Research ArticleHeat Transfer Analysis of MHD Water Functionalized CarbonNanotube Flow over a StaticMoving Wedge

Waqar A Khan1 Richard Culham1 and Rizwan Ul Haq2

1Department of Mechanical and Mechatronics Engineering University of Waterloo Waterloo ON Canada N2L 3G12Department of Mathematics Quaid-i-Azam University Islamabad 45320 Pakistan

Correspondence should be addressed to Waqar A Khan wkhan 2000yahoocom

Received 6 November 2014 Accepted 2 March 2015

Academic Editor Theodorian Borca-Tasciuc

Copyright copy 2015 Waqar A Khan et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The MHD flow and heat transfer from water functionalized CNTs over a staticmoving wedge are studied numerically Thermalconductivity and viscosity of both single and multiple wall carbon nanotubes (CNTs) within a base fluid (water) of similarvolume are investigated to determine the impact of these properties on thermofluid performanceThe governing partial differentialequations are converted into nonlinear ordinary and coupled differential equations and are solved using an implicit finite differencemethod with quasi-linearization techniques The effects of volume fraction of CNTs and magnetic and wedge parameters areinvestigated and presented graphically The numerical results are compared with the published data and are found to be in goodagreement It is shown that the magnetic field reduces boundary layer thickness and increases skin friction and Nusselt numbersDue to higher density and thermal conductivity SWCNTs offer higher skin friction and Nusselt numbers

1 Introduction

Many manufacturing procedures consist of continuousstretching surfaces cooled by an external stream along theproduction line Boundary layer flow behavior over a mov-ing continuous solid surface is a significant type of flowhappening in several industrial processes The examples arethe thermal processing of sheet-like materials which is anecessary operation in the production of paper linoleumpolymeric sheets wire drawing drawing of plastic filmsmetal spinning roofing shingles insulating materials fine-fiber matts cooling of films or sheets conveyor belts metal-lic plates and cylinders In virtually all such processingoperations the sheet moves parallel to its own plane Themoving sheet may induce motion in the neighboring fluidor alternatively the fluid may have an independent forcedconvection motion that is parallel to that of the sheet Boththe kinematics of stretching and the simultaneous heating orcooling during such processes have a decisive influence on thequality of the final products In virtually all such processingoperations the sheet moves parallel to its own plane

The pioneering works of Falkner and Skan [1] Hartree[2] and Koh and Hartnett [3] provided the fundamentalsolutions for wedge problems and extended wedge problemin different directions They solved governing equations forlaminar flow of conventional fluids over permeable wedgeswith suction including isothermal and variable wall tem-perature distributions and calculated the heat transfer andskin friction for laminar flow Takhar and Pop [4] studied theproblem of MHD heat transfer from a wedge with a uniformsurface temperature in a fluid of large Prandtl numbersTheyobtained a closed form solution and compared their resultswith published data H-T Lin and L-K Lin [5] proposeda similarity solution method to determine accurate solu-tions for laminar forced convection heat transfer from bothisothermal and uniform-flux wedges to fluids of any PrandtlnumberThey presented a simple correlation equation for anywedge and any Prandtl number Watanabe [6] investigatedtheoretically the behavior of forced flow over a wedge withuniform suction or injectionThe resulting integral equationswere solved by iterative numerical quadratures Yih [7]investigated the effects of viscous dissipation and stress work

Hindawi Publishing CorporationJournal of NanomaterialsVolume 2015 Article ID 934367 13 pageshttpdxdoiorg1011552015934367

2 Journal of Nanomaterials

on the MHD forced convection adjacent to a nonisothermalwedge They used the Keller box method and presented thenumerical results for the local friction coefficient and thelocal Nusselt number

Hossain et al [8] considered forced flow and heat trans-fer of a viscous incompressible fluid having temperature-dependent viscosity and thermal conductivity past a wedgewith a uniform surface heat fluxThey presented the local skinfriction coefficient and rate of heat transfer for various valuesof the governing parameters Pal and Mondal [9] analyzedheat transfer characteristics of an incompressible Newto-nian electrically conducting and heat generatingabsorbingfluid having temperature-dependent viscosity over a non-isothermal wedge in the presence of thermal radiation Theyincluded the effects of viscous dissipation Joule heatingstress work and thermal radiation and obtained numericalresults for the velocity and temperature profiles as well as thelocal skin friction coefficient and local Nusselt number Ishaket al [10] investigatedMHDboundary layer flowpast a wedgewith constant surface heat flux immersed in an incompress-ible micropolar fluid They employed the Keller box methodto obtain numerical results and showed thatmicropolar fluidsdisplay drag reduction and consequently reduce the heattransfer rate at the surface Su et al [11] proposed a newanalytical method named DTM-BF and applied their studyto MHD mixed convective flow and heat transfer of an elec-trically conducting fluid over a stretching permeable wedgein the presence of thermal radiation and Ohmic heatingThey presented the effects of various embedded parametersand dimensionless parameters on the skin friction and heattransfer Prasad et al [12] obtained numerical solutions forthe hydromagnetic mixed convection boundary layer flow ofan electrically conducting fluid over a nonisothermal wedgeThey investigated various effects of the embeddeddimension-less parameters Rahman et al [13] discovered the effects ofthermophoresis on an unsteadyMHD forced convective heatandmass transfer flow of an electrically conducting fluid overa wedge with variable electrical conductivity They solvedlocally similar ordinary differential equations by applyingNachtsheim-Swigert shooting iteration technique along witha sixth order Runge-Kutta integration scheme They foundthat the skin friction coefficient Nusselt number and Sher-wood number are higher for the fluids of constant electricalconductivity than those of the variable electrical conductivityMuhaimin et al [14] analyzed the effects of thermophoresisparticle deposition and temperature-dependent viscosity onunsteady MHD mixed convective heat and mass transfer ofan electrically conducting fluid past a porous wedge in thepresence of a chemical reaction They compared their resultswith those known from the literature and found excellentagreement They found that the flow field is influencedappreciably by the appliedmagnetic field and thermophoresisparticle deposition Abbasbandy and Hayat [15] determinedthe solution of the Falkner-Skan flow by using a HomotopyAnalysis method They obtained the skin friction coefficientand compared their results with published data Fang andZhang [16] obtained an exact analytical solution of the famousFalkner-Skan equation which involves the boundary layerflow over a moving wall with mass transfer in the presence

of a free stream They analyzed the effects of mass transferand wall stretching

It is well known that convectional heat transfer fluids likewater or oil are poor heat transfer fluids due to lower thermalconductivity To enhance heat transfer thermal conductiv-ity of these fluids is increased by adding nanomicrosizedparticles of higher thermal conductivity These fluids areknown as nanofluids Following the introduction of thisidea by Choi [17] several researchers including [18ndash23] usednanofluids to improve heat transfer fromdifferent geometriesunder different thermal boundary conditions Khan and Pop[24] studied flow and heat transfer from a moving wedgein a water-based nanofluid They investigated the effectsof the governing parameters on the dimensionless velocitytemperature and nanoparticle volume fraction and presentedtheir results graphically Yacob et al [25] studied the problemof the steady flow and heat transfer over a static or movingwedge with a prescribed surface heat flux in a nanofluidTheyanalyzed and discussed three different types of nanoparticleswith water as the base fluid They found that the skin frictioncoefficient and the heat transfer rate are highest for a copperndashwater nanofluid compared to the other nanofluids consideredin their study Rahman et al [26] investigated numerically theheat transfer characteristics of water-based nanofluids over awedge with slip and a convective surface They showed thatthe nanofluid velocity is lower than the velocity of the basefluid and the existence of the nanofluid leads to the thinningof the hydrodynamic boundary layerThey also found that therate of heat transfer increases with an increase of the surfaceconvection and the slip parameters Later on Rahman et al[27] extended their study to include thermal radiation andshowed that the rate of heat transfer in a nanofluid in thepresence of thermal radiation significantly depends on thesurface convection parameter Chamkha et al [28] studiedmixed convection boundary layer flow over an isothermalvertical wedge embedded in a porous medium saturated witha nanofluid They included thermal radiation and nanofluidparameters in the analysis They presented numerical resultsfor the velocity temperature and volume fraction the localNusselt and Sherwood numbers Kameswaran et al [29]presented a mathematical model for the two-dimensionalsteady incompressible convective heat and mass transferfrom an isothermal wedge immersed in a nanofluid Theyconsidered two types of water-based nanofluids and includedthe effects of the volume fraction parameter

While there are many materials that can be used toform nanoparticles carbon is very promising due to itshigh thermal electrical and mechanical properties [30] Acarbon nanotube is composed of a single wall (SWCNT)or multiwall (MWCNT) The thermal properties of carbonnanotubes CNT suspensions are found to be much higherthan those of other nanoparticles with the same volumefraction [31ndash33]They noticed enhancement in thermal prop-erties of nanofluids with temperature and volume fractionDing et al [34] investigated the heat transfer behavior ofaqueous suspensions of multiwalled carbon nanotubes (CNTnanofluids) flowing through a horizontal tubeThey observedsignificant enhancement of the convective heat transfer Lateron Halelfadl et al [35] extended these findings to include

Journal of Nanomaterials 3

SWCNT

1-2nm

02ndash5120583m u

Ω = 120573120587

xy

ue(x)

Tinfin

Uw

(a)

MWCNT u

Ω = 120573120587

y x

ue(x)

Tinfin

Uw

(b)

Figure 1 Geometry of the problem

a coaxial heat exchanger under a laminar flow regime withinthe range of Reynolds numbers 500ndash2500 and measuredthermal and rheological properties of the nanofluids Theystudied the effects of the aspect ratio of carbon nanotubesthe type of base fluid and surfactant on viscosity thermalconductivity and laminar convective heat transfer Khan et al[36] employed homogeneous flow model to study the flowand heat transfer of carbon nanotubes (CNTs) along a flatplate subjected toNavier slip and uniform heat flux boundaryconditions They investigated the effects of the governingparameters on the dimensionless velocity temperature skinfriction and Nusselt numbers and presented numericalresults in graphical and tabular forms Recently Ul Haq et al[37] studied thermophysical effects of carbon nanotubes onMHD flow over a stretching sheet They considered threetypes of base fluids specifically water ethylene glycol andengine oil with single and multiwalled carbon nanotubes andfound that the engine oil provides the highest skin frictionand heat transfer rates

The above literature review reveals that there are nostudies that investigate thermal performance of carbon nan-otubes over static or moving wedges The objective of thisstudy is to explore the heat transfer properties of water-basednanofluids containing SWCNTs andMWCNTs To the best ofour knowledge the results of this paper are original and theyhave not been published before

2 Mathematical Model

Consider steady two-dimensional boundary layer flow pasta static or a moving wedge in a water-based nanofluidcontaining SWCNTs and MWCNTsThe physical model andgeometrical coordinates are depicted in Figure 1

The nanofluid is assumed incompressible and the flowis assumed to be steady and laminar It is also assumedthat the base fluid (ie water) and the nanoparticles are inthermal equilibrium and no slip occurs between them The

Table 1 Thermophysical properties of water and CNTs [41 42]

Thermophysicalproperties

Base fluid Carbon nanotubesWater SWCNT MWCNT

120588 (kgm3) 997 2600 1600119888119901 (Jkg-K) 4179 425 796119896 (Wm-K) 0613 6600 3000Pr 62 mdash mdash

thermophysical properties of the fluid and nanoparticles aregiven in Table 1 Further it is assumed that the velocity ofthe free stream (inviscid flow) is 119906119890(119909) = 119880infin119909

119898 and thatof the moving wedge is 119906119908(119909) = 119880119908119909

119898 where 119880infin 119880119908

and 119898 are constants with 0 le 119898 le 1 We consider atwo-dimensional Cartesian coordinate system (119909 119910) where119909 and 119910 are the coordinates measured along the surface ofthe wedge and normal to it respectively Under the boundarylayer approximations the basic steady conservation of massmomentum and energy equations can be written as

120597119906

120597119909+

120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= 119880119890

119889119880119890

119889119909+ ]nf

1205972119906

1205972119910+

1205901198612

119900

120588nf(119880119890 minus 119906) (2)

119906120597119879

120597119909+ V

120597119879

120597119910= 120572nf

1205972119879

1205971199112+

]nf(120588119888119901)nf

(120597119906

120597119910)

2

(3)

Subject to the boundary conditions

(1) static wedge

V = 0 119906 = 0 119879 = 119879119908 at 119910 = 0

119906 = 119906119890 (119909) 119879 = 119879119908 as 119910 997888rarr infin

(4)

4 Journal of Nanomaterials

(2) moving wedge

V = 0 119906 = 119906119908 (119909) 119879 = 119879119908 at 119910 = 0

119906 = 119906119890 (119909) 119879 = 119879119908 as 119910 997888rarr infin

(5)

Here 119906 and V are the velocity components along the 119909- and 119910-axes respectively 119879 is the temperature of the base fluid ]nf isviscosity of the nanofluid 120588nf is the density of the nanofluidand120572nf is the thermal diffusivity of nanofluid which are givenby

]nf =120583nf120588nf

120583nf =120583119891

(1 minus 120601)25

120572nf =119896nf

120588nf (119862119901)nf

120588nf = (1 minus 120601) 120588119891 + 120601120588CNT

(120588119862119901)nf= (1 minus 120601) (120588119862119901)119891

+ 120601 (120588119862119901)CNT

119896nf119896119891

= (1 minus 120601 + 2120601119896CNT

119896CNT minus 119896119891

ln119896CNT + 119896119891

2119896119891

)

sdot (1 minus 120601 + 2120601119896119891

119896CNT minus 119896119891

ln119896CNT + 119896119891

2119896119891

)

minus1

(6)

where 120583119891 is the viscosity of base fluid 120601 is the nanoparticlefraction (120588119862119901)nf is the effective heat capacity of CNTs119896nf is the thermal conductivity of nanofluid 119896119891 and 119896CNTare the thermal conductivities of the base fluid and CNTsrespectively and 120588119891 and 120588CNT are the thermal conductivitiesof the base fluid and CNTs respectively Introducing thefollowing similarity transformations

120595 = [2]119891119909119906119890(119909)

119898 + 1]

12

119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908 minus 119879infin

120578 = [(119898 + 1)119906119890(119909)

2]119891119909]

12

119910

(7)

where 120595 is the stream function and is defined in the usualway as 119906 = 120597120595120597119910 and V = minus120597120595120597119909 so as to identicallysatisfy (1) and ]119891 is the kinematic viscosity of the base fluidOn substituting (7) into (2) and (3) along with the boundaryconditions defined in (4) and (5) we obtain the followingordinary differential equations

1

(1 minus 120601)25

(1 minus 120601 + 120601120588CNT120588119891)

119891101584010158401015840

+ 11989111989110158401015840

+ (2119898

119898 + 1) (1 minus 119891

10158402)

+1198722

(1 minus 120601 + 120601120588CNT120588119891)(1 minus 119891

10158402) = 0

(8)

1

Pr119896nf119896119891

1 minus 120601 + 120601 (120588119888119901)CNT (120588119888119901)119891

12057910158401015840

+ 1198911205791015840

+119864119888

(1 minus 120601)25

119891101584010158402

= 0

(9)

subject to the boundary conditions

119891 (0) = 0 1198911015840

(0) = 120582 120579 (0) = 1

1198911015840

997888rarr 1 120579 997888rarr 0 as 120578 997888rarr infin

(10)

where the primes denote differentiationwith respect to 120578Theparameters 120582 120573 and Pr are defined as

120582 =119880119908

119880infin

120573 =2119898

119898 + 1 Pr =

]119891120572119891

(11)

where 120582 is the constant moving wedge parameter with 120582 gt

0 and 120582 lt 0 correspond to a moving wedge in the sameand opposite directions to the free stream respectively while120582 = 0 corresponds to a static wedge Further 120573 is the Hartreepressure gradient parameter which corresponds to 120573 = Ω120587

for a total wedge angle Ω According to White [38] positivevalue of120573means the pressure gradient is negative or favorableand negative values of 120573 denote an unfavorable pressuregradient Further 120573 = 0 (119898 = 0) represents the boundarylayer flow past a horizontal flat plate while 120573 = 1 (119898 = 1)corresponds to the boundary layer flow near the stagnationpoint of a vertical flat plate

It is important to note that (8) in the absence of amagneticfield (119872 = 0) and CNTs (120601 = 0) reduces to the same equationas obtained by Fang and Zhang [16] who provided an exactanalytical solution of the famous Falkner-Skan equation for aconventional fluid when 120573 = minus1

The skin friction coefficient 119862119891 and local Nusselt numberNu119909 are defined as

119862 =120591119908

1205881198911199062119890

Nu119909 =119909119902119908

119896119891 (119879119908 minus 119879infin) (12)

where 120591119908 and 119902119908 are the shear stress and heat flux at the wallwhich is given by

120591119908 = 120583nf (120597119906

120597119910)

119910=0

119902119908 = minus119896nf (120597119879

120597119910)

119910=0

(13)

Substituting (7) into (13) and (14) we obtain

Re12119909

119862119891 = radic119898 + 1

2

1

(1 minus 120601)25

11989110158401015840

(0)

Reminus12119909

Nu119909 = radic119898 + 1

2

119896nf119896119891

1205791015840

(0)

(14)

where Re119909 = 119906119890119909]119891 is the local Reynolds number

Journal of Nanomaterials 5

Table 2 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 White [38] Yacob et al [43] Yih [44] Khan and Pop [24] Present results0 04696 04696 04696 04696 04696111 06550 06550 06550 06550 0655015 08021 08021 08021 08021 0802113 09277 09277 09276 09277 0927712 10389 10389 mdash 10389 103891 12326 12326 12326 12326 12326

3 Numerical Method

Nonlinear ordinary differential equations (8) and (9) withboundary conditions (12) are solved by using an implicit finitedifference method with a linearization technique The detailsof thismethod are omitted here to save space but can be foundin [39 40] The step size is taken as Δ120578 = 0001 and theconvergence criteria are set to 10minus6The asymptotic boundaryconditions given by (12) were replaced by using a value of 10for the similarity variable 120578max as follows

120578max = 10 1198911015840

(10) = 1 120579 (10) = 0 (15)

The choice of 120578max = 10 ensures that all numerical solutionsapproached the asymptotic values correctly

4 Results and Discussion

MHD flow of water-based carbon nanotubes (CNTs) overstatic and moving wedges is investigated numerically Thethermophysical properties of water and both CNTs arepresented in Table 1 The numerical results are compared inTables 2 and 3 for skin friction and in Tables 4 and 5 forheat transfer in some special cases The results are foundto be in excellent agreement with the published data Theeffects of the magnetic field and volume fraction of SWCNTson the dimensionless velocity are displayed in Figures 2 and3 for flow past horizontal and near the stagnation point ofvertical stationary and moving flat plates respectively It canbe seen that the hydrodynamic boundary layer thickness islarger in the absence of a magnetic field in each case Infact the magnetic field tends to reduce the boundary layerthickness Figures 2(a) and 2(b) depict velocity profiles for theflow of SWCNTs over a stationary horizontal plate (119898 = 0)and near the stagnation point of a vertical stationary plate(119898 = 1) respectively In the absence of magnetic field theeffect of volume fraction of SWCNTs on the dimensionlessvelocity can be noticed but as the magnetic field increasesits effect diminishes When the horizontal and vertical platesmove in the free stream direction the dimensionless velocityprofiles are shown in Figures 3(a) and 3(b) respectively Itis important to note that the boundary layer thickness isreduced in this case The effects of volume fraction of CNTsare found to be negligible and the velocity boundary layerthickness reduces with an increase in magnetic field in bothcases

Table 3 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for aconventional fluid (120601 = 0)

120573 Kuo [45] Rajagopal et al [46] Present results0 04696 04696 04696005 05317 05311 0531101 05879 05870 0587002 06876 06867 0686703 07755 07748 0774804 08549 08544 0854405 09279 09277 0927706 09958 09958 0995807 10594 mdash 1059808 11196 11203 1120309 11767 mdash 117771 12313 12326 123262 16831 mdash 16872

Table 4 Comparison of dimensionless heat transfer minus1205791015840(0) when

119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 Kuo [45] Blasius [47] Khan and Pop [24] Present results0 08673 08673 08769 087691 11147 11152 11279 11280

The effects of magnetic field and volume fraction ofSWCNTs are illustrated in Figure 4 for stationary platesand in Figure 5 for moving plates in opposite direction tofree stream respectively In each case the magnetic fieldtends to reduce the thermal boundary layer thickness Thecomparison of Figures 4(a) and 4(b) shows that the thermalboundary layer thickness is smaller in the case of flowover a horizontal plate and hence less thermal resistance isoffered in this case However when the plates are moving inthe direction of the free stream the magnetic field has noappreciable effect on the thermal boundary layer thickness (asshown in Figures 5(a) and 5(b)) Conventional fluids (phi =0) have a smaller thermal boundary layer thickness and as thevolume fraction of SWCNTs increases the thermal boundary

6 Journal of Nanomaterials

Table 5 Comparison of dimensionless heat transfer minus1205791015840(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

Pr 120573 = 0 120573 = 03 120573 = 1

White [38] Kuo [45] Present results White [38] Kuo [45] Present results White [38] Kuo [45] Present results1 04696 04696 04696 05195 05200 05195 05705 05708 057052 05972 05972 05972 06691 06696 06690 07437 07441 074373 06860 06860 06860 07734 07741 07734 08652 08657 086526 08673 08673 08673 09873 09881 09873 11147 11152 1114710 10297 10297 10297 11791 11800 11791 13388 13394 1338830 14873 14873 14873 17198 17210 17198 19706 19714 1970660 18746 18746 18746 21776 21791 21776 25054 25063 25054100 22229 22229 22229 25892 25910 25892 29863 29874 29863

0 1 2 3 4 50

02

04

06

08

1

01

M

m = 0

120601 = 0 01 02

f998400 (120578)

120578

120582 = 0 Ec = 05

(a)

0 1 2 3 4

01

M

m = 1

120601 = 0 01 02

0

02

04

06

08

1

f998400 (120578)

120578

120582 = 0 Ec = 05

(b)

Figure 2 Effect of magnetic field on dimensionless velocity of water-based SWCNTs for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

layer thickness also increases due to an increase in the densityof CNTs

The variation of skin friction with magnetic field andvolume fraction of CNTs is presented in Figure 6 for thehorizontal plate moving in an opposite direction to thefree stream In Figure 6(a) SWCNTs are used whereas inFigure 6(b) MWCNTs are used to show the comparison ofskin friction in the presence of a magnetic field It is noticedthat the skin friction increases with volume fraction of CNTsin both cases However skin friction of SWCNTs is foundto be higher than MWCNTs It is due to higher density ofSWCNTs than MWCNTs (Table 1) The effect of magneticfield creates a Lorentz force which acts against the flow andthus enhances the skin friction As the horizontal platemovesin the opposing flow the hydraulic resistance also increases

which helps in an increase in skin frictionWhen the horizon-tal platemoves in an assisting flow Figures 7(a) and 7(b) showthe variation of skin friction with volume fraction of CNTsand magnetic field In this case the resistance decreases andhence the skin friction decreases with an increase in the platevelocity The same behavior was observed when a verticalplate moves in opposing flow For a vertical plate moving inopposite direction to the free stream the variation of skinfriction with magnetic field and volume fraction of CNTs isillustrated in Figure 8 for both CNTs in the presence of amagnetic field Similar to the horizontal plate skin frictionalso increases with plate velocity as shown in Figures 8(a)and 8(b) Figures 9(a) and 9(b) present the variation of skinfriction of CNTs along wedges moving in opposite directionto the free stream in the presence of a magnetic field Due to

Journal of Nanomaterials 7

0 1 2 3 4

01

M

120601 = 0 01 02

f998400 (120578)

120578

m = 0

05

06

07

08

09

1

11

120582 = Ec = 05

(a)

0 1 2 3

01

m = 1

f998400 (120578)

120578

M

120601 = 0 01 02

05

06

07

08

09

1

11

120582 = Ec = 05

(b)

Figure 3 Effect of magnetic field on dimensionless velocity of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 1 2 3

01

M

120601 = 0 01 02

0

02

04

06

08

1

120578

m = 0

120579(120578)

120582 = 0 Ec = 05

(a)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

0

02

04

06

08

1

120579(120578) 120582 = 0 Ec = 05

(b)

Figure 4 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

the increase in density with volume fraction of CNTs the skinfriction increases with volume friction of CNTs and wedgeparameter For conventional fluid (water) the skin friction isthe same for both CNTs Again the skin friction is found tobe higher for SWCNTs for both wedges

The variation of Nusselt numbers with volume fractionof CNTs magnetic and wedge parameters is reported inFigures 10ndash13 for special cases Due to higher thermalconductivity SWCNTs possess higher Nusselt numbers ineach case The effects of magnetic field and the motion

8 Journal of Nanomaterials

0 05 1 15 2

01

M

120601 = 0 01 02

120578

m = 0

0

02

04

06

08

1120579(120578) 120582 = Ec = 05

(a)

0

1

120579(120578)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

02

04

06

08

120582 = Ec = 05

(b)

Figure 5 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 01 02

3

1M

120601

120582 = minus01 minus03 minus05

005 01505

1

15

15

2

25

Re12

xCfx

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

m = 0 Ec = 005

(b) MWCNT

Figure 6 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

of the horizontal plate in the direction opposite to thefree stream are shown in Figure 10(a) for SWCNTs and inFigure 10(b) forMWCNTs It is noticed that Nusselt numbersdecrease with an increase in the plate velocity However

magnetic field helps in increasing Nusselt numbers for bothCNTs The same behavior can be observed in Figure 11for a vertical plate and in Figure 12 for a wedge movingin opposite direction to the free stream However when

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

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MaterialsJournal of

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

2 Journal of Nanomaterials

on the MHD forced convection adjacent to a nonisothermalwedge They used the Keller box method and presented thenumerical results for the local friction coefficient and thelocal Nusselt number

Hossain et al [8] considered forced flow and heat trans-fer of a viscous incompressible fluid having temperature-dependent viscosity and thermal conductivity past a wedgewith a uniform surface heat fluxThey presented the local skinfriction coefficient and rate of heat transfer for various valuesof the governing parameters Pal and Mondal [9] analyzedheat transfer characteristics of an incompressible Newto-nian electrically conducting and heat generatingabsorbingfluid having temperature-dependent viscosity over a non-isothermal wedge in the presence of thermal radiation Theyincluded the effects of viscous dissipation Joule heatingstress work and thermal radiation and obtained numericalresults for the velocity and temperature profiles as well as thelocal skin friction coefficient and local Nusselt number Ishaket al [10] investigatedMHDboundary layer flowpast a wedgewith constant surface heat flux immersed in an incompress-ible micropolar fluid They employed the Keller box methodto obtain numerical results and showed thatmicropolar fluidsdisplay drag reduction and consequently reduce the heattransfer rate at the surface Su et al [11] proposed a newanalytical method named DTM-BF and applied their studyto MHD mixed convective flow and heat transfer of an elec-trically conducting fluid over a stretching permeable wedgein the presence of thermal radiation and Ohmic heatingThey presented the effects of various embedded parametersand dimensionless parameters on the skin friction and heattransfer Prasad et al [12] obtained numerical solutions forthe hydromagnetic mixed convection boundary layer flow ofan electrically conducting fluid over a nonisothermal wedgeThey investigated various effects of the embeddeddimension-less parameters Rahman et al [13] discovered the effects ofthermophoresis on an unsteadyMHD forced convective heatandmass transfer flow of an electrically conducting fluid overa wedge with variable electrical conductivity They solvedlocally similar ordinary differential equations by applyingNachtsheim-Swigert shooting iteration technique along witha sixth order Runge-Kutta integration scheme They foundthat the skin friction coefficient Nusselt number and Sher-wood number are higher for the fluids of constant electricalconductivity than those of the variable electrical conductivityMuhaimin et al [14] analyzed the effects of thermophoresisparticle deposition and temperature-dependent viscosity onunsteady MHD mixed convective heat and mass transfer ofan electrically conducting fluid past a porous wedge in thepresence of a chemical reaction They compared their resultswith those known from the literature and found excellentagreement They found that the flow field is influencedappreciably by the appliedmagnetic field and thermophoresisparticle deposition Abbasbandy and Hayat [15] determinedthe solution of the Falkner-Skan flow by using a HomotopyAnalysis method They obtained the skin friction coefficientand compared their results with published data Fang andZhang [16] obtained an exact analytical solution of the famousFalkner-Skan equation which involves the boundary layerflow over a moving wall with mass transfer in the presence

of a free stream They analyzed the effects of mass transferand wall stretching

It is well known that convectional heat transfer fluids likewater or oil are poor heat transfer fluids due to lower thermalconductivity To enhance heat transfer thermal conductiv-ity of these fluids is increased by adding nanomicrosizedparticles of higher thermal conductivity These fluids areknown as nanofluids Following the introduction of thisidea by Choi [17] several researchers including [18ndash23] usednanofluids to improve heat transfer fromdifferent geometriesunder different thermal boundary conditions Khan and Pop[24] studied flow and heat transfer from a moving wedgein a water-based nanofluid They investigated the effectsof the governing parameters on the dimensionless velocitytemperature and nanoparticle volume fraction and presentedtheir results graphically Yacob et al [25] studied the problemof the steady flow and heat transfer over a static or movingwedge with a prescribed surface heat flux in a nanofluidTheyanalyzed and discussed three different types of nanoparticleswith water as the base fluid They found that the skin frictioncoefficient and the heat transfer rate are highest for a copperndashwater nanofluid compared to the other nanofluids consideredin their study Rahman et al [26] investigated numerically theheat transfer characteristics of water-based nanofluids over awedge with slip and a convective surface They showed thatthe nanofluid velocity is lower than the velocity of the basefluid and the existence of the nanofluid leads to the thinningof the hydrodynamic boundary layerThey also found that therate of heat transfer increases with an increase of the surfaceconvection and the slip parameters Later on Rahman et al[27] extended their study to include thermal radiation andshowed that the rate of heat transfer in a nanofluid in thepresence of thermal radiation significantly depends on thesurface convection parameter Chamkha et al [28] studiedmixed convection boundary layer flow over an isothermalvertical wedge embedded in a porous medium saturated witha nanofluid They included thermal radiation and nanofluidparameters in the analysis They presented numerical resultsfor the velocity temperature and volume fraction the localNusselt and Sherwood numbers Kameswaran et al [29]presented a mathematical model for the two-dimensionalsteady incompressible convective heat and mass transferfrom an isothermal wedge immersed in a nanofluid Theyconsidered two types of water-based nanofluids and includedthe effects of the volume fraction parameter

While there are many materials that can be used toform nanoparticles carbon is very promising due to itshigh thermal electrical and mechanical properties [30] Acarbon nanotube is composed of a single wall (SWCNT)or multiwall (MWCNT) The thermal properties of carbonnanotubes CNT suspensions are found to be much higherthan those of other nanoparticles with the same volumefraction [31ndash33]They noticed enhancement in thermal prop-erties of nanofluids with temperature and volume fractionDing et al [34] investigated the heat transfer behavior ofaqueous suspensions of multiwalled carbon nanotubes (CNTnanofluids) flowing through a horizontal tubeThey observedsignificant enhancement of the convective heat transfer Lateron Halelfadl et al [35] extended these findings to include

Journal of Nanomaterials 3

SWCNT

1-2nm

02ndash5120583m u

Ω = 120573120587

xy

ue(x)

Tinfin

Uw

(a)

MWCNT u

Ω = 120573120587

y x

ue(x)

Tinfin

Uw

(b)

Figure 1 Geometry of the problem

a coaxial heat exchanger under a laminar flow regime withinthe range of Reynolds numbers 500ndash2500 and measuredthermal and rheological properties of the nanofluids Theystudied the effects of the aspect ratio of carbon nanotubesthe type of base fluid and surfactant on viscosity thermalconductivity and laminar convective heat transfer Khan et al[36] employed homogeneous flow model to study the flowand heat transfer of carbon nanotubes (CNTs) along a flatplate subjected toNavier slip and uniform heat flux boundaryconditions They investigated the effects of the governingparameters on the dimensionless velocity temperature skinfriction and Nusselt numbers and presented numericalresults in graphical and tabular forms Recently Ul Haq et al[37] studied thermophysical effects of carbon nanotubes onMHD flow over a stretching sheet They considered threetypes of base fluids specifically water ethylene glycol andengine oil with single and multiwalled carbon nanotubes andfound that the engine oil provides the highest skin frictionand heat transfer rates

The above literature review reveals that there are nostudies that investigate thermal performance of carbon nan-otubes over static or moving wedges The objective of thisstudy is to explore the heat transfer properties of water-basednanofluids containing SWCNTs andMWCNTs To the best ofour knowledge the results of this paper are original and theyhave not been published before

2 Mathematical Model

Consider steady two-dimensional boundary layer flow pasta static or a moving wedge in a water-based nanofluidcontaining SWCNTs and MWCNTsThe physical model andgeometrical coordinates are depicted in Figure 1

The nanofluid is assumed incompressible and the flowis assumed to be steady and laminar It is also assumedthat the base fluid (ie water) and the nanoparticles are inthermal equilibrium and no slip occurs between them The

Table 1 Thermophysical properties of water and CNTs [41 42]

Thermophysicalproperties

Base fluid Carbon nanotubesWater SWCNT MWCNT

120588 (kgm3) 997 2600 1600119888119901 (Jkg-K) 4179 425 796119896 (Wm-K) 0613 6600 3000Pr 62 mdash mdash

thermophysical properties of the fluid and nanoparticles aregiven in Table 1 Further it is assumed that the velocity ofthe free stream (inviscid flow) is 119906119890(119909) = 119880infin119909

119898 and thatof the moving wedge is 119906119908(119909) = 119880119908119909

119898 where 119880infin 119880119908

and 119898 are constants with 0 le 119898 le 1 We consider atwo-dimensional Cartesian coordinate system (119909 119910) where119909 and 119910 are the coordinates measured along the surface ofthe wedge and normal to it respectively Under the boundarylayer approximations the basic steady conservation of massmomentum and energy equations can be written as

120597119906

120597119909+

120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= 119880119890

119889119880119890

119889119909+ ]nf

1205972119906

1205972119910+

1205901198612

119900

120588nf(119880119890 minus 119906) (2)

119906120597119879

120597119909+ V

120597119879

120597119910= 120572nf

1205972119879

1205971199112+

]nf(120588119888119901)nf

(120597119906

120597119910)

2

(3)

Subject to the boundary conditions

(1) static wedge

V = 0 119906 = 0 119879 = 119879119908 at 119910 = 0

119906 = 119906119890 (119909) 119879 = 119879119908 as 119910 997888rarr infin

(4)

4 Journal of Nanomaterials

(2) moving wedge

V = 0 119906 = 119906119908 (119909) 119879 = 119879119908 at 119910 = 0

119906 = 119906119890 (119909) 119879 = 119879119908 as 119910 997888rarr infin

(5)

Here 119906 and V are the velocity components along the 119909- and 119910-axes respectively 119879 is the temperature of the base fluid ]nf isviscosity of the nanofluid 120588nf is the density of the nanofluidand120572nf is the thermal diffusivity of nanofluid which are givenby

]nf =120583nf120588nf

120583nf =120583119891

(1 minus 120601)25

120572nf =119896nf

120588nf (119862119901)nf

120588nf = (1 minus 120601) 120588119891 + 120601120588CNT

(120588119862119901)nf= (1 minus 120601) (120588119862119901)119891

+ 120601 (120588119862119901)CNT

119896nf119896119891

= (1 minus 120601 + 2120601119896CNT

119896CNT minus 119896119891

ln119896CNT + 119896119891

2119896119891

)

sdot (1 minus 120601 + 2120601119896119891

119896CNT minus 119896119891

ln119896CNT + 119896119891

2119896119891

)

minus1

(6)

where 120583119891 is the viscosity of base fluid 120601 is the nanoparticlefraction (120588119862119901)nf is the effective heat capacity of CNTs119896nf is the thermal conductivity of nanofluid 119896119891 and 119896CNTare the thermal conductivities of the base fluid and CNTsrespectively and 120588119891 and 120588CNT are the thermal conductivitiesof the base fluid and CNTs respectively Introducing thefollowing similarity transformations

120595 = [2]119891119909119906119890(119909)

119898 + 1]

12

119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908 minus 119879infin

120578 = [(119898 + 1)119906119890(119909)

2]119891119909]

12

119910

(7)

where 120595 is the stream function and is defined in the usualway as 119906 = 120597120595120597119910 and V = minus120597120595120597119909 so as to identicallysatisfy (1) and ]119891 is the kinematic viscosity of the base fluidOn substituting (7) into (2) and (3) along with the boundaryconditions defined in (4) and (5) we obtain the followingordinary differential equations

1

(1 minus 120601)25

(1 minus 120601 + 120601120588CNT120588119891)

119891101584010158401015840

+ 11989111989110158401015840

+ (2119898

119898 + 1) (1 minus 119891

10158402)

+1198722

(1 minus 120601 + 120601120588CNT120588119891)(1 minus 119891

10158402) = 0

(8)

1

Pr119896nf119896119891

1 minus 120601 + 120601 (120588119888119901)CNT (120588119888119901)119891

12057910158401015840

+ 1198911205791015840

+119864119888

(1 minus 120601)25

119891101584010158402

= 0

(9)

subject to the boundary conditions

119891 (0) = 0 1198911015840

(0) = 120582 120579 (0) = 1

1198911015840

997888rarr 1 120579 997888rarr 0 as 120578 997888rarr infin

(10)

where the primes denote differentiationwith respect to 120578Theparameters 120582 120573 and Pr are defined as

120582 =119880119908

119880infin

120573 =2119898

119898 + 1 Pr =

]119891120572119891

(11)

where 120582 is the constant moving wedge parameter with 120582 gt

0 and 120582 lt 0 correspond to a moving wedge in the sameand opposite directions to the free stream respectively while120582 = 0 corresponds to a static wedge Further 120573 is the Hartreepressure gradient parameter which corresponds to 120573 = Ω120587

for a total wedge angle Ω According to White [38] positivevalue of120573means the pressure gradient is negative or favorableand negative values of 120573 denote an unfavorable pressuregradient Further 120573 = 0 (119898 = 0) represents the boundarylayer flow past a horizontal flat plate while 120573 = 1 (119898 = 1)corresponds to the boundary layer flow near the stagnationpoint of a vertical flat plate

It is important to note that (8) in the absence of amagneticfield (119872 = 0) and CNTs (120601 = 0) reduces to the same equationas obtained by Fang and Zhang [16] who provided an exactanalytical solution of the famous Falkner-Skan equation for aconventional fluid when 120573 = minus1

The skin friction coefficient 119862119891 and local Nusselt numberNu119909 are defined as

119862 =120591119908

1205881198911199062119890

Nu119909 =119909119902119908

119896119891 (119879119908 minus 119879infin) (12)

where 120591119908 and 119902119908 are the shear stress and heat flux at the wallwhich is given by

120591119908 = 120583nf (120597119906

120597119910)

119910=0

119902119908 = minus119896nf (120597119879

120597119910)

119910=0

(13)

Substituting (7) into (13) and (14) we obtain

Re12119909

119862119891 = radic119898 + 1

2

1

(1 minus 120601)25

11989110158401015840

(0)

Reminus12119909

Nu119909 = radic119898 + 1

2

119896nf119896119891

1205791015840

(0)

(14)

where Re119909 = 119906119890119909]119891 is the local Reynolds number

Journal of Nanomaterials 5

Table 2 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 White [38] Yacob et al [43] Yih [44] Khan and Pop [24] Present results0 04696 04696 04696 04696 04696111 06550 06550 06550 06550 0655015 08021 08021 08021 08021 0802113 09277 09277 09276 09277 0927712 10389 10389 mdash 10389 103891 12326 12326 12326 12326 12326

3 Numerical Method

Nonlinear ordinary differential equations (8) and (9) withboundary conditions (12) are solved by using an implicit finitedifference method with a linearization technique The detailsof thismethod are omitted here to save space but can be foundin [39 40] The step size is taken as Δ120578 = 0001 and theconvergence criteria are set to 10minus6The asymptotic boundaryconditions given by (12) were replaced by using a value of 10for the similarity variable 120578max as follows

120578max = 10 1198911015840

(10) = 1 120579 (10) = 0 (15)

The choice of 120578max = 10 ensures that all numerical solutionsapproached the asymptotic values correctly

4 Results and Discussion

MHD flow of water-based carbon nanotubes (CNTs) overstatic and moving wedges is investigated numerically Thethermophysical properties of water and both CNTs arepresented in Table 1 The numerical results are compared inTables 2 and 3 for skin friction and in Tables 4 and 5 forheat transfer in some special cases The results are foundto be in excellent agreement with the published data Theeffects of the magnetic field and volume fraction of SWCNTson the dimensionless velocity are displayed in Figures 2 and3 for flow past horizontal and near the stagnation point ofvertical stationary and moving flat plates respectively It canbe seen that the hydrodynamic boundary layer thickness islarger in the absence of a magnetic field in each case Infact the magnetic field tends to reduce the boundary layerthickness Figures 2(a) and 2(b) depict velocity profiles for theflow of SWCNTs over a stationary horizontal plate (119898 = 0)and near the stagnation point of a vertical stationary plate(119898 = 1) respectively In the absence of magnetic field theeffect of volume fraction of SWCNTs on the dimensionlessvelocity can be noticed but as the magnetic field increasesits effect diminishes When the horizontal and vertical platesmove in the free stream direction the dimensionless velocityprofiles are shown in Figures 3(a) and 3(b) respectively Itis important to note that the boundary layer thickness isreduced in this case The effects of volume fraction of CNTsare found to be negligible and the velocity boundary layerthickness reduces with an increase in magnetic field in bothcases

Table 3 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for aconventional fluid (120601 = 0)

120573 Kuo [45] Rajagopal et al [46] Present results0 04696 04696 04696005 05317 05311 0531101 05879 05870 0587002 06876 06867 0686703 07755 07748 0774804 08549 08544 0854405 09279 09277 0927706 09958 09958 0995807 10594 mdash 1059808 11196 11203 1120309 11767 mdash 117771 12313 12326 123262 16831 mdash 16872

Table 4 Comparison of dimensionless heat transfer minus1205791015840(0) when

119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 Kuo [45] Blasius [47] Khan and Pop [24] Present results0 08673 08673 08769 087691 11147 11152 11279 11280

The effects of magnetic field and volume fraction ofSWCNTs are illustrated in Figure 4 for stationary platesand in Figure 5 for moving plates in opposite direction tofree stream respectively In each case the magnetic fieldtends to reduce the thermal boundary layer thickness Thecomparison of Figures 4(a) and 4(b) shows that the thermalboundary layer thickness is smaller in the case of flowover a horizontal plate and hence less thermal resistance isoffered in this case However when the plates are moving inthe direction of the free stream the magnetic field has noappreciable effect on the thermal boundary layer thickness (asshown in Figures 5(a) and 5(b)) Conventional fluids (phi =0) have a smaller thermal boundary layer thickness and as thevolume fraction of SWCNTs increases the thermal boundary

6 Journal of Nanomaterials

Table 5 Comparison of dimensionless heat transfer minus1205791015840(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

Pr 120573 = 0 120573 = 03 120573 = 1

White [38] Kuo [45] Present results White [38] Kuo [45] Present results White [38] Kuo [45] Present results1 04696 04696 04696 05195 05200 05195 05705 05708 057052 05972 05972 05972 06691 06696 06690 07437 07441 074373 06860 06860 06860 07734 07741 07734 08652 08657 086526 08673 08673 08673 09873 09881 09873 11147 11152 1114710 10297 10297 10297 11791 11800 11791 13388 13394 1338830 14873 14873 14873 17198 17210 17198 19706 19714 1970660 18746 18746 18746 21776 21791 21776 25054 25063 25054100 22229 22229 22229 25892 25910 25892 29863 29874 29863

0 1 2 3 4 50

02

04

06

08

1

01

M

m = 0

120601 = 0 01 02

f998400 (120578)

120578

120582 = 0 Ec = 05

(a)

0 1 2 3 4

01

M

m = 1

120601 = 0 01 02

0

02

04

06

08

1

f998400 (120578)

120578

120582 = 0 Ec = 05

(b)

Figure 2 Effect of magnetic field on dimensionless velocity of water-based SWCNTs for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

layer thickness also increases due to an increase in the densityof CNTs

The variation of skin friction with magnetic field andvolume fraction of CNTs is presented in Figure 6 for thehorizontal plate moving in an opposite direction to thefree stream In Figure 6(a) SWCNTs are used whereas inFigure 6(b) MWCNTs are used to show the comparison ofskin friction in the presence of a magnetic field It is noticedthat the skin friction increases with volume fraction of CNTsin both cases However skin friction of SWCNTs is foundto be higher than MWCNTs It is due to higher density ofSWCNTs than MWCNTs (Table 1) The effect of magneticfield creates a Lorentz force which acts against the flow andthus enhances the skin friction As the horizontal platemovesin the opposing flow the hydraulic resistance also increases

which helps in an increase in skin frictionWhen the horizon-tal platemoves in an assisting flow Figures 7(a) and 7(b) showthe variation of skin friction with volume fraction of CNTsand magnetic field In this case the resistance decreases andhence the skin friction decreases with an increase in the platevelocity The same behavior was observed when a verticalplate moves in opposing flow For a vertical plate moving inopposite direction to the free stream the variation of skinfriction with magnetic field and volume fraction of CNTs isillustrated in Figure 8 for both CNTs in the presence of amagnetic field Similar to the horizontal plate skin frictionalso increases with plate velocity as shown in Figures 8(a)and 8(b) Figures 9(a) and 9(b) present the variation of skinfriction of CNTs along wedges moving in opposite directionto the free stream in the presence of a magnetic field Due to

Journal of Nanomaterials 7

0 1 2 3 4

01

M

120601 = 0 01 02

f998400 (120578)

120578

m = 0

05

06

07

08

09

1

11

120582 = Ec = 05

(a)

0 1 2 3

01

m = 1

f998400 (120578)

120578

M

120601 = 0 01 02

05

06

07

08

09

1

11

120582 = Ec = 05

(b)

Figure 3 Effect of magnetic field on dimensionless velocity of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 1 2 3

01

M

120601 = 0 01 02

0

02

04

06

08

1

120578

m = 0

120579(120578)

120582 = 0 Ec = 05

(a)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

0

02

04

06

08

1

120579(120578) 120582 = 0 Ec = 05

(b)

Figure 4 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

the increase in density with volume fraction of CNTs the skinfriction increases with volume friction of CNTs and wedgeparameter For conventional fluid (water) the skin friction isthe same for both CNTs Again the skin friction is found tobe higher for SWCNTs for both wedges

The variation of Nusselt numbers with volume fractionof CNTs magnetic and wedge parameters is reported inFigures 10ndash13 for special cases Due to higher thermalconductivity SWCNTs possess higher Nusselt numbers ineach case The effects of magnetic field and the motion

8 Journal of Nanomaterials

0 05 1 15 2

01

M

120601 = 0 01 02

120578

m = 0

0

02

04

06

08

1120579(120578) 120582 = Ec = 05

(a)

0

1

120579(120578)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

02

04

06

08

120582 = Ec = 05

(b)

Figure 5 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 01 02

3

1M

120601

120582 = minus01 minus03 minus05

005 01505

1

15

15

2

25

Re12

xCfx

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

m = 0 Ec = 005

(b) MWCNT

Figure 6 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

of the horizontal plate in the direction opposite to thefree stream are shown in Figure 10(a) for SWCNTs and inFigure 10(b) forMWCNTs It is noticed that Nusselt numbersdecrease with an increase in the plate velocity However

magnetic field helps in increasing Nusselt numbers for bothCNTs The same behavior can be observed in Figure 11for a vertical plate and in Figure 12 for a wedge movingin opposite direction to the free stream However when

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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MaterialsJournal of

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Journal of Nanomaterials 3

SWCNT

1-2nm

02ndash5120583m u

Ω = 120573120587

xy

ue(x)

Tinfin

Uw

(a)

MWCNT u

Ω = 120573120587

y x

ue(x)

Tinfin

Uw

(b)

Figure 1 Geometry of the problem

a coaxial heat exchanger under a laminar flow regime withinthe range of Reynolds numbers 500ndash2500 and measuredthermal and rheological properties of the nanofluids Theystudied the effects of the aspect ratio of carbon nanotubesthe type of base fluid and surfactant on viscosity thermalconductivity and laminar convective heat transfer Khan et al[36] employed homogeneous flow model to study the flowand heat transfer of carbon nanotubes (CNTs) along a flatplate subjected toNavier slip and uniform heat flux boundaryconditions They investigated the effects of the governingparameters on the dimensionless velocity temperature skinfriction and Nusselt numbers and presented numericalresults in graphical and tabular forms Recently Ul Haq et al[37] studied thermophysical effects of carbon nanotubes onMHD flow over a stretching sheet They considered threetypes of base fluids specifically water ethylene glycol andengine oil with single and multiwalled carbon nanotubes andfound that the engine oil provides the highest skin frictionand heat transfer rates

The above literature review reveals that there are nostudies that investigate thermal performance of carbon nan-otubes over static or moving wedges The objective of thisstudy is to explore the heat transfer properties of water-basednanofluids containing SWCNTs andMWCNTs To the best ofour knowledge the results of this paper are original and theyhave not been published before

2 Mathematical Model

Consider steady two-dimensional boundary layer flow pasta static or a moving wedge in a water-based nanofluidcontaining SWCNTs and MWCNTsThe physical model andgeometrical coordinates are depicted in Figure 1

The nanofluid is assumed incompressible and the flowis assumed to be steady and laminar It is also assumedthat the base fluid (ie water) and the nanoparticles are inthermal equilibrium and no slip occurs between them The

Table 1 Thermophysical properties of water and CNTs [41 42]

Thermophysicalproperties

Base fluid Carbon nanotubesWater SWCNT MWCNT

120588 (kgm3) 997 2600 1600119888119901 (Jkg-K) 4179 425 796119896 (Wm-K) 0613 6600 3000Pr 62 mdash mdash

thermophysical properties of the fluid and nanoparticles aregiven in Table 1 Further it is assumed that the velocity ofthe free stream (inviscid flow) is 119906119890(119909) = 119880infin119909

119898 and thatof the moving wedge is 119906119908(119909) = 119880119908119909

119898 where 119880infin 119880119908

and 119898 are constants with 0 le 119898 le 1 We consider atwo-dimensional Cartesian coordinate system (119909 119910) where119909 and 119910 are the coordinates measured along the surface ofthe wedge and normal to it respectively Under the boundarylayer approximations the basic steady conservation of massmomentum and energy equations can be written as

120597119906

120597119909+

120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910= 119880119890

119889119880119890

119889119909+ ]nf

1205972119906

1205972119910+

1205901198612

119900

120588nf(119880119890 minus 119906) (2)

119906120597119879

120597119909+ V

120597119879

120597119910= 120572nf

1205972119879

1205971199112+

]nf(120588119888119901)nf

(120597119906

120597119910)

2

(3)

Subject to the boundary conditions

(1) static wedge

V = 0 119906 = 0 119879 = 119879119908 at 119910 = 0

119906 = 119906119890 (119909) 119879 = 119879119908 as 119910 997888rarr infin

(4)

4 Journal of Nanomaterials

(2) moving wedge

V = 0 119906 = 119906119908 (119909) 119879 = 119879119908 at 119910 = 0

119906 = 119906119890 (119909) 119879 = 119879119908 as 119910 997888rarr infin

(5)

Here 119906 and V are the velocity components along the 119909- and 119910-axes respectively 119879 is the temperature of the base fluid ]nf isviscosity of the nanofluid 120588nf is the density of the nanofluidand120572nf is the thermal diffusivity of nanofluid which are givenby

]nf =120583nf120588nf

120583nf =120583119891

(1 minus 120601)25

120572nf =119896nf

120588nf (119862119901)nf

120588nf = (1 minus 120601) 120588119891 + 120601120588CNT

(120588119862119901)nf= (1 minus 120601) (120588119862119901)119891

+ 120601 (120588119862119901)CNT

119896nf119896119891

= (1 minus 120601 + 2120601119896CNT

119896CNT minus 119896119891

ln119896CNT + 119896119891

2119896119891

)

sdot (1 minus 120601 + 2120601119896119891

119896CNT minus 119896119891

ln119896CNT + 119896119891

2119896119891

)

minus1

(6)

where 120583119891 is the viscosity of base fluid 120601 is the nanoparticlefraction (120588119862119901)nf is the effective heat capacity of CNTs119896nf is the thermal conductivity of nanofluid 119896119891 and 119896CNTare the thermal conductivities of the base fluid and CNTsrespectively and 120588119891 and 120588CNT are the thermal conductivitiesof the base fluid and CNTs respectively Introducing thefollowing similarity transformations

120595 = [2]119891119909119906119890(119909)

119898 + 1]

12

119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908 minus 119879infin

120578 = [(119898 + 1)119906119890(119909)

2]119891119909]

12

119910

(7)

where 120595 is the stream function and is defined in the usualway as 119906 = 120597120595120597119910 and V = minus120597120595120597119909 so as to identicallysatisfy (1) and ]119891 is the kinematic viscosity of the base fluidOn substituting (7) into (2) and (3) along with the boundaryconditions defined in (4) and (5) we obtain the followingordinary differential equations

1

(1 minus 120601)25

(1 minus 120601 + 120601120588CNT120588119891)

119891101584010158401015840

+ 11989111989110158401015840

+ (2119898

119898 + 1) (1 minus 119891

10158402)

+1198722

(1 minus 120601 + 120601120588CNT120588119891)(1 minus 119891

10158402) = 0

(8)

1

Pr119896nf119896119891

1 minus 120601 + 120601 (120588119888119901)CNT (120588119888119901)119891

12057910158401015840

+ 1198911205791015840

+119864119888

(1 minus 120601)25

119891101584010158402

= 0

(9)

subject to the boundary conditions

119891 (0) = 0 1198911015840

(0) = 120582 120579 (0) = 1

1198911015840

997888rarr 1 120579 997888rarr 0 as 120578 997888rarr infin

(10)

where the primes denote differentiationwith respect to 120578Theparameters 120582 120573 and Pr are defined as

120582 =119880119908

119880infin

120573 =2119898

119898 + 1 Pr =

]119891120572119891

(11)

where 120582 is the constant moving wedge parameter with 120582 gt

0 and 120582 lt 0 correspond to a moving wedge in the sameand opposite directions to the free stream respectively while120582 = 0 corresponds to a static wedge Further 120573 is the Hartreepressure gradient parameter which corresponds to 120573 = Ω120587

for a total wedge angle Ω According to White [38] positivevalue of120573means the pressure gradient is negative or favorableand negative values of 120573 denote an unfavorable pressuregradient Further 120573 = 0 (119898 = 0) represents the boundarylayer flow past a horizontal flat plate while 120573 = 1 (119898 = 1)corresponds to the boundary layer flow near the stagnationpoint of a vertical flat plate

It is important to note that (8) in the absence of amagneticfield (119872 = 0) and CNTs (120601 = 0) reduces to the same equationas obtained by Fang and Zhang [16] who provided an exactanalytical solution of the famous Falkner-Skan equation for aconventional fluid when 120573 = minus1

The skin friction coefficient 119862119891 and local Nusselt numberNu119909 are defined as

119862 =120591119908

1205881198911199062119890

Nu119909 =119909119902119908

119896119891 (119879119908 minus 119879infin) (12)

where 120591119908 and 119902119908 are the shear stress and heat flux at the wallwhich is given by

120591119908 = 120583nf (120597119906

120597119910)

119910=0

119902119908 = minus119896nf (120597119879

120597119910)

119910=0

(13)

Substituting (7) into (13) and (14) we obtain

Re12119909

119862119891 = radic119898 + 1

2

1

(1 minus 120601)25

11989110158401015840

(0)

Reminus12119909

Nu119909 = radic119898 + 1

2

119896nf119896119891

1205791015840

(0)

(14)

where Re119909 = 119906119890119909]119891 is the local Reynolds number

Journal of Nanomaterials 5

Table 2 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 White [38] Yacob et al [43] Yih [44] Khan and Pop [24] Present results0 04696 04696 04696 04696 04696111 06550 06550 06550 06550 0655015 08021 08021 08021 08021 0802113 09277 09277 09276 09277 0927712 10389 10389 mdash 10389 103891 12326 12326 12326 12326 12326

3 Numerical Method

Nonlinear ordinary differential equations (8) and (9) withboundary conditions (12) are solved by using an implicit finitedifference method with a linearization technique The detailsof thismethod are omitted here to save space but can be foundin [39 40] The step size is taken as Δ120578 = 0001 and theconvergence criteria are set to 10minus6The asymptotic boundaryconditions given by (12) were replaced by using a value of 10for the similarity variable 120578max as follows

120578max = 10 1198911015840

(10) = 1 120579 (10) = 0 (15)

The choice of 120578max = 10 ensures that all numerical solutionsapproached the asymptotic values correctly

4 Results and Discussion

MHD flow of water-based carbon nanotubes (CNTs) overstatic and moving wedges is investigated numerically Thethermophysical properties of water and both CNTs arepresented in Table 1 The numerical results are compared inTables 2 and 3 for skin friction and in Tables 4 and 5 forheat transfer in some special cases The results are foundto be in excellent agreement with the published data Theeffects of the magnetic field and volume fraction of SWCNTson the dimensionless velocity are displayed in Figures 2 and3 for flow past horizontal and near the stagnation point ofvertical stationary and moving flat plates respectively It canbe seen that the hydrodynamic boundary layer thickness islarger in the absence of a magnetic field in each case Infact the magnetic field tends to reduce the boundary layerthickness Figures 2(a) and 2(b) depict velocity profiles for theflow of SWCNTs over a stationary horizontal plate (119898 = 0)and near the stagnation point of a vertical stationary plate(119898 = 1) respectively In the absence of magnetic field theeffect of volume fraction of SWCNTs on the dimensionlessvelocity can be noticed but as the magnetic field increasesits effect diminishes When the horizontal and vertical platesmove in the free stream direction the dimensionless velocityprofiles are shown in Figures 3(a) and 3(b) respectively Itis important to note that the boundary layer thickness isreduced in this case The effects of volume fraction of CNTsare found to be negligible and the velocity boundary layerthickness reduces with an increase in magnetic field in bothcases

Table 3 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for aconventional fluid (120601 = 0)

120573 Kuo [45] Rajagopal et al [46] Present results0 04696 04696 04696005 05317 05311 0531101 05879 05870 0587002 06876 06867 0686703 07755 07748 0774804 08549 08544 0854405 09279 09277 0927706 09958 09958 0995807 10594 mdash 1059808 11196 11203 1120309 11767 mdash 117771 12313 12326 123262 16831 mdash 16872

Table 4 Comparison of dimensionless heat transfer minus1205791015840(0) when

119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 Kuo [45] Blasius [47] Khan and Pop [24] Present results0 08673 08673 08769 087691 11147 11152 11279 11280

The effects of magnetic field and volume fraction ofSWCNTs are illustrated in Figure 4 for stationary platesand in Figure 5 for moving plates in opposite direction tofree stream respectively In each case the magnetic fieldtends to reduce the thermal boundary layer thickness Thecomparison of Figures 4(a) and 4(b) shows that the thermalboundary layer thickness is smaller in the case of flowover a horizontal plate and hence less thermal resistance isoffered in this case However when the plates are moving inthe direction of the free stream the magnetic field has noappreciable effect on the thermal boundary layer thickness (asshown in Figures 5(a) and 5(b)) Conventional fluids (phi =0) have a smaller thermal boundary layer thickness and as thevolume fraction of SWCNTs increases the thermal boundary

6 Journal of Nanomaterials

Table 5 Comparison of dimensionless heat transfer minus1205791015840(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

Pr 120573 = 0 120573 = 03 120573 = 1

White [38] Kuo [45] Present results White [38] Kuo [45] Present results White [38] Kuo [45] Present results1 04696 04696 04696 05195 05200 05195 05705 05708 057052 05972 05972 05972 06691 06696 06690 07437 07441 074373 06860 06860 06860 07734 07741 07734 08652 08657 086526 08673 08673 08673 09873 09881 09873 11147 11152 1114710 10297 10297 10297 11791 11800 11791 13388 13394 1338830 14873 14873 14873 17198 17210 17198 19706 19714 1970660 18746 18746 18746 21776 21791 21776 25054 25063 25054100 22229 22229 22229 25892 25910 25892 29863 29874 29863

0 1 2 3 4 50

02

04

06

08

1

01

M

m = 0

120601 = 0 01 02

f998400 (120578)

120578

120582 = 0 Ec = 05

(a)

0 1 2 3 4

01

M

m = 1

120601 = 0 01 02

0

02

04

06

08

1

f998400 (120578)

120578

120582 = 0 Ec = 05

(b)

Figure 2 Effect of magnetic field on dimensionless velocity of water-based SWCNTs for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

layer thickness also increases due to an increase in the densityof CNTs

The variation of skin friction with magnetic field andvolume fraction of CNTs is presented in Figure 6 for thehorizontal plate moving in an opposite direction to thefree stream In Figure 6(a) SWCNTs are used whereas inFigure 6(b) MWCNTs are used to show the comparison ofskin friction in the presence of a magnetic field It is noticedthat the skin friction increases with volume fraction of CNTsin both cases However skin friction of SWCNTs is foundto be higher than MWCNTs It is due to higher density ofSWCNTs than MWCNTs (Table 1) The effect of magneticfield creates a Lorentz force which acts against the flow andthus enhances the skin friction As the horizontal platemovesin the opposing flow the hydraulic resistance also increases

which helps in an increase in skin frictionWhen the horizon-tal platemoves in an assisting flow Figures 7(a) and 7(b) showthe variation of skin friction with volume fraction of CNTsand magnetic field In this case the resistance decreases andhence the skin friction decreases with an increase in the platevelocity The same behavior was observed when a verticalplate moves in opposing flow For a vertical plate moving inopposite direction to the free stream the variation of skinfriction with magnetic field and volume fraction of CNTs isillustrated in Figure 8 for both CNTs in the presence of amagnetic field Similar to the horizontal plate skin frictionalso increases with plate velocity as shown in Figures 8(a)and 8(b) Figures 9(a) and 9(b) present the variation of skinfriction of CNTs along wedges moving in opposite directionto the free stream in the presence of a magnetic field Due to

Journal of Nanomaterials 7

0 1 2 3 4

01

M

120601 = 0 01 02

f998400 (120578)

120578

m = 0

05

06

07

08

09

1

11

120582 = Ec = 05

(a)

0 1 2 3

01

m = 1

f998400 (120578)

120578

M

120601 = 0 01 02

05

06

07

08

09

1

11

120582 = Ec = 05

(b)

Figure 3 Effect of magnetic field on dimensionless velocity of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 1 2 3

01

M

120601 = 0 01 02

0

02

04

06

08

1

120578

m = 0

120579(120578)

120582 = 0 Ec = 05

(a)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

0

02

04

06

08

1

120579(120578) 120582 = 0 Ec = 05

(b)

Figure 4 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

the increase in density with volume fraction of CNTs the skinfriction increases with volume friction of CNTs and wedgeparameter For conventional fluid (water) the skin friction isthe same for both CNTs Again the skin friction is found tobe higher for SWCNTs for both wedges

The variation of Nusselt numbers with volume fractionof CNTs magnetic and wedge parameters is reported inFigures 10ndash13 for special cases Due to higher thermalconductivity SWCNTs possess higher Nusselt numbers ineach case The effects of magnetic field and the motion

8 Journal of Nanomaterials

0 05 1 15 2

01

M

120601 = 0 01 02

120578

m = 0

0

02

04

06

08

1120579(120578) 120582 = Ec = 05

(a)

0

1

120579(120578)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

02

04

06

08

120582 = Ec = 05

(b)

Figure 5 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 01 02

3

1M

120601

120582 = minus01 minus03 minus05

005 01505

1

15

15

2

25

Re12

xCfx

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

m = 0 Ec = 005

(b) MWCNT

Figure 6 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

of the horizontal plate in the direction opposite to thefree stream are shown in Figure 10(a) for SWCNTs and inFigure 10(b) forMWCNTs It is noticed that Nusselt numbersdecrease with an increase in the plate velocity However

magnetic field helps in increasing Nusselt numbers for bothCNTs The same behavior can be observed in Figure 11for a vertical plate and in Figure 12 for a wedge movingin opposite direction to the free stream However when

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

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MaterialsJournal of

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Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

4 Journal of Nanomaterials

(2) moving wedge

V = 0 119906 = 119906119908 (119909) 119879 = 119879119908 at 119910 = 0

119906 = 119906119890 (119909) 119879 = 119879119908 as 119910 997888rarr infin

(5)

Here 119906 and V are the velocity components along the 119909- and 119910-axes respectively 119879 is the temperature of the base fluid ]nf isviscosity of the nanofluid 120588nf is the density of the nanofluidand120572nf is the thermal diffusivity of nanofluid which are givenby

]nf =120583nf120588nf

120583nf =120583119891

(1 minus 120601)25

120572nf =119896nf

120588nf (119862119901)nf

120588nf = (1 minus 120601) 120588119891 + 120601120588CNT

(120588119862119901)nf= (1 minus 120601) (120588119862119901)119891

+ 120601 (120588119862119901)CNT

119896nf119896119891

= (1 minus 120601 + 2120601119896CNT

119896CNT minus 119896119891

ln119896CNT + 119896119891

2119896119891

)

sdot (1 minus 120601 + 2120601119896119891

119896CNT minus 119896119891

ln119896CNT + 119896119891

2119896119891

)

minus1

(6)

where 120583119891 is the viscosity of base fluid 120601 is the nanoparticlefraction (120588119862119901)nf is the effective heat capacity of CNTs119896nf is the thermal conductivity of nanofluid 119896119891 and 119896CNTare the thermal conductivities of the base fluid and CNTsrespectively and 120588119891 and 120588CNT are the thermal conductivitiesof the base fluid and CNTs respectively Introducing thefollowing similarity transformations

120595 = [2]119891119909119906119890(119909)

119898 + 1]

12

119891 (120578)

120579 (120578) =119879 minus 119879infin

119879119908 minus 119879infin

120578 = [(119898 + 1)119906119890(119909)

2]119891119909]

12

119910

(7)

where 120595 is the stream function and is defined in the usualway as 119906 = 120597120595120597119910 and V = minus120597120595120597119909 so as to identicallysatisfy (1) and ]119891 is the kinematic viscosity of the base fluidOn substituting (7) into (2) and (3) along with the boundaryconditions defined in (4) and (5) we obtain the followingordinary differential equations

1

(1 minus 120601)25

(1 minus 120601 + 120601120588CNT120588119891)

119891101584010158401015840

+ 11989111989110158401015840

+ (2119898

119898 + 1) (1 minus 119891

10158402)

+1198722

(1 minus 120601 + 120601120588CNT120588119891)(1 minus 119891

10158402) = 0

(8)

1

Pr119896nf119896119891

1 minus 120601 + 120601 (120588119888119901)CNT (120588119888119901)119891

12057910158401015840

+ 1198911205791015840

+119864119888

(1 minus 120601)25

119891101584010158402

= 0

(9)

subject to the boundary conditions

119891 (0) = 0 1198911015840

(0) = 120582 120579 (0) = 1

1198911015840

997888rarr 1 120579 997888rarr 0 as 120578 997888rarr infin

(10)

where the primes denote differentiationwith respect to 120578Theparameters 120582 120573 and Pr are defined as

120582 =119880119908

119880infin

120573 =2119898

119898 + 1 Pr =

]119891120572119891

(11)

where 120582 is the constant moving wedge parameter with 120582 gt

0 and 120582 lt 0 correspond to a moving wedge in the sameand opposite directions to the free stream respectively while120582 = 0 corresponds to a static wedge Further 120573 is the Hartreepressure gradient parameter which corresponds to 120573 = Ω120587

for a total wedge angle Ω According to White [38] positivevalue of120573means the pressure gradient is negative or favorableand negative values of 120573 denote an unfavorable pressuregradient Further 120573 = 0 (119898 = 0) represents the boundarylayer flow past a horizontal flat plate while 120573 = 1 (119898 = 1)corresponds to the boundary layer flow near the stagnationpoint of a vertical flat plate

It is important to note that (8) in the absence of amagneticfield (119872 = 0) and CNTs (120601 = 0) reduces to the same equationas obtained by Fang and Zhang [16] who provided an exactanalytical solution of the famous Falkner-Skan equation for aconventional fluid when 120573 = minus1

The skin friction coefficient 119862119891 and local Nusselt numberNu119909 are defined as

119862 =120591119908

1205881198911199062119890

Nu119909 =119909119902119908

119896119891 (119879119908 minus 119879infin) (12)

where 120591119908 and 119902119908 are the shear stress and heat flux at the wallwhich is given by

120591119908 = 120583nf (120597119906

120597119910)

119910=0

119902119908 = minus119896nf (120597119879

120597119910)

119910=0

(13)

Substituting (7) into (13) and (14) we obtain

Re12119909

119862119891 = radic119898 + 1

2

1

(1 minus 120601)25

11989110158401015840

(0)

Reminus12119909

Nu119909 = radic119898 + 1

2

119896nf119896119891

1205791015840

(0)

(14)

where Re119909 = 119906119890119909]119891 is the local Reynolds number

Journal of Nanomaterials 5

Table 2 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 White [38] Yacob et al [43] Yih [44] Khan and Pop [24] Present results0 04696 04696 04696 04696 04696111 06550 06550 06550 06550 0655015 08021 08021 08021 08021 0802113 09277 09277 09276 09277 0927712 10389 10389 mdash 10389 103891 12326 12326 12326 12326 12326

3 Numerical Method

Nonlinear ordinary differential equations (8) and (9) withboundary conditions (12) are solved by using an implicit finitedifference method with a linearization technique The detailsof thismethod are omitted here to save space but can be foundin [39 40] The step size is taken as Δ120578 = 0001 and theconvergence criteria are set to 10minus6The asymptotic boundaryconditions given by (12) were replaced by using a value of 10for the similarity variable 120578max as follows

120578max = 10 1198911015840

(10) = 1 120579 (10) = 0 (15)

The choice of 120578max = 10 ensures that all numerical solutionsapproached the asymptotic values correctly

4 Results and Discussion

MHD flow of water-based carbon nanotubes (CNTs) overstatic and moving wedges is investigated numerically Thethermophysical properties of water and both CNTs arepresented in Table 1 The numerical results are compared inTables 2 and 3 for skin friction and in Tables 4 and 5 forheat transfer in some special cases The results are foundto be in excellent agreement with the published data Theeffects of the magnetic field and volume fraction of SWCNTson the dimensionless velocity are displayed in Figures 2 and3 for flow past horizontal and near the stagnation point ofvertical stationary and moving flat plates respectively It canbe seen that the hydrodynamic boundary layer thickness islarger in the absence of a magnetic field in each case Infact the magnetic field tends to reduce the boundary layerthickness Figures 2(a) and 2(b) depict velocity profiles for theflow of SWCNTs over a stationary horizontal plate (119898 = 0)and near the stagnation point of a vertical stationary plate(119898 = 1) respectively In the absence of magnetic field theeffect of volume fraction of SWCNTs on the dimensionlessvelocity can be noticed but as the magnetic field increasesits effect diminishes When the horizontal and vertical platesmove in the free stream direction the dimensionless velocityprofiles are shown in Figures 3(a) and 3(b) respectively Itis important to note that the boundary layer thickness isreduced in this case The effects of volume fraction of CNTsare found to be negligible and the velocity boundary layerthickness reduces with an increase in magnetic field in bothcases

Table 3 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for aconventional fluid (120601 = 0)

120573 Kuo [45] Rajagopal et al [46] Present results0 04696 04696 04696005 05317 05311 0531101 05879 05870 0587002 06876 06867 0686703 07755 07748 0774804 08549 08544 0854405 09279 09277 0927706 09958 09958 0995807 10594 mdash 1059808 11196 11203 1120309 11767 mdash 117771 12313 12326 123262 16831 mdash 16872

Table 4 Comparison of dimensionless heat transfer minus1205791015840(0) when

119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 Kuo [45] Blasius [47] Khan and Pop [24] Present results0 08673 08673 08769 087691 11147 11152 11279 11280

The effects of magnetic field and volume fraction ofSWCNTs are illustrated in Figure 4 for stationary platesand in Figure 5 for moving plates in opposite direction tofree stream respectively In each case the magnetic fieldtends to reduce the thermal boundary layer thickness Thecomparison of Figures 4(a) and 4(b) shows that the thermalboundary layer thickness is smaller in the case of flowover a horizontal plate and hence less thermal resistance isoffered in this case However when the plates are moving inthe direction of the free stream the magnetic field has noappreciable effect on the thermal boundary layer thickness (asshown in Figures 5(a) and 5(b)) Conventional fluids (phi =0) have a smaller thermal boundary layer thickness and as thevolume fraction of SWCNTs increases the thermal boundary

6 Journal of Nanomaterials

Table 5 Comparison of dimensionless heat transfer minus1205791015840(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

Pr 120573 = 0 120573 = 03 120573 = 1

White [38] Kuo [45] Present results White [38] Kuo [45] Present results White [38] Kuo [45] Present results1 04696 04696 04696 05195 05200 05195 05705 05708 057052 05972 05972 05972 06691 06696 06690 07437 07441 074373 06860 06860 06860 07734 07741 07734 08652 08657 086526 08673 08673 08673 09873 09881 09873 11147 11152 1114710 10297 10297 10297 11791 11800 11791 13388 13394 1338830 14873 14873 14873 17198 17210 17198 19706 19714 1970660 18746 18746 18746 21776 21791 21776 25054 25063 25054100 22229 22229 22229 25892 25910 25892 29863 29874 29863

0 1 2 3 4 50

02

04

06

08

1

01

M

m = 0

120601 = 0 01 02

f998400 (120578)

120578

120582 = 0 Ec = 05

(a)

0 1 2 3 4

01

M

m = 1

120601 = 0 01 02

0

02

04

06

08

1

f998400 (120578)

120578

120582 = 0 Ec = 05

(b)

Figure 2 Effect of magnetic field on dimensionless velocity of water-based SWCNTs for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

layer thickness also increases due to an increase in the densityof CNTs

The variation of skin friction with magnetic field andvolume fraction of CNTs is presented in Figure 6 for thehorizontal plate moving in an opposite direction to thefree stream In Figure 6(a) SWCNTs are used whereas inFigure 6(b) MWCNTs are used to show the comparison ofskin friction in the presence of a magnetic field It is noticedthat the skin friction increases with volume fraction of CNTsin both cases However skin friction of SWCNTs is foundto be higher than MWCNTs It is due to higher density ofSWCNTs than MWCNTs (Table 1) The effect of magneticfield creates a Lorentz force which acts against the flow andthus enhances the skin friction As the horizontal platemovesin the opposing flow the hydraulic resistance also increases

which helps in an increase in skin frictionWhen the horizon-tal platemoves in an assisting flow Figures 7(a) and 7(b) showthe variation of skin friction with volume fraction of CNTsand magnetic field In this case the resistance decreases andhence the skin friction decreases with an increase in the platevelocity The same behavior was observed when a verticalplate moves in opposing flow For a vertical plate moving inopposite direction to the free stream the variation of skinfriction with magnetic field and volume fraction of CNTs isillustrated in Figure 8 for both CNTs in the presence of amagnetic field Similar to the horizontal plate skin frictionalso increases with plate velocity as shown in Figures 8(a)and 8(b) Figures 9(a) and 9(b) present the variation of skinfriction of CNTs along wedges moving in opposite directionto the free stream in the presence of a magnetic field Due to

Journal of Nanomaterials 7

0 1 2 3 4

01

M

120601 = 0 01 02

f998400 (120578)

120578

m = 0

05

06

07

08

09

1

11

120582 = Ec = 05

(a)

0 1 2 3

01

m = 1

f998400 (120578)

120578

M

120601 = 0 01 02

05

06

07

08

09

1

11

120582 = Ec = 05

(b)

Figure 3 Effect of magnetic field on dimensionless velocity of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 1 2 3

01

M

120601 = 0 01 02

0

02

04

06

08

1

120578

m = 0

120579(120578)

120582 = 0 Ec = 05

(a)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

0

02

04

06

08

1

120579(120578) 120582 = 0 Ec = 05

(b)

Figure 4 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

the increase in density with volume fraction of CNTs the skinfriction increases with volume friction of CNTs and wedgeparameter For conventional fluid (water) the skin friction isthe same for both CNTs Again the skin friction is found tobe higher for SWCNTs for both wedges

The variation of Nusselt numbers with volume fractionof CNTs magnetic and wedge parameters is reported inFigures 10ndash13 for special cases Due to higher thermalconductivity SWCNTs possess higher Nusselt numbers ineach case The effects of magnetic field and the motion

8 Journal of Nanomaterials

0 05 1 15 2

01

M

120601 = 0 01 02

120578

m = 0

0

02

04

06

08

1120579(120578) 120582 = Ec = 05

(a)

0

1

120579(120578)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

02

04

06

08

120582 = Ec = 05

(b)

Figure 5 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 01 02

3

1M

120601

120582 = minus01 minus03 minus05

005 01505

1

15

15

2

25

Re12

xCfx

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

m = 0 Ec = 005

(b) MWCNT

Figure 6 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

of the horizontal plate in the direction opposite to thefree stream are shown in Figure 10(a) for SWCNTs and inFigure 10(b) forMWCNTs It is noticed that Nusselt numbersdecrease with an increase in the plate velocity However

magnetic field helps in increasing Nusselt numbers for bothCNTs The same behavior can be observed in Figure 11for a vertical plate and in Figure 12 for a wedge movingin opposite direction to the free stream However when

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CeramicsJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Journal of Nanomaterials 5

Table 2 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 White [38] Yacob et al [43] Yih [44] Khan and Pop [24] Present results0 04696 04696 04696 04696 04696111 06550 06550 06550 06550 0655015 08021 08021 08021 08021 0802113 09277 09277 09276 09277 0927712 10389 10389 mdash 10389 103891 12326 12326 12326 12326 12326

3 Numerical Method

Nonlinear ordinary differential equations (8) and (9) withboundary conditions (12) are solved by using an implicit finitedifference method with a linearization technique The detailsof thismethod are omitted here to save space but can be foundin [39 40] The step size is taken as Δ120578 = 0001 and theconvergence criteria are set to 10minus6The asymptotic boundaryconditions given by (12) were replaced by using a value of 10for the similarity variable 120578max as follows

120578max = 10 1198911015840

(10) = 1 120579 (10) = 0 (15)

The choice of 120578max = 10 ensures that all numerical solutionsapproached the asymptotic values correctly

4 Results and Discussion

MHD flow of water-based carbon nanotubes (CNTs) overstatic and moving wedges is investigated numerically Thethermophysical properties of water and both CNTs arepresented in Table 1 The numerical results are compared inTables 2 and 3 for skin friction and in Tables 4 and 5 forheat transfer in some special cases The results are foundto be in excellent agreement with the published data Theeffects of the magnetic field and volume fraction of SWCNTson the dimensionless velocity are displayed in Figures 2 and3 for flow past horizontal and near the stagnation point ofvertical stationary and moving flat plates respectively It canbe seen that the hydrodynamic boundary layer thickness islarger in the absence of a magnetic field in each case Infact the magnetic field tends to reduce the boundary layerthickness Figures 2(a) and 2(b) depict velocity profiles for theflow of SWCNTs over a stationary horizontal plate (119898 = 0)and near the stagnation point of a vertical stationary plate(119898 = 1) respectively In the absence of magnetic field theeffect of volume fraction of SWCNTs on the dimensionlessvelocity can be noticed but as the magnetic field increasesits effect diminishes When the horizontal and vertical platesmove in the free stream direction the dimensionless velocityprofiles are shown in Figures 3(a) and 3(b) respectively Itis important to note that the boundary layer thickness isreduced in this case The effects of volume fraction of CNTsare found to be negligible and the velocity boundary layerthickness reduces with an increase in magnetic field in bothcases

Table 3 Comparison of skin friction 11989110158401015840

(0) when 119872 = 120582 = 0 for aconventional fluid (120601 = 0)

120573 Kuo [45] Rajagopal et al [46] Present results0 04696 04696 04696005 05317 05311 0531101 05879 05870 0587002 06876 06867 0686703 07755 07748 0774804 08549 08544 0854405 09279 09277 0927706 09958 09958 0995807 10594 mdash 1059808 11196 11203 1120309 11767 mdash 117771 12313 12326 123262 16831 mdash 16872

Table 4 Comparison of dimensionless heat transfer minus1205791015840(0) when

119872 = 120582 = 0 for a conventional fluid (120601 = 0)

119898 Kuo [45] Blasius [47] Khan and Pop [24] Present results0 08673 08673 08769 087691 11147 11152 11279 11280

The effects of magnetic field and volume fraction ofSWCNTs are illustrated in Figure 4 for stationary platesand in Figure 5 for moving plates in opposite direction tofree stream respectively In each case the magnetic fieldtends to reduce the thermal boundary layer thickness Thecomparison of Figures 4(a) and 4(b) shows that the thermalboundary layer thickness is smaller in the case of flowover a horizontal plate and hence less thermal resistance isoffered in this case However when the plates are moving inthe direction of the free stream the magnetic field has noappreciable effect on the thermal boundary layer thickness (asshown in Figures 5(a) and 5(b)) Conventional fluids (phi =0) have a smaller thermal boundary layer thickness and as thevolume fraction of SWCNTs increases the thermal boundary

6 Journal of Nanomaterials

Table 5 Comparison of dimensionless heat transfer minus1205791015840(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

Pr 120573 = 0 120573 = 03 120573 = 1

White [38] Kuo [45] Present results White [38] Kuo [45] Present results White [38] Kuo [45] Present results1 04696 04696 04696 05195 05200 05195 05705 05708 057052 05972 05972 05972 06691 06696 06690 07437 07441 074373 06860 06860 06860 07734 07741 07734 08652 08657 086526 08673 08673 08673 09873 09881 09873 11147 11152 1114710 10297 10297 10297 11791 11800 11791 13388 13394 1338830 14873 14873 14873 17198 17210 17198 19706 19714 1970660 18746 18746 18746 21776 21791 21776 25054 25063 25054100 22229 22229 22229 25892 25910 25892 29863 29874 29863

0 1 2 3 4 50

02

04

06

08

1

01

M

m = 0

120601 = 0 01 02

f998400 (120578)

120578

120582 = 0 Ec = 05

(a)

0 1 2 3 4

01

M

m = 1

120601 = 0 01 02

0

02

04

06

08

1

f998400 (120578)

120578

120582 = 0 Ec = 05

(b)

Figure 2 Effect of magnetic field on dimensionless velocity of water-based SWCNTs for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

layer thickness also increases due to an increase in the densityof CNTs

The variation of skin friction with magnetic field andvolume fraction of CNTs is presented in Figure 6 for thehorizontal plate moving in an opposite direction to thefree stream In Figure 6(a) SWCNTs are used whereas inFigure 6(b) MWCNTs are used to show the comparison ofskin friction in the presence of a magnetic field It is noticedthat the skin friction increases with volume fraction of CNTsin both cases However skin friction of SWCNTs is foundto be higher than MWCNTs It is due to higher density ofSWCNTs than MWCNTs (Table 1) The effect of magneticfield creates a Lorentz force which acts against the flow andthus enhances the skin friction As the horizontal platemovesin the opposing flow the hydraulic resistance also increases

which helps in an increase in skin frictionWhen the horizon-tal platemoves in an assisting flow Figures 7(a) and 7(b) showthe variation of skin friction with volume fraction of CNTsand magnetic field In this case the resistance decreases andhence the skin friction decreases with an increase in the platevelocity The same behavior was observed when a verticalplate moves in opposing flow For a vertical plate moving inopposite direction to the free stream the variation of skinfriction with magnetic field and volume fraction of CNTs isillustrated in Figure 8 for both CNTs in the presence of amagnetic field Similar to the horizontal plate skin frictionalso increases with plate velocity as shown in Figures 8(a)and 8(b) Figures 9(a) and 9(b) present the variation of skinfriction of CNTs along wedges moving in opposite directionto the free stream in the presence of a magnetic field Due to

Journal of Nanomaterials 7

0 1 2 3 4

01

M

120601 = 0 01 02

f998400 (120578)

120578

m = 0

05

06

07

08

09

1

11

120582 = Ec = 05

(a)

0 1 2 3

01

m = 1

f998400 (120578)

120578

M

120601 = 0 01 02

05

06

07

08

09

1

11

120582 = Ec = 05

(b)

Figure 3 Effect of magnetic field on dimensionless velocity of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 1 2 3

01

M

120601 = 0 01 02

0

02

04

06

08

1

120578

m = 0

120579(120578)

120582 = 0 Ec = 05

(a)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

0

02

04

06

08

1

120579(120578) 120582 = 0 Ec = 05

(b)

Figure 4 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

the increase in density with volume fraction of CNTs the skinfriction increases with volume friction of CNTs and wedgeparameter For conventional fluid (water) the skin friction isthe same for both CNTs Again the skin friction is found tobe higher for SWCNTs for both wedges

The variation of Nusselt numbers with volume fractionof CNTs magnetic and wedge parameters is reported inFigures 10ndash13 for special cases Due to higher thermalconductivity SWCNTs possess higher Nusselt numbers ineach case The effects of magnetic field and the motion

8 Journal of Nanomaterials

0 05 1 15 2

01

M

120601 = 0 01 02

120578

m = 0

0

02

04

06

08

1120579(120578) 120582 = Ec = 05

(a)

0

1

120579(120578)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

02

04

06

08

120582 = Ec = 05

(b)

Figure 5 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 01 02

3

1M

120601

120582 = minus01 minus03 minus05

005 01505

1

15

15

2

25

Re12

xCfx

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

m = 0 Ec = 005

(b) MWCNT

Figure 6 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

of the horizontal plate in the direction opposite to thefree stream are shown in Figure 10(a) for SWCNTs and inFigure 10(b) forMWCNTs It is noticed that Nusselt numbersdecrease with an increase in the plate velocity However

magnetic field helps in increasing Nusselt numbers for bothCNTs The same behavior can be observed in Figure 11for a vertical plate and in Figure 12 for a wedge movingin opposite direction to the free stream However when

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

6 Journal of Nanomaterials

Table 5 Comparison of dimensionless heat transfer minus1205791015840(0) when 119872 = 120582 = 0 for a conventional fluid (120601 = 0)

Pr 120573 = 0 120573 = 03 120573 = 1

White [38] Kuo [45] Present results White [38] Kuo [45] Present results White [38] Kuo [45] Present results1 04696 04696 04696 05195 05200 05195 05705 05708 057052 05972 05972 05972 06691 06696 06690 07437 07441 074373 06860 06860 06860 07734 07741 07734 08652 08657 086526 08673 08673 08673 09873 09881 09873 11147 11152 1114710 10297 10297 10297 11791 11800 11791 13388 13394 1338830 14873 14873 14873 17198 17210 17198 19706 19714 1970660 18746 18746 18746 21776 21791 21776 25054 25063 25054100 22229 22229 22229 25892 25910 25892 29863 29874 29863

0 1 2 3 4 50

02

04

06

08

1

01

M

m = 0

120601 = 0 01 02

f998400 (120578)

120578

120582 = 0 Ec = 05

(a)

0 1 2 3 4

01

M

m = 1

120601 = 0 01 02

0

02

04

06

08

1

f998400 (120578)

120578

120582 = 0 Ec = 05

(b)

Figure 2 Effect of magnetic field on dimensionless velocity of water-based SWCNTs for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

layer thickness also increases due to an increase in the densityof CNTs

The variation of skin friction with magnetic field andvolume fraction of CNTs is presented in Figure 6 for thehorizontal plate moving in an opposite direction to thefree stream In Figure 6(a) SWCNTs are used whereas inFigure 6(b) MWCNTs are used to show the comparison ofskin friction in the presence of a magnetic field It is noticedthat the skin friction increases with volume fraction of CNTsin both cases However skin friction of SWCNTs is foundto be higher than MWCNTs It is due to higher density ofSWCNTs than MWCNTs (Table 1) The effect of magneticfield creates a Lorentz force which acts against the flow andthus enhances the skin friction As the horizontal platemovesin the opposing flow the hydraulic resistance also increases

which helps in an increase in skin frictionWhen the horizon-tal platemoves in an assisting flow Figures 7(a) and 7(b) showthe variation of skin friction with volume fraction of CNTsand magnetic field In this case the resistance decreases andhence the skin friction decreases with an increase in the platevelocity The same behavior was observed when a verticalplate moves in opposing flow For a vertical plate moving inopposite direction to the free stream the variation of skinfriction with magnetic field and volume fraction of CNTs isillustrated in Figure 8 for both CNTs in the presence of amagnetic field Similar to the horizontal plate skin frictionalso increases with plate velocity as shown in Figures 8(a)and 8(b) Figures 9(a) and 9(b) present the variation of skinfriction of CNTs along wedges moving in opposite directionto the free stream in the presence of a magnetic field Due to

Journal of Nanomaterials 7

0 1 2 3 4

01

M

120601 = 0 01 02

f998400 (120578)

120578

m = 0

05

06

07

08

09

1

11

120582 = Ec = 05

(a)

0 1 2 3

01

m = 1

f998400 (120578)

120578

M

120601 = 0 01 02

05

06

07

08

09

1

11

120582 = Ec = 05

(b)

Figure 3 Effect of magnetic field on dimensionless velocity of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 1 2 3

01

M

120601 = 0 01 02

0

02

04

06

08

1

120578

m = 0

120579(120578)

120582 = 0 Ec = 05

(a)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

0

02

04

06

08

1

120579(120578) 120582 = 0 Ec = 05

(b)

Figure 4 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

the increase in density with volume fraction of CNTs the skinfriction increases with volume friction of CNTs and wedgeparameter For conventional fluid (water) the skin friction isthe same for both CNTs Again the skin friction is found tobe higher for SWCNTs for both wedges

The variation of Nusselt numbers with volume fractionof CNTs magnetic and wedge parameters is reported inFigures 10ndash13 for special cases Due to higher thermalconductivity SWCNTs possess higher Nusselt numbers ineach case The effects of magnetic field and the motion

8 Journal of Nanomaterials

0 05 1 15 2

01

M

120601 = 0 01 02

120578

m = 0

0

02

04

06

08

1120579(120578) 120582 = Ec = 05

(a)

0

1

120579(120578)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

02

04

06

08

120582 = Ec = 05

(b)

Figure 5 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 01 02

3

1M

120601

120582 = minus01 minus03 minus05

005 01505

1

15

15

2

25

Re12

xCfx

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

m = 0 Ec = 005

(b) MWCNT

Figure 6 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

of the horizontal plate in the direction opposite to thefree stream are shown in Figure 10(a) for SWCNTs and inFigure 10(b) forMWCNTs It is noticed that Nusselt numbersdecrease with an increase in the plate velocity However

magnetic field helps in increasing Nusselt numbers for bothCNTs The same behavior can be observed in Figure 11for a vertical plate and in Figure 12 for a wedge movingin opposite direction to the free stream However when

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Journal of Nanomaterials 7

0 1 2 3 4

01

M

120601 = 0 01 02

f998400 (120578)

120578

m = 0

05

06

07

08

09

1

11

120582 = Ec = 05

(a)

0 1 2 3

01

m = 1

f998400 (120578)

120578

M

120601 = 0 01 02

05

06

07

08

09

1

11

120582 = Ec = 05

(b)

Figure 3 Effect of magnetic field on dimensionless velocity of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 1 2 3

01

M

120601 = 0 01 02

0

02

04

06

08

1

120578

m = 0

120579(120578)

120582 = 0 Ec = 05

(a)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

0

02

04

06

08

1

120579(120578) 120582 = 0 Ec = 05

(b)

Figure 4 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical stationary flat plates

the increase in density with volume fraction of CNTs the skinfriction increases with volume friction of CNTs and wedgeparameter For conventional fluid (water) the skin friction isthe same for both CNTs Again the skin friction is found tobe higher for SWCNTs for both wedges

The variation of Nusselt numbers with volume fractionof CNTs magnetic and wedge parameters is reported inFigures 10ndash13 for special cases Due to higher thermalconductivity SWCNTs possess higher Nusselt numbers ineach case The effects of magnetic field and the motion

8 Journal of Nanomaterials

0 05 1 15 2

01

M

120601 = 0 01 02

120578

m = 0

0

02

04

06

08

1120579(120578) 120582 = Ec = 05

(a)

0

1

120579(120578)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

02

04

06

08

120582 = Ec = 05

(b)

Figure 5 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 01 02

3

1M

120601

120582 = minus01 minus03 minus05

005 01505

1

15

15

2

25

Re12

xCfx

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

m = 0 Ec = 005

(b) MWCNT

Figure 6 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

of the horizontal plate in the direction opposite to thefree stream are shown in Figure 10(a) for SWCNTs and inFigure 10(b) forMWCNTs It is noticed that Nusselt numbersdecrease with an increase in the plate velocity However

magnetic field helps in increasing Nusselt numbers for bothCNTs The same behavior can be observed in Figure 11for a vertical plate and in Figure 12 for a wedge movingin opposite direction to the free stream However when

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 Journal of Nanomaterials

0 05 1 15 2

01

M

120601 = 0 01 02

120578

m = 0

0

02

04

06

08

1120579(120578) 120582 = Ec = 05

(a)

0

1

120579(120578)

0 05 1 15 2

01

M

m = 1

120601 = 0 01 02

120578

02

04

06

08

120582 = Ec = 05

(b)

Figure 5 Effect of magnetic field on dimensionless temperature of water-based nanofluids for (a) flow past horizontal and (b) flow near thestagnation point of vertical flat plates moving in the same direction

0 01 02

3

1M

120601

120582 = minus01 minus03 minus05

005 01505

1

15

15

2

25

Re12

xCfx

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

m = 0 Ec = 005

(b) MWCNT

Figure 6 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

of the horizontal plate in the direction opposite to thefree stream are shown in Figure 10(a) for SWCNTs and inFigure 10(b) forMWCNTs It is noticed that Nusselt numbersdecrease with an increase in the plate velocity However

magnetic field helps in increasing Nusselt numbers for bothCNTs The same behavior can be observed in Figure 11for a vertical plate and in Figure 12 for a wedge movingin opposite direction to the free stream However when

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Journal of Nanomaterials 9

35

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

115

M

0 01 02

3

120601

005 01505

1

15

2

25

Re12

xCfx

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 7 Variation of skin friction with volume fraction of nanoparticles and magnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

35

45

115

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

115

35

45

0 01 02

3

4

5

120601

005 01515

2

25

Re12

xCfx

m = 1 Ec = 005

(b) MWCNT

Figure 8Variation of skin frictionwith volume fraction of nanoparticles andmagnetic field along a vertical platemoving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

the horizontal plate moves in the direction of free streamthe Nusselt numbers increase with plate velocity This isshown in Figure 13(a) for SWCNTs and in Figure 13(b) forMWCNTs

5 Conclusions

The steady MHD flow and heat transfer from water-basedCNTs over a staticmoving wedge are studied numerically

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

10 Journal of Nanomaterials

m1613

120582 = minus01 minus03 minus05

0 01 02120601

005 0151

15

2

25Re

12

xCfx

M = 1 Ec = 005

(a) SWCNT

m

120582 = minus01 minus03 minus05

1613

0 01 02120601

005 0151

15

2

25

Re12

xCfx

M = 1 Ec = 005

(b) MWCNT

Figure 9 Variation of skin friction with volume fraction of nanoparticles and wedge parameter along a wedge moving in opposite directionto the free stream for (a) SWCNTs and (b) MWCNTs

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

14

12

1

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

14

12

1

08

06

04

Reminus12

xNu x

m = 0 Ec = 005

(b) MWCNT

Figure 10 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

This problem reduces to the classical Falkner-Skanrsquos [1]problem of the boundary layer flow of a viscous (Newtonian)fluid past a fixed wedge when 119872 = 120601 = 0 The effects ofvolume fraction of CNTs magnetic and wedge parameters

are investigated and presented graphically It is found that thefollowing happens

(i) A magnetic field increases skin friction and heattransfer rates

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Journal of Nanomaterials 11

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(a) SWCNT

M

120582 = minus01 minus03 minus05

0 01 02120601

005 015

15

1

Reminus12

xNu x

2

17

14

11

08

05

m = 1 Ec = 005

(b) MWCNT

Figure 11 Variation of Nusselt number with volume fraction of nanoparticles and magnetic field along a vertical plate moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(a) SWCNT

1613

m

120582 = minus01 minus03 minus05

0 01 02120601

005 015

Reminus12

xNu x

14

11

08

05M = 1 Ec = 005

(b) MWCNT

Figure 12 Variation of Nusselt number with volume fraction of nanoparticles and wedge parameter along a wedge moving in oppositedirection to the free stream for (a) SWCNTs and (b) MWCNTs

(ii) Velocity boundary layer thickness is smaller for theflow near the stagnation point of vertical station-arymoving flat plates

(iii) Thermal boundary layer thickness is smaller for theflow past horizontal stationarymoving flat plates

(iv) MWCNTs offer less friction in each case(v) Heat transfer rates increase when plates move in the

free stream direction(vi) Heat transfer rates decrease when plates move in

opposite direction to the free stream

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

12 Journal of Nanomaterials

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(a) SWCNT

M

0 01 02120601

005 015

15

1

Reminus12

xNu x

3

27

24

21

18

15

120582 = 01 03 05

m = 0 Ec = 005

(b) MWCNT

Figure 13 Variation of Nusselt number with volume fraction of nanoparticles andmagnetic field along a horizontal plate moving in the samedirection to the free stream for (a) SWCNT and (b) MWCNT

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] V M Falkner and S W Skan ldquoSome approximate solutions ofthe boundary layer equationsrdquo Philosophical Magazine vol 12no 80 pp 865ndash896 1931

[2] D R Hartree ldquoOn an equation occurring in Falkner andSkanrsquos approximate treatment of the equations of the boundarylayerrdquoMathematical Proceedings of the Cambridge PhilosophicalSociety vol 33 no 2 pp 223ndash239 1937

[3] J C Y Koh and J P Hartnett ldquoSkin friction and heat transfer forincompressible laminar flow over porous wedges with suctionand variable wall temperaturerdquo International Journal of Heatand Mass Transfer vol 2 no 3 pp 185ndash198 1961

[4] H S Takhar and I Pop ldquoOn MHD heat transfer from a wedgeat large prandtl numbersrdquoMechanics ResearchCommunicationsvol 11 no 3 pp 191ndash194 1984

[5] H-T Lin and L-K Lin ldquoSimilarity solutions for laminar forcedconvection heat transfer from wedges to fluids of any Prandtlnumberrdquo International Journal of Heat and Mass Transfer vol30 no 6 pp 1111ndash1118 1987

[6] T Watanabe ldquoThermal boundary layers over a wedge withuniform suction or injection in forced flowrdquo Acta Mechanicavol 83 no 3-4 pp 119ndash126 1990

[7] K A Yih ldquoMHD forced convection flow adjacent to a non-isothermal wedgerdquo International Communications in Heat andMass Transfer vol 26 no 6 pp 819ndash827 1999

[8] A Hossain S Munir and D A S Rees ldquoFlow of viscousincompressible fluid with temperature dependent viscosity and

thermal conductivity past a permeable wedge with uniformsurface heat fluxrdquo International Journal ofThermal Sciences vol39 no 6 pp 635ndash644 2000

[9] D Pal and H Mondal ldquoInfluence of temperature-dependentviscosity and thermal radiation on MHD forced convectionover a non-isothermal wedgerdquo Applied Mathematics and Com-putation vol 212 no 1 pp 194ndash208 2009

[10] A Ishak R Nazar and I Pop ldquoMHD boundary-layer flowof a micropolar fluid past a wedge with constant wall heatfluxrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 1 pp 109ndash118 2009

[11] X Su L Zheng X Zhang and J Zhang ldquoMHD mixedconvective heat transfer over a permeable stretchingwedgewiththermal radiation and ohmic heatingrdquo Chemical EngineeringScience vol 78 pp 1ndash8 2012

[12] K V Prasad P S Datti and K Vajravelu ldquoMHD mixedconvection flow over a permeable non-isothermal wedgerdquoJournal of King Saud UniversitymdashScience vol 25 no 4 pp 313ndash324 2013

[13] A T M M Rahman M S Alam M A Alim and M KChowdhury ldquoUnsteady MHD forced convective heat and masstransfer flow along a wedge with variable electric conductivityand thermophoresisrdquo Procedia Engineering vol 5 pp 531ndash5372013

[14] I Muhaimin R Kandasamy A B Khamis and R RoslanldquoEffect of thermophoresis particle deposition and chemicalreaction on unsteady MHD mixed convective flow over aporouswedge in the presence of temperature-dependent viscos-ityrdquo Nuclear Engineering and Design vol 261 pp 95ndash106 2013

[15] S Abbasbandy and T Hayat ldquoSolution of the MHD Falkner-Skan flow by homotopy analysis methodrdquo Communications inNonlinear Science and Numerical Simulation vol 14 no 9-10pp 3591ndash3598 2009

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Journal of Nanomaterials 13

[16] T Fang and J Zhang ldquoAn exact analytical solution of theFalkner-Skan equation with mass transfer and wall stretchingrdquoInternational Journal of Non-LinearMechanics vol 43 no 9 pp1000ndash1006 2008

[17] U S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlesrdquo in Proceedings of the ASME Fluids EngineeringDivision vol 231 pp 99ndash103 San Francisco Calif USANovember 1995

[18] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for natural convective boundary-layer flow in a porousmedium saturated by a nanofluidrdquo International Journal of Heatand Mass Transfer vol 52 no 25-26 pp 5792ndash5795 2009

[19] A V Kuznetsov and D A Nield ldquoDouble-diffusive naturalconvective boundary-layer flow of a nanofluid past a verticalplaterdquo International Journal of Thermal Sciences vol 50 no 5pp 712ndash717 2011

[20] D A Nield and A V Kuznetsov ldquoThe Cheng-Minkowyczproblem for the double-diffusive natural convective boundarylayer flow in a porous medium saturated by a nanofluidrdquoInternational Journal of Heat andMass Transfer vol 54 no 1ndash3pp 374ndash378 2011

[21] P Cheng andW JMinkowycz ldquoFree convection about a verticalflat plate embedded in a porous medium with application toheat transfer from a dikerdquo Journal of Geophysical Research vol82 no 14 pp 2040ndash2044 1977

[22] AV Kuznetsov andDANield ldquoNatural convective boundary-layer flow of a nanofluid past a vertical platerdquo InternationalJournal of Thermal Sciences vol 49 no 2 pp 243ndash247 2010

[23] M Narayana and P Sibanda ldquoLaminar flow of a nanoliquid filmover an unsteady stretching sheetrdquo International Journal of Heatand Mass Transfer vol 55 no 25-26 pp 7552ndash7560 2012

[24] W A Khan and I Pop ldquoBoundary layer flow past a wedgemoving in a nanofluidrdquoMathematical Problems in Engineeringvol 2013 Article ID 637285 7 pages 2013

[25] N A Yacob A Ishak R Nazar and I Pop ldquoFalkner-Skanproblem for a static and moving wedge with prescribed surfaceheat flux in a nanofluidrdquo International Communications in Heatand Mass Transfer vol 38 no 2 pp 149ndash153 2011

[26] M M Rahman M A Al-Lawatia I A Eltayeb and NAl-Salti ldquoHydromagnetic slip flow of water based nanofluidspast a wedge with convective surface in the presence of heatgeneration (or) absorptionrdquo International Journal of ThermalSciences vol 57 pp 172ndash182 2012

[27] M M Rahman W A Al-Mazroui F S Al-Hatmi M A Al-Lawatia and I A Eltayeb ldquoThe role of a convective surfacein models of the radiative heat transfer in nanofluidsrdquo NuclearEngineering and Design vol 275 pp 382ndash392 2014

[28] A J Chamkha S Abbasbandy A M Rashad and K VajraveluldquoRadiation effects onmixed convection over a wedge embeddedin a porousmediumfilledwith a nanofluidrdquoTransport in PorousMedia vol 91 no 1 pp 261ndash279 2012

[29] P K Kameswaran M Narayana S Shaw and P Sibanda ldquoHeatand mass transfer from an isothermal wedge in nanofluids withSoret effectrdquoThe European Physical Journal Plus vol 129 no 7pp 154ndash164 2014

[30] S Halelfadl T Mare and P Estelle ldquoEfficiency of carbonnanotubes water based nanofluids as coolantsrdquo ExperimentalThermal and Fluid Science vol 53 pp 104ndash110 2014

[31] TMare S Halelfadl O Sow P Estelle S Duret and F BazantayldquoComparison of the thermal performances of two nanofluidsat low temperature in a plate heat exchangerrdquo ExperimentalThermal and Fluid Science vol 35 no 8 pp 1535ndash1543 2011

[32] M-S Liu M Ching-Cheng Lin I-T Huang and C-C WangldquoEnhancement of thermal conductivity with carbon nanotubefor nanofluidsrdquo International Communications inHeat andMassTransfer vol 32 no 9 pp 1202ndash1210 2005

[33] H Xie H Lee W Youn and M Choi ldquoNanofluids containingmultiwalled carbon nanotubes and their enhanced thermalconductivitiesrdquo Journal of Applied Physics vol 94 no 8 pp4967ndash4971 2003

[34] Y Ding H Alias D Wen and R A Williams ldquoHeat transfer ofaqueous suspensions of carbon nanotubes (CNT nanofluids)rdquoInternational Journal of Heat and Mass Transfer vol 49 no 1-2pp 240ndash250 2006

[35] S Halelfadl P Estelle and T Mare ldquoHeat transfer properties ofaqueous carbon nanotubes nanofluids in coaxial heat exchangerunder laminar regimerdquo Experimental Thermal and Fluid Sci-ence vol 55 pp 174ndash180 2014

[36] W A Khan Z H Khan and M Rahi ldquoFluid flow and heattransfer of carbon nanotubes along a flat plate with Navier slipboundaryrdquo Applied Nanoscience vol 4 no 5 pp 633ndash641 2014

[37] RUlHaq ZHKhan andWAKhan ldquoThermophysical effectsof carbon nanotubes on MHD flow over a stretching surfacerdquoPhysica E Low-dimensional Systems andNanostructures vol 63pp 215ndash222 2014

[38] F M White Viscous Fluid Flow McGraw-Hill NewYork NYUSA 2nd edition 1991

[39] R E Bellman and R E Kalaba Quasilinearization and Non-Linear Boundary Value Problems American Elsevier Publish-ing New York NY USA 1965

[40] K Inouye and A Tate ldquoFinite differences version of quasi-linearization applied to boundary layer equationsrdquo AIAA Jour-nal vol 12 no 4 pp 558ndash560 1974

[41] J Hone ldquoCarbon nanotubes thermal propertiesrdquo in DekkerEncyclopedia of Nanoscience and Nanotechnology pp 603ndash6102004

[42] R S Ruoff and D C Lorents ldquoMechanical and thermalproperties of carbon nanotubesrdquo Carbon vol 33 no 7 pp 925ndash930 1995

[43] N A Yacob A Ishak and I Pop ldquoFalkner-Skan problem for astatic or moving wedge in nanofluidsrdquo International Journal ofThermal Sciences vol 50 no 2 pp 133ndash139 2011

[44] K A Yih ldquoUniform suctionblowing effect on forced convec-tion about a wedge uniformheat fluxrdquoActaMechanica vol 128no 3-4 pp 173ndash181 1998

[45] B-L Kuo ldquoHeat transfer analysis for the Falkner-Skan wedgeflow by the differential transformation methodrdquo InternationalJournal of Heat and Mass Transfer vol 48 no 23-24 pp 5036ndash5046 2005

[46] K R Rajagopal A S Gupta and T Y Na ldquoA note on theFalkner-Skan flows of a non-Newtonian fluidrdquo InternationalJournal of Non-Linear Mechanics vol 18 no 4 pp 313ndash3201983

[47] H Blasius ldquoGrenzschichten in Flussigkeiten mit kleiner Rei-bungrdquo Zentralblatt Mathematical Physics vol 56 pp 1ndash37 1908

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials