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ORINIGAL ARTICLE

Thermo-diffusion effects on MHD stagnationpoint flow towards a stretching sheetin a nanofluid

Umar Khana, Naveed Ahmeda, Sheikh Irfan Ullah Khana,b,Syed Tauseef Mohyud-dina,n

aDepartment of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, PakistanbCOMSATS Institute of Information Technology, University Road, Abbottabad, Pakistan

Received 25 March 2014; accepted 3 July 2014

KEYWORDS

Nanofluid;Stretching sheet;Thermodiffusion;Numerical solution;Nusselt number;Sherwood number;Nanofluid Sherwoodnumber

ational Laboratory f16/j.jppr.2014.07.00

or: Tel.: þ92 32355

edtauseefs@hotmaild-din).

esponsibility of Natia.

le as: Umar Khan,er Research (2014),

Abstract Thermodiffusion effects on stagnation point flow of a nanofluid towards astretching surface with applied magnetic field is presented. Similarity transforms are appliedto reduce the equations that govern the flow to a system of nonlinear ordinary differentialequations. Runge-Kutta-Fehlberg method is applied to solve the system. Results are comparedwith existing solutions that are special cases to our problem. Concrete graphical analysis iscarried out to study the effects of different emerging parameters such as stretching ratio A,magnetic influence parameter M, Prandtl number Pr, Lewis number Le, Brownian motionparameter Nb, thermophoresis parameter Nt, nanofluid Lewis number Ln, modified Dufourparameter Nd and Dufour solutal number Ld coupled with comprehensive discussions.Numerical effects of local Nusselt number, local Sherwood number and nanofluid Sherwoodnumber are also discussed.& 2014 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V.

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et al., Thermo-diffusion effects onhttp://dx.doi.org/10.1016/j.jppr.2014.

1. Introduction

Over the years boundary layer flows have been of muchinterest to researchers because of their real world applicationssuch as engineering melt spinning, manufacturing of rubbersheets, glass fiber production, and so on. One of theseboundary layer flows is the stagnation point flow. Cooling of

duction and hosting by Elsevier B.V. All rights reserved.

MHD stagnation point flow towards a stretching sheet in a nanofluid,07.006

Nomenclature

A velocity ratio parameterB0 applied magnetic fieldcf skin friction coefficientCw solutal concentration at the surfaceTw temperature at the surfaceϕw nanoparticle volume fraction at the surfaceC1 ambient solutal concentrationT1 ambient temperatureϕ1 ambient nanoparticle volume fractionU1 free stream velocityu, v velocity components along x and y directionsM magnetic parameterf dimensionless stream functions dimensionless concentrationDT thermophoresis diffusion coefficientDS solutal diffusivityDB Brownian motion diffusion coefficientDTC Dufour diffusivityDCT Soret diffusivityPr Prandtl numberNb Brownian motion parameterNt thermophoresis parameterNd modified Dufour paramterLe Lewis numberLd Dofour solutal Lewis numberLn nano Lewis number

Nux local Nusselt numberShx Sherwood numberShrx nanofluid Sherwood numberNur reduced Nusselt numberShr reduced Sherwod numberShrn reduced nanofluid Sherwood number

Greek symbols

η dimsensionless similarity variableμ dynamic viscosity of the fluidυ kinematic viscosity of the fluidðρÞf density of the base fluidðρcÞf heat capacity of the base fluidðρcÞp effective heat capacity of the nanoparticleψ stream functionαm thermal diffusivityσ electric conductivityθ dimensionless temperatureϕ dimensionless nanoparticle volume fractionτ parameter defined by ðρcÞp=ðρcÞf

Subscripts

1 condition at free streamw condition at the wall

Umar Khan et al.2

electronic devices by fans, cooling of nuclear reactors duringemergency shutdowns, etc. are the applications of stagnationflows. Crane [1] first studied this problem for stagnationpoint flow towards a solid surface. After this seminal work,many researchers explored various aspects of boundary layerflows incorporating innumerable physical configurations[2–6].In real world, most of the fluids such as water, kerosene

oil, ethylene, glycol, and others are poor conductors of heatdue to their lower values of thermal conductivity. To copeup with this problem and to enhance the thermal conduc-tivity or other thermal properties of these fluids, a newlydeveloped technique is used which includes, addition ofnano-sized particles of good conductors such as copper,aluminum, titanium, iron and other oxides to the fluids.Choi [7] was the first one to come up with this idea. Choiet al. [8] also showed that the thermal conductivity ofconventional fluids can be doubled by adding nano particlesto base fluids that also incorporate other thermal properties.These enhancements can be used practically in electroniccooling, heat exchangers, double plane windows, etc.Buongiorno [9] presented a more comprehensive modelfor the nanotechnology based fluids that unveils the thermalproperties superior to base fluids. He discussed all theconvective properties of nanofluids by developing a moregeneral model. After these developments in nanofluids,Khan and Pop [10] were the first ones to study boundarylayer flow over a stretching sheet by using the model of

Please cite this article as: Umar Khan, et al., Thermo-diffusion effects onPropulsion and Power Research (2014), http://dx.doi.org/10.1016/j.jppr.2014.

Nield and Kuznetsov [11,12]. Mustafa et al. presented firststudy on stagnation point flow of a nanofluid [13]. Theypresented both Brownian motion and thermophoresis effectson transport equations by reducing them to a nonlinearboundary value problem. One can easily find enoughliterature on nano fluid flows, however some of the studiesare reported in [14–18] and references therein.

Study of electrically conducting fluids hold importancedue to applications in modern metallurgy and metal work-ing processes. Magnetic nanofluids are used to regulate theflow and heat transfer by controlling the fluid velocity.Mahaputra [19] studied MHD stagnation point flow over astretching sheet using numerical simulations. MHD stagna-tion point flow for nanofluid was presented by Ibrahim et al.[20] employing fourth order Runge-Kutta technique. Someothers studies regarding magneto-nanofluids are presentedin [21–23] and references therein.

In all presented studies, Soret and Dufour effects wereneglected. It is a well-known fact that the temperature andconcentration gradients present mass and energy fluxes,respectively. Concentration gradients result in Dufour effect(diffusion-thermo) while Soret effect (thermal-diffusion) isdue to temperature gradients. Such effects play a significantrole when there are density differences in the flow. For theflows of mixture of gases with light molecular weights(He, H2) and moderate weights (N2, air), Soret and Dufoureffects cannot be neglected. Thermo-diffusion effects on theflow over a stretching sheet are examined by Awad et al.

MHD stagnation point flow towards a stretching sheet in a nanofluid,07.006

Thermo-diffusion effects on MHD stagnation point flow towards a stretching sheet in a nanofluid 3

[24]. Same idea is extended for power law stretching sheetby Goyal et al. in a recent study [25].

To the best of our knowledge, no one has ever attemptedto study the thermo-diffusion effects for stagnation pointflow towards a stretching sheet. Main objective of this studyis to analyze Soret and Dufour effects for stagnation pointflow of electrically conducting nanofluid. In mathematicalformulation of the problem, combined effects of thermo-phoresis, Brownian motion and diffusion thermo are pre-sented. Resulting system of non-linear equations is solvedby employing a well-known numerical technique called theRunge-Kutta-Fehlberg method after converting the bound-ary value problem into an initial value problem usingshooting technique. Influence of emerging parameters onvelocity, temperature, solute and nanoparticle concentrationprofiles are presented with the help of graphs. A compre-hensive discussion is also included. Also, a comparison ofcurrent study to the previous ones is provided to validateour solutions.

2. Governing equations

In this problem, a two-dimensional stagnation point flowof a nanofluid towards a stretching sheet is taken intoaccount. Sheet is kept at a constant temperature Tw, solutalconcentration Cw and nanoparticle concentration ϕw. At alarge distance from the sheet, temperature, solutal concen-tration and nanoparticle concentration are represented byT1, C1 and ϕ1, respectively. A constant transversmagnetic field is applied to the flow in y-direction. Strengthof magnetic field is taken to be B¼B0. Induced magneticfield is assumed to be small as compared to appliedmagnetic field and is neglected. Fluid phase and nanopar-ticles are assumed to be in thermal equilibrium and there isno slip between them. A coordinate system is chosen suchthat the x-axis is along stretching sheet and y-axis is normalto it. Under assumptions mentioned above, governingequations for conservation of mass, momentum, energy,solute and nanoparticles can be written as [20]:

∂u∂x

þ ∂v∂y

¼ 0; ð1Þ

u∂u∂x

þ v∂u∂y

¼ � 1ρf

∂p∂x

þ U1∂U1∂x

þυ∂2u∂x2

þ ∂2u∂y2

� �þ σB2

0

ρfðU1�uÞ; ð2Þ

u∂v∂x

þ v∂v∂y

¼ � 1ρf

∂p∂y

þ υ∂2v∂x2

þ ∂2v∂y2

� �� σB2

0

ρfv; ð3Þ

u∂T∂x

þ v∂T∂y

¼ αm∂2T∂x2

þ ∂2T∂y2

� �

Please cite this article as: Umar Khan, et al., Thermo-diffusion effects onPropulsion and Power Research (2014), http://dx.doi.org/10.1016/j.jppr.2014.

þτ DT∂j∂x

∂T∂x þ ∂j

∂y∂T∂y

� �þ DT

T1∂T∂x

� �2 þ ∂T∂y

� �2� �

þDKTC∂2C∂x2

þ ∂2C∂y2

� �; ð4Þ

u∂C∂x

þ v∂C∂y

¼DS∂2C∂x2

þ ∂2C∂y2

� �

þDKCT∂2T∂x2

þ ∂2T∂y2

� �; ð5Þ

u∂ϕ∂x

þ v∂ϕ∂y

¼DB∂2ϕ∂x2

þ ∂2ϕ∂y2

!þ DT

T1

∂2T∂x2

þ ∂2T∂y2

� �; ð6Þ

with boundary conditions

u¼ uw¼ ax; v¼ 0; T ¼ Tw; C¼Cw; φ ¼ φw at y¼ 0;

u¼U1 ¼ bx; v-0; T-T1;C-C1; φ-φ1 at y-1; ð7Þ

where, u and v are velocity components along x and y-axis,respectively, U1 is the free stream velocity, ρf is density ofbase fluid, υ¼ μ=ρf is kinematic viscosity, p is pressure, σis electrical conductivity, B0 is magnetic field flux density, αis thermal diffusivity, DB is Brownian motion diffusioncoefficient, DT is thermo-phoresis diffusion coefficient, DTC

and DCT are Soret and Dufour diffusivities, DS is solutaldiffusivity, T is fluid temperature, C is solutal concentration,ϕ is the nanoparticle volume fraction, ðρcÞf and ðρcÞp areheat capacity of fluid and effective heat capacity ofnanoparticle material respectively, τ is the parameterdefined by ðρcÞf =ðρcÞp. Using boundary layer approxima-tions, Eqs. (1)–(6) reduce to

∂u∂x

þ ∂v∂y

¼ 0; ð8Þ

u∂u∂x

þ v∂u∂y

¼U1∂U1∂x

þ υ∂2u∂y2

� �þ σB2

0

ρfðU1�uÞ; ð9Þ

u∂T∂x

þ v∂T∂y

¼ αm∂2T∂y2

� �

þτ DT∂ϕ∂y

∂T∂y

!þ DT

T1þ ∂T

∂y

� �2( )" #

þ DTC∂2C∂y2

� �;

ð10Þ

u∂C∂x

þ v∂C∂y

¼DS∂2C∂y2

� �þ DCT

∂2T∂y2

� �; ð11Þ

MHD stagnation point flow towards a stretching sheet in a nanofluid,07.006

Figure 1 Variation of f 0ðηÞ for different values of A.

Figure 2 Variation of f 0ðηÞ for different values of M.

Umar Khan et al.4

u∂ϕ∂x

þ v∂ϕ∂y

¼DB∂2ϕ∂y2

!þ DT

T1

∂2T∂y2

� �: ð12Þ

For mathematical analysis of the problem, we usefollowing similarity transforms

η¼ffiffiffia

ν

ry; ψ ¼ ffiffiffiffiffi

aνp

f ðηÞ; θðηÞ ¼ T�T1Tw�T1

;

sðηÞ ¼ C�C1Cw�C1

; φðηÞ ¼ φ� φ1φw� φ1

ð13Þ

where stream function ψ is defined in a usual way asu¼ ∂ψ=∂y and v¼ �ð∂ψ=∂xÞ:After the implementation of Eq. (13), Eq. (8) is identi-

cally satisfied and from Eqs. (9)–(12), we get a system offollowing nonlinear differential equations,

f ″0 þ f f 0 � f 02 þMðA� f 0Þ þ A2 ¼ 0; ð14Þ

θ″þ Prf θ0 þ PrNbθ0φ0 þ PrNtθ02 þ Nds″¼ 0; ð15Þ

s″þ Lef s0 þ Ldθ″¼ 0: ð16Þ

ϕ″þ Lnfϕ0 þ Nt

Nbθ″¼ 0: ð17Þ

Using Eq. (13), boundary conditions Eq. (7) reduce to

f ð0Þ ¼ 0; f 0ð0Þ ¼ 1; f 0ð1Þ-A;

θð0Þ ¼ 1; θð1Þ-0;sð0Þ ¼ 1; sð1Þ-0;φð0Þ ¼ 1; φð1Þ-0; ð18Þ

where primes denote the differentiation with respect to η.Besides,

M ¼ σB02

ρf a; A¼ b

a; Pr¼ ν

αm; Le¼ αm

DS; Nb¼ τDBðϕw

4�ϕ1

4Þ

ν;

Nt¼ τDBðTw�T1ÞT1ν

; Ln¼ ν

DB;

Nd¼ DTCðCw�C1ÞαmðTw�T1Þ and Ld¼ DCT ðTw�T1Þ

αmðCw�C1Þ ;

represent the magnetic number, velocity ratio, Prandtlnumber, Lewis number, Brownian motion parameter, ther-mophoresis parameter, nano Lewis number, modifiedDufour parameter and Dufour solutal Lewis numberrespectively.The quantities of engineering interest are local Nusselt

number Nux, Sherwood number Shx and nanofluid Sher-wood number Shx;n. These parameters characterize the wallheat, regular and nano mass transfer rates, respectively andare defined by

Nux ¼ �x

Tw�T1

∂T∂y

� � y ¼ 0

¼ �Re1=2x θ0ð0Þ;

Please cite this article as: Umar Khan, et al., Thermo-diffusion effects onPropulsion and Power Research (2014), http://dx.doi.org/10.1016/j.jppr.2014.

Shx ¼�x

Cw�C1

∂C∂y

� � y ¼ 0

¼ �Re1=2x s0ð0Þ;

Shx;n ¼�x

φw� φ1

∂φ∂y

� � y ¼ 0

¼ �Re1=2x φ0ð0Þ:

Reduced form of the Nusselt, Sherwood and nanofluidSherwood number is given by

Nur¼ Nux

Re1=2x

; Shr¼ Shx

Re1=2x

and Shrn¼ Shx;n

Re1=2x

:

3. Numerical solution

System of coupled nonlinear Eqs. (14)–(17) are solvednumerically by using an efficient numerical technique Runge-Kutta-Fehlberg method. System of boundary value problemEqs. (14)–(17) are first converted into an initial value problemby using shooting technique. We choose ηmax such that oursolution converges for all the parameters. A fifth decimal placeaccuracy is restricted for the sake of convergence.

4. Results and discussions

In this section, effects of all the emerging parameters onvelocity, temperature, solute and nanoparticles concentration

MHD stagnation point flow towards a stretching sheet in a nanofluid,07.006

Thermo-diffusion effects on MHD stagnation point flow towards a stretching sheet in a nanofluid 5

profiles are highlighted with the help of graphs and tables.Figures 1–11 are plotted to discuss the effects of theseparameters. In Figure 1, behavior of A (ratio of free streamvelocity to the velocity of stretching sheet) is depicted onvelocity profile f 0ðηÞ. It can be observed that when free velocityis less than the velocity of stretching sheet, there is a decrease invelocity of the fluid and boundary layer thickness. On the otherhand, when free stream velocity is greater than the velocity ofstretching sheet, there is an increase in velocity of the fluid,while boundary layer thickness decreases. Figure 2 depicts thebehavior of magnetic number M on the velocity field. Rise inmagnetic number M for the case when velocity of stretching

Figure 5 Variation of θðηÞ for different values of Nb.

Figure 4 Variation of θðηÞ for different values of Nt.

Figure 3 Variation of θðηÞ for different values of Pr.

Please cite this article as: Umar Khan, et al., Thermo-diffusion effects onPropulsion and Power Research (2014), http://dx.doi.org/10.1016/j.jppr.2014.

sheet is greater than the main stream velocity, has deceleratingeffect on main flow velocity. That is, with increasing values ofM, velocity of the fluid decreases. This is because of theretardation force created by the Lorentz force that opposes theflow. Also, boundary layer thickness is a decreasing functionfor increasing M. However, increase in velocity is observed forthe case when main stream velocity is greater than the velocityof the stretching sheet. In short, magnetic field can be used tocontrol boundary layer separation in several physicalphenomenon.

Effects of Prandtl number Pr, Brownian motion para-meter Nb, thermophoresis parameter Nt and modified

Figure 6 Variation of θðηÞ for different values of Nd.

Figure 7 Variation of sðηÞ for different values of Le.

Figure 8 Variation of sðηÞ for different values of Ld.

MHD stagnation point flow towards a stretching sheet in a nanofluid,07.006

Table 1 Comparison of values of f ″ð0Þ with existing solutionsfor M¼0.

A↓ f ″ð0Þ

Present results Hayat [13] Mahaputra [19] Ibrahim [13]

0.01 �0.9980 �0.9982 �0.9980 �0.99800.1 �0.9694 �0.9695 �0.9694 �0.96940.2 �0.9181 �0.9181 �0.9181 �0.91810.5 �0.6673 �0.6673 �0.6673 �0.66732.0 2.0175 2.0176 2.0175 2.0175

Figure 9 Variation of ϕðηÞ for different values of Ln.

Figure 10 Variation of ϕðηÞ for different values of Nt.

Figure 11 Variation of ϕðηÞ for different values of Nb.

Umar Khan et al.6

Dufour parameter Nd on the temperature profile aredemonstrated in Figures 3–6.From Figure 3, it can be observed that increase in Pr

decreases temperature quite significantly. Since, Prandtlnumber is the ratio of viscous diffusion rate to the thermaldiffusion rate, therefore increase in Pr results in decrease ofthermal diffusivity. Figure 4 depicts the behavior oftemperature profile for increasing values of thermophoresisparameter Nt. The thermophoretic force generated by tempe-rature gradient results in a fast flow away from the stretchingsurface, therefore, increasing Nt results in increased tempera-ture. In Figures 5 and 6, effects of Brownian motion

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parameter Nb and modified Dufour solute parameter Nd ontemperature profile are plotted. It is clear that with anincrease in Nb and Nd, the temperature of the fluid increaseconsiderably.

Effects of Lewis number Le and Dufour-solute Lewisnumber Ld on concentration profile are shown in Figures 7and 8, respectively. Figure 7 depicts that the increase in Ledecreases sðηÞ quite significantly, i.e. there is a rapiddecrease in mass. Ld makes opposite effects on concentra-tion profile as compared to Le that is increasing Ld givesincreased sðηÞ. As a result the concentration boundary layerthickness increase.

Behavior of nanofluid Lewis number Ln, thermophoresisparameter Nt, Brownian motion parameter Nb, and onnanomass volume fraction profile ϕðηÞ are depicted inFigures 9–11. Figure 9 considers the behavior of Ln onϕðηÞ.

Ln is the ratio between thermal diffusivity and massdiffusivity; it is used to analyze the fluid flows wheresimultaneous effects of heat and mass transfer by convec-tion are considered.

Increment in nanofluid Lewis number Ln decreases ϕðηÞ.This is due to the fact that Brownian diffusion decreaseswith an increase in Ln that forces the concentration todecrease. Influences of thermophoresis parameter Nt onϕðηÞ are plotted in Figure 10. Rise in Nt increases ϕðηÞrapidly. This happens because of the fact that increasing Ntcorresponds to greater mass flux (due to temperaturegradient) that increase the concentration appreciably.

Increasing values of Brownian motion parameter Nb areplotted for ϕðηÞ in Figure 11. A considerable drop in ϕðηÞ isobserved for increasing values of Nb. Rise in Nb is due toincrease in Brownian motion that increases temperature,which in result is account for the fall in concentration.

Tables 1 and 2, are drawn to compare the results obtainedby Runge-Kutta-Fehlberg method with already existingresults in literature. From Table 1, it can be observed thatour results are in exact agreement to the ones obtained inprevious studies. In Table 2, numerical values of localNusselt number for the case of a regular Newtonian fluidare compared with the existing solutions. Again, excellentagreement is seen between the solutions.

Tables 3 and 4 are drawn to study the effects of differentparameters on reduced Nusselt number, reduced Sherwoodnumber and reduced nanofluid Sherwood number. From

MHD stagnation point flow towards a stretching sheet in a nanofluid,07.006

Table 2 Comparison of values local Nusselt number �θ0ð0Þwith existing solutions for Nb¼Nt¼Nd¼0.

Pr↓ A↓ �θ0ð0Þ

Present results Hayat [13] Mahaputra [19] Ibrahim [20]

1.0 0.1 0.6022 0.6021 0.603 0.60220.3 0.6255 0.6244 0.625 0.62550.5 0.6924 0.6924 0.692 0.6924

1.5 0.1 0.7768 0.7768 0.777 0.77680.3 0.7971 0.7971 0.797 0.79710.5 0.8648 0.8647 0.863 0.8648

Table 3 Variation of Nur and Shrn for different values of Nt andNb.

Nb¼0.3 Nb¼0.5 Nb¼0.7

Nt↓ Nur Shrn Nur Shrn Nur Shrn

0.1 0.3430 0.3157 0.3060 0.3491 0.2713 0.36300.2 0.3193 0.2020 0.2839 0.2938 0.2506 0.33190.3 0.6968 0.1164 0.2628 0.2532 0.2309 0.31030.7 0.2780 0.0271 0.2456 0.2018 0.2152 0.2837

Table 4 Variation of Shr with varying Ld and Le.

Le↓ Ld¼0.1 Ld¼0.3 Ld¼0.5Shr Shr Shr

1.0 0.1359 0.1277 0.11863.0 0.2676 0.2697 0.2663

Thermo-diffusion effects on MHD stagnation point flow towards a stretching sheet in a nanofluid 7

Table 3, one can clearly observe that with an increase inBrownian motion parameter Nb, decrease in values ofreduced Nusselt number is seen and reduced nanofluidSherwood number is found to be an increasing function ofNb. Thermophoresis parameter Nt effects the Nusseltnumber and nanofluid Sherwood number in a same manneras Brownian motion parameter Nt. That is, decrease inNusselt number and increase in nanofluid Sherwood num-ber is observed for increasing values of Nt.

Table 4 gives us the numerical values of reduced Sher-wood number for increasing values of Dufour solutalnumber Ld and Lewis number Le. A slight increase inSherwood number is seen for increasing values of Le. Whileincrease in Ld decreases the numerical values of Sherwoodnumber.

5. Conclusions

Analysis of thermodiffusion effects on stagnation pointflow of magneto nanofluid is presented. Governing equa-tions after implementing similar transforms are convertedinto system of non-linear ordinary differential equations.Numerical solution using Runge-Kutta-Fehlberg method is

Please cite this article as: Umar Khan, et al., Thermo-diffusion effects onPropulsion and Power Research (2014), http://dx.doi.org/10.1016/j.jppr.2014.

carried out. Emerging parameters are analyzed graphically.Comparison with existing solutions is also made to checkthe efficiency of numerical technique applied. It is con-cluded that increase in Brownian motion parameter Nb andthermophoresis parameter Nt result in decreasing values ofreduced Nusselt number and increasing reduced nanofluidSherwood number. Also, numerical values of reducedSherwood number increase slightly with increase in Dufoursolutal number Ld. Slight increase in Sherwood number isseen for increasing values of Le. Decrease in Sherwoodnumber us seen for increasing values of Lewis number Le.This work can be extended for non-Newtonian nanofluidsfor the better understanding of different properties ofnanofluids.

Acknowledgement

Authors are thankful to the anonymous reviews andeditor for their worthy comments that really helped toimprove the quality of presented work.

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