3080]0.2 7.0 0.2223 [0.2248] 0.0534 [0.0547] 1.0138 [1.0114] 2.6216 [2.6202]0.3 0.2229 [0.2261] 0.0533 [0.0546] 1.0015 [0.9808] 2.6085 [2.5871]3.0 0.2266 [0.2288] 0.0522 [0.0537] 0.9207 [0.9185] 2.5210 [2.5194]

10.0 0.2274 [0.2297] 0.0520 [0.0534] 0.9007 [0.8985] 2.4989 [2.4973]20.0 0.2276 [0.2299] 0.0519 [0.0534] 0.8956 [0.8933] 2.4932 [2.4916]

Table 4Comparison of Nusselt number ð�θ0ð0ÞÞ when Nb¼Nt ¼M¼ Ec¼ 0 and n¼ 1.

Pr Khan and Pop [4] Wang [18] Gorla and Sidawi [19] Present result

0.07 0.0663 0.0656 0.0656 0.06650.20 0.1691 0.1691 0.1691 0.16910.70 0.4539 0.4539 0.4539 0.45392.00 0.9113 0.9114 0.9114 0.91147.00 1.8954 1.8954 1.8905 1.8954

20.00 3.3539 3.3539 3.3539 3.353970.00 6.4621 6.4622 6.4622 6.4622

Table 5Calculation of skin friction coefficient, Nusselt and Sherwood number for variousvalues of Ec and M when Pr¼ 6:2; Le¼ 5; Nb¼Nt ¼ 0:1 and n¼ 2; η1 ¼ 10.

Ec M � f ″ð0Þ �θ0ð0Þ �ϕ0ð0Þ

0 0 1.10102 1.06719 1.077190.1 0.88199 1.223450.2 0.70998 1.370780.3 0.52953 1.519190.5 0.16484 1.819330 0.5 1.30989 1.04365 1.010900.1 0.81055 1.206050.2 0.57564 1.402790.3 0.33889 1.601150.5 0.14022 2.002820 1 1.48912 1.02337 0.954950.1 0.74058 1.194960.2 0.45543 1.437060.3 0.16789 1.681280.5 0.41451 2.17623

f' ()

0 2 4 6 80

0.2

0.4

0.6

0.8

1

0

1

n = 0, 1, 2

M

Pr = 6.2, Le = 5, Nb = Nt = Ec = 0.1

Fig. 2. Effects of M and n on dimensionless velocity.

F. Mabood et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 569–576572

()

0 1 2 30

0.2

0.4

0.6

0.8

1

0

5

n = 0, 1, 2

M

Pr = 6.2, Le = 10,Nb = Nt = Ec = 0.1

()

0 1 2 30

0.2

0.4

0.6

0.8

1

0

5

n = -4, -3, -2M

Pr = 6.2, Le = 10,Nb = Nt = Ec = 0.1

Fig. 3. Effects of M and n on dimensionless temperature.

()

0 2 4

0

0.2

0.4

0.6

0.8

1

0

5

M

Pr = 6.2, Le = 5,Nb = Nt = Ec = 0.1

n = 0, 1, 2

()

0 2 4

0

0.2

0.4

0.6

0.8

1

0

5

M

Pr = 6.2, Le = 5,Nb = Nt = Ec = 0.1

n = -4, -3, -2

Fig. 4. Effect of n and M on dimensionless concentration.

()

0 1 2 30

0.2

0.4

0.6

0.8

1

0.1

0.2

Nb

Pr = 6.2, Le = 5, M = 0.2, Ec = 0.1, n = 2

Nt = 0.1, 0.3, 0.5

Fig. 5. Effects of Nb and Nt on dimensionless temperature.

()

0 2 4

0

0.25

0.5

0.75

1

0.3

0.5

Nb

Pr = 6.2, Le = 5, Ec = n = 2, M = 1, Ec = 0.1

Nt = 0.1, 0.2, 0.3

Fig. 6. Effects of Nt and Nb on dimensionless concentration.

F. Mabood et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 569–576 573

displayed in Fig. 10. It is also observed that heat transfer rates aresignificantly reduced with magnetic parameter M, while for aslight reduction in heat transfer rate is also observed with Eckertnumber Ec and nonlinear stretching sheet parameter n. The effects

()

0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

0

0.3

Ec

Pr = 6.2, n = 2, M = Nb = Nt = 0.5

Le = 5, 10, 50

Fig. 7. Effects of Le and Ec on dimensionless temperature.

()

0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

0

0.3

Ec

Pr = 6.2, n = 2, M = Nb = Nt = 0.1

Le = 5, 10, 50

Fig. 8. Effects of Le and Ec on dimensionless concentration.

M

-f'' (0)

0 1 2 3 4 50.5

1

1.5

2

2.5

n = 0, 1, 2

Pr = 6.2, Le = 5, Nt = Nb = Ec = 0.1

Fig. 9. Effects of M and n on skin friction coefficient.

M

-' (0)

0 1 2 3 4 50.2

0.4

0.6

0.8

1

5

10

Pr = 6.2, Nb = Nt = Ec = 0.1

Le

n = 0, 1, 2

Fig. 10. Variation of heat transfer rate with M and n.

Nt

-' (0)

0.1 0.2 0.3 0.4 0.50

0.15

0.3

0.45

0.6

0.75

0.9

1.05

0.1

0.3

Pr = 6.2, Le = 5, M = 0, n = 2

Nb

Ec = 0, 0.1, 0.2

Fig. 11. Variation of heat transfer rate with Nt, Ec and Nb.

Nt

-' (0)

0.1 0.2 0.3 0.4 0.51.56

1.6

1.64

1.68

1.72

1.76

0.6

0.7

Nb

Pr =6.2, Le = 5, M = 0.1, n = 2

Ec = 0, 0.1, 0.2

Fig. 12. Variation of dimensionless concentration with Nt, Ec and Nb.

F. Mabood et al. / Journal of Magnetism and Magnetic Materials 374 (2015) 569–576574

of Brownian motion, thermophoresis and viscous dissipationparameters on the reduced Nusselt number are depicted inFig. 11. We noticed that rate of heat transfer reduces with allparameters. As Ec increases, it can be seen that maximum of rateof heat transfer decreases in magnitude. The strength of Brownianmotion and thermophoretic effects will be greater for the largevalues of Nb and Nt. It is further noticed from Fig. 11 that for theweaker Brownian motion ðNb¼ 0:1Þ the change in the Nt parameterhas a minor impact on the reduced Nusselt number. However, whenthe value of Brownian motion parameter Nb is increased from 0.1 to0.3 the reduced Nusselt number effectively decline due to thermo-phoresis parameter Nt. This reduction occurs because of the nano-particles movement from stretching wall to the quiescent fluid. InFig. 12, the combined influence of Brownian motion and thermophor-esis parameters on the reduced Sherwood number is observed in thepresence of magnetic field for the different values of viscous dissipa-tion parameter. We found a monotonic increase in the reducedSherwood number when Nt and Ec are increased. In case of Brownianmotion parameter Nb, the reduced Sherwood number showed asignificant decrease far away from the sheet. Finally, in Fig. 13, thedimensionless Sherwood number is found to be decreases slightlywithmagnetic parameterM and nonlinear stretching parameter n, butincreases monotonically with Lewis number Le. This happens since alarger Lewis number will correspond to smaller mass diffusivity, andconsequently a thinner concentration boundary layer. The net effect isa larger concentration gradient at the wall which would then enhancethe mass transfer rate.

4. Conclusion

MHD boundary layer flow and heat transfer of a water-basednanofluid over a nonlinear stretching sheet with viscous dissipa-tion have been investigated numerically. A similarity solution ispresented which depends on Prandtl and Lewis numbers, mag-netic, viscous dissipation, nonlinearity of stretching sheet, Brow-nian motion and thermophoresis parameters. The effects ofgoverning parameters on the flow, concentration and heat transfercharacteristics are presented graphically and quantitatively. Themain observations of the present study are as follows:

� The skin friction coefficient increases, whereas the reduced Nusseltand Sherwood numbers decrease with magnetic parameter

� Rate of heat transfer decreases, whereas mass transfer rateincreases with viscous dissipation parameter

� The increase in Brownian motion parameter Nb and thermo-phoresis parameter Nb is to increase the temperature in thethermal boundary layer which consequently reduces the heattransfer rate at the surface

� The heat transfer rate decreases and mass transfer rateincreases with Lewis number Le.

Acknowledgments

We acknowledge the financial support of Universiti SainsMalaysia RU grant 1001/PMATHS/811252. The first authoracknowledges the USM fellowship.

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Le

-' (0)

5 6 7 8 9 10

1.65

1.8

1.95

2.1

2.25

2.4

2.55

2.7

0

1

Pr = 6.2, Le = 5, Ec = 0.1, Nb = Nt = 0.7

n = 0, 1, 2

M

Fig. 13. Variation of dimensionless concentration with Le, M and n.

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