Thermophysical effects of carbon nanotubes on MHD flow overa stretching surface

Rizwan Ul Haq a,b,n, Zafar Hayat Khan c,d, Waqar Ahmed Khan e

a Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistanb Mechanical and Materials Engineering, Spencer Engineering Building, Room 3055, University of Western Ontario, London, Ontario, Canadac School of Mathematical Sciences, Peking University, Beijing 100871, P.R. Chinad Department of Mathematics, University of Malakand, Dir (Lower), Khyber Pakhtunkhwa, Pakistane Department of Engineering, University of Waterloo 200, University of West Waterloo, Ontario, Canada N2L 3G1

H I G H L I G H T S

� MHD and CNTs effects are incorpo-rated within the base fluids.

� Three different types of base fluidshave been considered.

� Hot fluid along the wall is also con-sidered.

� Obtained coupled ordinary differentialequations are investigated numerically.

� Xue model has the best approach toguess the superb thermal conductivityas compare to rest of models.

G R A P H I C A L A B S T R A C T

Heat transfer difference for different base fluids in the presence of SWCNTs and MWCNTs. Xue modelhas the best approach to guess the superb thermal conductivity as compare to rest of models.

a r t i c l e i n f o

Article history:Received 1 May 2014Received in revised form2 June 2014Accepted 4 June 2014Available online 12 June 2014

Keywords:Carbon nanotubeHeat transferMHD flowStretching sheetNanofluidNumerical solution

a b s t r a c t

This article is intended for investigating the effects of magnetohydrodynamics (MHD) and volumefraction of carbon nanotubes (CNTs) on the flow and heat transfer in two lateral directions over astretching sheet. For this purpose, three types of base fluids specifically water, ethylene glycol andengine oil with single and multi-walled carbon nanotubes are used in the analysis. The convectiveboundary condition in the presence of CNTs is presented first time and not been explored so far.The transformed nonlinear differential equations are solved by the Runge–Kutta–Fehlberg method witha shooting technique. The dimensionless velocity and shear stress are obtained in both directions. Thedimensionless heat transfer is determined on the surface. Three different models of thermal conductivityare comparable for both CNTs and it is found that the Xue [1] model gives the best approach to guess thesuperb thermal conductivity in comparison with the Maxwell [2] and Hamilton and Crosser [3] models.And finally, another finding suggests the engine oil provides the highest skin friction and heattransfer rates.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Flow due to a stretching sheet has gained considerable attentionbecause of its practical applications in industries and engineeringfields such as shrinking wrapping, bundle wrapping, hot rolling, andextrusion of sheet. Initially, Sakiadis [4,5] presented the pioneering

concept of boundary layer two dimensional flows over a continuouslystretching surface with a constant speed. Crane [6] extended thisconcept and found a closed form solution for the stretching surfaces.Since then many authors have extended the idea of stretching sheetfor different fluid flow models [7–11].

Wang [12] presented the three-dimensional axisymmetricboundary layer flow of a Newtonian fluid caused by a stretchingflat surface in two lateral directions. Devi et al. [13] studiednumerically the transient three-dimensional Newtonian flow from

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Physica E

http://dx.doi.org/10.1016/j.physe.2014.06.0041386-9477/& 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: + 92 51 90642182, +1 519 6971969.E-mail addresses: ideal_riz@hotmail.com, rulhaq@uwo.ca (R. Ul Haq).

Physica E 63 (2014) 215–222

a stretching surface. Takhar et al. [14] studied the un-steady MHDflow of an incompressible viscous electrically conducting fluid,caused by an impulsive stretching of the surface in two lateraldirections. They showed that the surface shear stresses inx-direction and y-direction and the surface heat transfer increaseswith the magnetic field. Ariel [15], in a short communication,presented the series solution of exponentially decaying functionsusing the Ackroyd method. More studies on three-dimensionalflow due to a stretching sheet can be found in Refs. [16–20].

The term “nano”was first introduced by Choi [21]. Nanofluids areactually homogeneous mixtures of nano-size particles in the basefluid. Sometimes the thermal conductivity of the base fluid is noteffective. An effective way of improving the thermal conductivity ofbase fluid is to suspend nanoparticles in the base fluids. Fewcommon base fluids include water, organic liquids (e.g. ethylene,tri-ethylene-glycols, refrigerant, etc.), kerosene, lubricants, bio-fluidsand polymeric solution. Nanofluids are created from differentmaterials, such as oxide ceramics (Al2O3, CuO), metal nitride (AlN,SiN), carbide ceramics (Sic, Tic), metals (Cu, Ag, Au); carbons invarious forms (e.g., diamond, graphite, carbon nanotubes, fullerene)and functionalized nanoparticles. Nanoparticles are heat transferliquids with dispersed nanoparticles and recent research has shownthat nanoparticles are capable of improving the thermal conductiv-ities and heat transport properties of the base fluid.

In recent developments, Nield and Kuznetsav [22,23] evaluatethe effects of nanoparticles on natural convection boundary layerflow past a vertical plate. Khan and Pop [24] present the concept ofboundary layer flow of nanofluid past a stretching sheet. Bachoket al. [25] discussed the boundary layer flow of nanofluid over amoving surface in a flowing fluid. For vertical sheet, Makinde andAziz [26,27] extended the literature on nanofluid. Nadeem and Lee[28] studied the effects of nanofluid on the boundary layer flowover an exponentially stretching surface. Makinde et al. [29]analyzed the effects of buoyancy force, magnetic field and con-vective heating on stagnation-point flow and heat transfer due tonanofluid flow towards a stretching sheet. The combined effects ofBrownian motion and thermophoresis on Jeffrey fluid over astretching sheet were investigated by Nadeem et al. [30]. Veryrecently homogeneous flow model is used by Khan et al. [31] tostudy the flow and heat transfer of carbon nanotubes (CNTs) along aflat plate subjected to Navier slip and uniform heat flux boundaryconditions. They suggested that the engine oil-based CNTs havehigher heat transfer rates than water and kerosene-based CNTs.Numerous authors discuss the study of fluid flow and heat transferin the presence of various nanoparticles [32–43].

Carbon nanotubes are found to have special thermal propertieswith very high thermal conductivity. Different models are availableto determine the effective thermal conductivity of nanotubes. Xue [1]proposed a theoretical model for the thermal conductivity of carbonnanotubes based on Maxwell theory considering rotational ellipticalnanotubes with very large axial ratio and compensating the effects ofthe space distribution on CNTs. We will use Xue [1] model and willconsider two different types of carbon nanotubes while taking intoconsideration, three different base fluids. In Section 2 we will presentthe mathematical model used under this study. In view of the samestudy, Section 3 will highlight the proposed numerical method usedto solve the nonlinear differential equations. Moving forward inSection 4, wewill present the physical interpretation and significanceof our results. And finally in Section 5, we will conclude our studyand point out the key findings.

2. Mathematical model

We consider a steady and incompressible flow of CNTs withthree different base fluids past a stretching sheet coinciding with

the plane z¼0. The flow has taken place at above the surfacewhich is connectively heated by a fluid at temperature Tf and heattransfer coefficient hf. The mainstream flow has the temperatureT1ð4Tf Þ (See Fig. 1). The sheet is assumed to stretch continuouslyalong x- and y-directions with linear velocities ax and by, respec-tively. Correspondingly, a uniform magnetic field B0 is appliedparallel to z-axis and the induced magnetic field is assumed to benegligible. The base fluids and the CNTs are assumed to be inthermal equilibrium. The viscous dissipation and radiation effectsare neglected in the energy equation. The ambient temperature isassumed to be constant. Using the order of magnitude analysis, thestandard boundary layer equations for this problem can be writtenas follows:

∂u∂x

þ∂v∂y

þ∂w∂z

¼ 0 ð1Þ

u∂u∂x

þv∂v∂y

þw∂w∂z

¼ νnf∂2u∂z2

�σB2

ρnfu ð2Þ

u∂u∂x

þv∂v∂y

þw∂w∂z

¼ νnf∂2v∂z2

�σB2

ρnfv ð3Þ

u∂T∂x

þv∂T∂y

þw∂T∂z

¼ αnf∂2T∂z2

ð4Þ

where u, v and w are the velocity components along the x-, y- andz-axes, T is the temperature, νnf and αnf are the effective kinematicviscosity and thermal diffusivity of nanofluids respectively. Theeffective properties of carbon nanotubes may be expressed interms of the properties of base fluid and carbon nanotubes andthe solid volume fraction of carbon nanotubes in the base fluids asfollows:

μnf ¼μf

ð1�ϕÞ2:5; ρnf ¼ ð1�ϕÞρf þϕρCNT

ðρcpÞnf ¼ ð1�ϕÞðρcpÞf þϕðρcpÞCNT ;νnf ¼ μnf

ρnf; αnf ¼ knf

ðρcpÞnf ;

knfkf¼

1�ϕþ2ϕ kCNTkCNT � kf

lnkCNT þ kf

2kf

1�ϕþ2ϕkf

kCNT � kfln

kCNT þ kf2kf

9>>>>>>>>>=>>>>>>>>>;

ð5Þ

where knf is the thermal conductivity of the carbon nanotubesproposed by Xue [1], ðρcpÞCNT is the heat capacity of carbonnanotubes and φ is the solid volume fraction of CNTs. Xue [1]

Fig. 1. Geometry of the problem.

R. Ul Haq et al. / Physica E 63 (2014) 215–222216

noticed that the existing models [2,3,35,36] are only valid forspherical or rotational elliptical particles with small axial ratio.Furthermore, these models do not account for the effect of thespace distribution of the CNTS on thermal conductivity.

The hydrodynamic and thermal boundary conditions for theproblem are given by the following equation:

v¼ VwðxÞ ¼ by; u¼ UwðxÞ ¼ ax; �knf ∂T∂y ¼ hnf ðTf �TÞ; at z¼ 0

u-0; v-0; T-T1; as z-1

)

ð6Þ

where a and b are the constants, UwðxÞ and VwðxÞ are the stretchingvelocities in the x- and y-directions, respectively.

We look for a similar solution of Eqs. (1)–(6) of the followingform:

u¼ axf 0ðηÞ; v¼ byg0ðηÞ; w¼ � νfa

� �1=2af ðηÞþbgðηÞ� �

η¼ aνf

� �1=2z; θðηÞ ¼ T�T1

Tf �T1

9>=>; ð7Þ

where η is the similarity variable. Employing the similarityvariables (7), Eqs. (1)–(4) reduce to the following nonlinear systemof ordinary differential equations:

1ð1�ϕÞ2:5ð1�ϕþϕρCNT=ρf Þ

f ″0 þðf þλgÞf ″� f 02� M2

ð1�ϕþϕρCNT=ρf Þf 0 ¼ 0

ð8Þ

1ð1�ϕÞ2:5ð1�ϕþϕρCNT=ρf Þ

g″0 þðf þλgÞg″�g02� M2

ð1�ϕþϕρCNT=ρf Þg0 ¼ 0

ð9Þ

1Pr

knf =kff1�ϕþϕðρcpÞCNT=ðρcpÞf g

θ″þðf þλgÞθ0 ¼ 0 ð10Þ

Subject to the boundary conditions

f ð0Þ ¼ 0; f 0ð0Þ ¼ 1; gð0Þ ¼ 0; g0ð0Þ ¼ λ;

θ0ð0Þ ¼ �Nc

1�ϕþ 2ϕkCNT

kCNT � kfln

kCNT þ kf2kf

1�ϕþ 2ϕkf

kCNT � kfln

kCNT þ kf2kf

0@

1A

� 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1�ϕÞ2:5ð1�ϕþϕρCNT=ρf Þ

p 1�θð0Þ� �f 0-0; g0-0; θ-0; as η-1

9>>>>>>>>=>>>>>>>>;

ð11Þ

here, primes denote derivative with respect to η, M2 ¼ σB20=ρf a is

the magnet parameter, Pr¼ νf =αf is the Prandtl number of the basefluid, Nc¼ hf =kf

ffiffiffiffiffiffiffiffiffiffiνf =a

pis the convective parameter and λ¼ b=a is

the stretching ratio of velocities in the y- and x-directions.The physical quantities of interest are the skin friction coeffi-

cients Cf x and Cfy along x- and y-directions, respectively, and thelocal Nusselt number Nux, which are defined as follows:

Cfx ¼τwx

ρf u2w; Cfy ¼

τwy

ρf u2w; Nux ¼ xqw

kfΔTð12Þ

where τwx and τwy are the surface shear stresses along the x- andy-directions, and qw is the wall heat flux, and are defined as follows:

τwx ¼ μnf∂u∂z

z ¼ 0

; τwy ¼ μnf∂v∂z

z ¼ 0

; qw ¼ �knf∂T∂z

z ¼ 0

ð13Þ

with mnf and knf being the dynamic viscosity and the thermalconductivity of the nanofluids, respectively. Using the similarity

variables (7), we obtain

Re1=2x Cf x ¼ 1ð1�ϕÞ2:5f ″ð0Þ

Re1=2y Cf y ¼ λðy=xÞ 1ð1�ϕÞ2:5g″ð0Þ

Re�1=2x Nux ¼ �knf

kfθ0ð0Þ

9>>>>=>>>>;

ð14Þ

where Rex is the local Reynolds number.

3. Numerical procedure

The coupled non-linear ordinary differential Eqs. (8)–(10)subjected to the boundary conditions (11) are solved numericallyusing the Runge–Kutta–Fehlberg method with a shooting techni-que for different values of governing parameters. Such includesmagnetic parameter M, stretching parameter λ, Prandtl number Pr,convective parameter Nc and CNTs volume fraction ϕ. The bound-ary value problem is initially transformed into an initial valueproblem (IVP). Then the IVP is solved by a systematic guessing for,g″ð0Þ and θ0ð0Þ until the far field boundary condition is achieved.The step size Δη¼ 0:01 is used to obtain the numerical solutionwith ηmax ¼ 12, where ηmax is the finite value of the similarityvariable η for the far field boundary conditions. The convergence isassured by taking error 10�6 in all cases. For pure fluid (φ¼0) andisothermal boundary condition ðNc-1Þ, the results for skinfriction (with Pr¼0.7) and Nusselt number (with Pr¼10) arecompared with those obtained by Nadeem et al. [37] and Wang[12] for different values of magnetic parameter M in Table 1. Wenotice that the comparison shows good agreement for each valueof M. Therefore, we are confident that the present results are veryaccurate.

4. Results and discussion

The effects of the solid volume fraction ϕ of CNTs, magneticparameter M, stretching ratio of velocities λ as well as theconvective parameter Nc on the dimensionless velocities withtemperature are analyzed for both CNTs with water as base fluid.The skin friction coefficients along x-direction and along they-direction and the heat transfer rates are analyzed for threedifferent base fluids. The solid volume fraction ϕ is used in therange of 0rϕr0.2. The thermo-physical properties of base fluidsand CNTs are presented in Table 2. Fig. 2(a) and (b) shows theeffects of CNTs volume fraction of both SWCNT and MWCNT on thedimensionless velocities for different values of magnetic para-meter. In the absence of magnetic field, the dimensionless velo-cities are found to be higher within the hydrodynamic boundarylayer for both pure fluid and water-based CNTs. It can be seen thatthe magnetic field reduces the hydrodynamic boundary layerthickness in both cases. It is important to note, for pure water(when ϕ¼0), the dimensionless velocities are the same in bothcases. They increase with a rise in the CNTs volume fraction ofboth SW and MW CNTs.

Table 1Comparison of results for skin friction and Nusselt number for pure fluid withNc-1 and λ¼ 1.

M Present results Nadeem et al. [37] Wang [12]

�g″ð0Þ �θ0ð0Þ �g″ð0Þ �θ0ð0Þ �g″ð0Þ

0 1.173721 3.306792 1.1748 3.3078 1.17372010 3.367240 2.830448 3.3667 2.8309 –

100 10.066473 1.680774 10.0663 1.5471 –

R. Ul Haq et al. / Physica E 63 (2014) 215–222 217

The effects of CNTs volume fraction on the dimensionlessvelocities are illustrated in Fig. 3(a) and (b) for stretching andshrinking sheets in the presence of a magnetic field. In both cases,these effects are investigated for water-based SWCNTs. It can beseen in Fig. 3(a) that no appreciable effect of φ or λ could be foundon f 0ðηÞ. However, the effects of stretching/shrinking parameterson the dimensionless velocity g0ðηÞ could be observed clearly inFig. 3(b). For both stretching and shrinking sheets, the dimension-less velocity g0ðηÞ converges at the same time.

Fig. 4(a) and (b) shows the effects of CNTs volume fraction,convective and magnetic parameters on the dimensionless tem-perature of water-based CNTs. It can be seen that the differencebetween the dimensions, temperatures of SWCNT and MWCNT are

almost negligible in both cases. The dimensionless temperature atthe surface is lower for a pure fluid (water) and for smaller valuesof the convective parameter. In the presence of a magnetic field,the surface temperature increases with CNTs volume fraction andconvective parameter, as shown in Fig. 4(a). Inside the thermalboundary layer, the dimensionless temperature increases withCNTs volume fraction and magnetic parameter.

The variation of skin friction along x-axis with CNTs volumefraction for different parameters is shown in Figs. 5 and 6 fordifferent base fluids. Since the thermo-physical properties of bothCNTs increase with volume fraction, the skin friction also increasesin each case. Fig. 5(a)–(c) shows the effects of magnetic field onskin friction for three different base fluids with SW and MW CNTs.In each case, the skin friction increases with magnetic field. Due tothe higher viscosity of engine oil, the skin friction is found to behigher in case of engine oil.

In Fig. 6(a)–(c), the effects of stretching parameter on the skinfriction along x-axis are depicted in the presence of a magneticfield. In each case, the skin friction is found to be higher forSWCNTs due to the higher density of SWCNTs. The effect ofstretching is to reduce friction in each case for both CNTs. Thisreduction is found to be higher for a higher volume fraction ofCNTs. The effects of same parameters on the skin friction alongy-axis are shown in Figs. 7 and 8 for three different base fluids

Table 2Thermophysical properties of base fluids and CNTs [38–43].

Physical properties Base fluids Nanoparticles

Water Ethylene glycol Engine oil SWCNT MWCNT

ρ (kg/m3) 997 1115 884 2600 1600cp (J/kg-K) 4179 2430 1910 425 796k (W/m-K) 0.613 0.253 0.144 6600 3000Pr 6.2 203 6450 – –

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1Water (Pr = 6.2)

SWCNT

MWCNT

= 0, 0.1, 0.2

M = 0

M = 2

Nc = = 0.5

0 1 2 3 4 5

0

0.1

0.2

0.3

0.4

0.5Water (Pr = 6.2)

SWCNT

MWCNT

= 0, 0.1, 0.2

M = 0

M = 2

Nc = = 0.5

Fig. 2. Effects of nanoparticle volume fraction and magnetic parameters on dimensionless velocities for CNTs.

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1SWCNT/Water (Pr = 6.2)

= 0, 0.1, 0.2

Nc = 0.5, M = 1-0.5

0.5

0 1 2 3 4-0.5

0

0.5

= 0, 0.1, 0.2

SWCNT/Water (Pr = 6.2)

= 0.5

= -0.5

Nc = 0.5, M = 1

Fig. 3. Effects of nanoparticle volume fraction and stretching parameters on dimensionless velocities for SWCNT.

R. Ul Haq et al. / Physica E 63 (2014) 215–222218

with both CNTs. The effects of magnetic fields on the skin frictionalong y-axis are found to be the same as before. The effects ofstretching parameter on the skin friction coefficient along y-axisare presented in Fig. 8(a)–(c), for three different base fluids in the

presence of both CNTs. It can be seen that the skin frictionincreases with stretching parameter in each case. The differencebetween skin friction of both CNTs increases with volume fractiondue to increase in density.

0 1 2 3 4

0

0.2

0.4

0.6

0.8

1

Bi = 10

Bi = 0.5

Water (Pr = 6.2)

SWCNT

MWCNT

= 0, 0.1, 0.2

M = 1, = 0.5

0 1 2 3 4 5

0

0.2

0.4

0.6

0.8

1

= 0, 0.1, 0.2

SWCNT

MWCNT

Water (Pr = 6.2)

Bi = 10, = 0.5

M = 0

M = 3

Fig. 4. Effects of nanoparticle volume fraction, convective and magnetic parameters on dimensionless temperature for CNTs.

0 0.05 0.1 0.15 0.20.8

1.2

1.6

2

2.4

Water (Pr = 6.2)

SWCNT

MWCNT

Nc= =0.5

M=0,1,2

0 0.05 0.1 0.15 0.20.8

1.2

1.6

2

2.4

Ethylene glycol (Pr = 203)

Nc= =0.5

M=0,1,2

SWCNT

MWCNT

0 0.05 0.1 0.15 0.20.8

1.2

1.6

2

2.4

2.8Nc= =0.5

M=0,1,2

SWCNT

MWCNT

Engine oil (Pr = 6450)

Fig. 5. Variation of skin friction coefficient along x-direction with CNTs volume fraction ϕ and magnetic parameter M.

0 0.05 0.1 0.15 0.21.2

1.5

1.8

2.1Nc = M = 0.5

SWCNT

MWCNT

Water (Pr = 6.2)

= -0.5, 0.5

0 0.05 0.1 0.15 0.21.2

1.5

1.8

2.1Ethylene glycol (Pr = 203)

Nc = M = 0.5

SWCNT

MWCNT

= -0.5, 0.5

0 0.05 0.1 0.15 0.21.2

1.5

1.8

2.1

Engine oil (Pr = 6450)

SWCNT

MWCNT

Nc = M = 0.5

= -0.5, 0.5

Fig. 6. Variation of skin friction coefficient along x-direction with CNTs volume fraction ϕ and stretching parameter λ.

R. Ul Haq et al. / Physica E 63 (2014) 215–222 219

The variation of Nusselt numbers with CNTs volume fraction fordifferent values of convective parameter is depicted in Fig. 9(a)–(c)for water, ethylene glycol and engine oil with both CNTs. It can be

observed that the Nusselt numbers increase with CNTs volumefraction and convective parameter in each case. This is due to the factthe thermal conductivity of each CNT rises with an increase in the

0 0.05 0.1 0.15 0.2

0.4

0.6

0.8

1

1.2

Ethylene glycol (Pr = 203)

Nc = = 0.5

M = 0, 1, 2

SWCNT

MWCNT

0 0.05 0.1 0.15 0.2

0.4

0.6

0.8

1

1.2

Water (Pr = 6.2)

SWCNT

MWCNT

Nc = = 0.5

M = 0, 1, 2

0 0.05 0.1 0.15 0.2

0.4

0.6

0.8

1

1.2Nc = = 0.5

M = 0, 1, 2

SWCNT

MWCNT

Engine oil (Pr = 6450)

Fig. 7. Variation of skin friction coefficient along y-direction with CNTs volume fraction ϕ and magnetic parameter M.

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

Water (Pr = 6.2)

SWCNT

MWCNT

Nc = M = 0.5

= 0.2, 0.5, 0.8

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

Ethylene glycol (Pr = 203)

Nc = M = 0.5

= 0.2, 0.5, 0.8

SWCNT

MWCNT

0 0.05 0.1 0.15 0.20

0.2

0.4

0.6

0.8

1

1.2

1.4

Nc = M = 0.5

= 0.2, 0.5, 0.8

SWCNT

MWCNT

Engine oil (Pr = 6450)

Fig. 8. Variation of skin friction coefficient along y-direction with CNTs volume fraction ϕ and stretching parameter λ.

0 0.05 0.1 0.15 0.20.6

0.8

1

1.2

1.4

1.6

1.8

2 SWCNTMWCNT

Water (Pr = 6.2)

Nc = 1.0

Nc = 1.5

Nc = 0.5

M = = 0.5

0 0.05 0.1 0.15 0.22

3

4

5

6

7 SWCNTMWCNT

Ethylene glycol (Pr = 203)

Nc = 0.5

Nc = 1.0

Nc = 1.5M = = 0.5

0 0.05 0.1 0.15 0.23

4.5

6

7.5

9

10.5

12SWCNTMWCNT

Engine oil (Pr = 6450)

Nc = 1.0

Nc = 1.5

Nc = 0.5

M = = 0.5

Fig. 9. Variation of Nusselt number with CNTs volume fraction ϕ and convective parameter Nc.

R. Ul Haq et al. / Physica E 63 (2014) 215–222220

CNTs volume fraction. Also, it is important to note the Nusseltnumbers increase in Prandtl numbers as we make a transition fromwater to engine oil. The increase in Prandtl numbers actually decreasesthe thermal boundary layer thickness, thus decreasing the thermalresistance and increasing the heat transfer rate.

Different models of thermal conductivity are compared in Fig. 10(a)–(c) for SWCNTs and in Table 3 for MWCNT, respectively. It can beverified that the Xue [1] model gives the highest heat transfer ratesfor both CNTs in comparison with Maxwell [2] and Hamilton andCrosser [3] models. This is due to the fact that Xue [1] modelconsiders rotational elliptical nanotubes with very large axial ratioand compensating the effects of the space distribution on CNTs.

5. Conclusions

In this article, we have presented similarity solutions for themomentum and energy equations governing a 3-D MHD flow andheat transfer of CNTs along a stretching sheet. Results for the localfriction factor and Nusselt number are presented with convectiveboundary condition for three different base fluids mixed withSWCNTs and MWCNTs. It is concluded that

� The hydro-thermal boundary layer thickness increase withCNTs volume fraction.

� The shear stresses in x- and y-direction and surface heattransfer increase with CNTs volume fraction.

� Engine oil-based CNTs have higher skin friction and heattransfer rate than water and ethylene glycol-based CNTs.

� Xue [1] model for the thermal conductivity of CNTs shows thehighest enhancement in heat transfer rates.

Acknowledgements

The first author (Rizwan Ul haq) is thankful to Higher Educa-tion Commission Pakistan for research funding through Interna-tional Research Support Initiative Program (IRSIP).

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0 0.05 0.1 0.15 0.20.5

1

1.5

2

2.5

M = = Nc = 0.5

SWCNT/Engine oil

Hamilton & Crosser model

Maxwell model

Xuem

odel

0 0.05 0.1 0.15 0.2

0.5

1

1.5SWCNT/Water

M = = Nc = 0.5

Xuemod

el

Hamilton & Crosser model

Maxwell model

0 0.05 0.1 0.15 0.20.5

1

1.5

2

2.5

SWCNT/Ethylene glycol

M = = Nc = 0.5

Xuemod

el

Hamilton & Crosser model

Maxwell model

Fig. 10. Comparison among the theoretical models for single-wall CNT with different base fluids. In the calculation, n¼6 for the Hamilton–Crosser model [3].

Table 3Comparison among three theoretical models for Nusselt numbers with MWCNTs suspended in three different base fluids, where Nc¼M¼λ¼0.5 and n¼6 for the Hamilton–Crosser model [3].

ϕ (%)↓ Water Ethylene glycol Engine oil

Xue H&C Maxwell Xue H&C Maxwell Xue H&C Maxwell

0.0 0.65077 0.65077 0.65077 1.90009 1.90009 1.90009 3.44808 3.44808 3.448080.02 0.66346 0.66681 0.67574 1.94042 1.94252 1.94894 3.50818 3.50887 3.511150.04 0.67640 0.68302 0.69807 1.98247 1.98662 1.99722 3.57156 3.57290 3.576600.06 0.68966 0.69950 0.71909 2.02639 2.03257 2.04617 3.63843 3.64042 3.645080.08 0.70328 0.71637 0.73949 2.07233 2.08055 2.09648 3.70904 3.71166 3.717020.1 0.71733 0.73371 0.75975 2.12044 2.13076 2.14858 3.78362 3.78688 3.79280

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