Research ArticleInfluence of Variable Thermal Conductivity on MHD BoundaryLayer Slip Flow of Ethylene-Glycol Based Cu Nanofluids over aStretching Sheet with Convective Boundary Condition

N Bhaskar Reddy1 T Poornima1 and P Sreenivasulu2

1 Department of Mathematics Sri Venkateswara University Tirupati 517502 India2Department of Mathematics Yogananda Institute of Technology and Science Tirupati 517520 India

Correspondence should be addressed to T Poornima poonimaanandgmailcom

Received 24 May 2014 Revised 2 October 2014 Accepted 2 October 2014 Published 6 November 2014

Academic Editor J A Tenreiro Machado

Copyright copy 2014 N Bhaskar Reddy et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

An analysis is carried out to investigate the influence of variable thermal conductivity and partial velocity slip on hydromagnetictwo-dimensional boundary layer flow of a nanofluid with Cu nanoparticles over a stretching sheet with convective boundarycondition Using similarity transformation the governing boundary layer equations along with the appropriate boundaryconditions are transformed to a set of ordinary differential equations Employing Runge-kutta fourth-order method along withshooting technique the resultant system of equations is solved The influence of various pertinent parameters such as nanofluidvolume fraction parameter the magnetic parameter radiation parameter thermal conductivity parameter velocity slip parameterBiot number and suction or injection parameter on the velocity of the flow field and heat transfer characteristics is computednumerically and illustrated graphically The present results are compared with the existing results for the case of regular fluid andfound an excellent agreement

1 Introduction

Theflow analysis of nanofluids has been the topic of extensiveresearch due to its enhanced thermal conductivity behaviorin heat transfer processes Nanofluid is a new class of heattransfer fluid (the term nanofluid was coined by Choi [1])that contains a base fluid and nanosized material particles(diameter less than 100 nm) or fibers suspended in theordinary fluids Nanoparticles are made from various mate-rials such as oxide ceramics (Al

2O3 CuO) nitride ceramics

(AlN SiN) carbide ceramics (SiC TiC) metals (Cu Ag Au)semiconductors (TiO

2 SiC) carbon nanotubes and com-

positematerials such as alloyed nanoparticles or nanoparticlecore-polymer shell composites According to Prodanovi et al[2] nanofluids containing ultrafine nanoparticle have thecapability of flowing in porous media and these flows canimprove oil recovery hence nanoparticles are able to controlthe processes of oil recovery To improve oil recovery ofviscous oils a fluid for example water is injected into the

porous medium to displace the oil since water viscosityis inferior to that of oil However increasing the injectedfluid viscosity using nanofluids would drastically increase therecovery efficiency Nanoparticles can also be used to deter-mine changes in fluid saturation and reservoir propertiesduring oil and gas production

Many studies on nanofluids are being conducted by scien-tists and engineers due to their diverse technical and biomed-ical applications Examples include nanofluid coolant (elec-tronics cooling vehicle cooling and so on) medical applica-tions (cancer therapy and safer surgery by cooling) processindustries (materials and chemicals detergency food anddrink oil and gas paper and printing and textiles) adv-ances in nanoelectronics nanophotonics and nanomagnet-ics ultrahigh performance cooling is necessary for manyindustrial technologies

The nanofluids are more stable and have acceptableviscosity and better wetting spreading and dispersion prop-erties on solid surface [3 4] The characteristic feature of

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2014 Article ID 905158 10 pageshttpdxdoiorg1011552014905158

2 International Journal of Engineering Mathematics

nanofluids is thermal conductivity enhancement a phe-nomenon observed by Masuda et al [5] This phenomenonsuggests the possibility of using nanofluids in advancednuclear systems (Buongiorno and Hu [6]) A benchmarkstudy on the thermal conductivity of nanofluids was made byBuongiorno et al [7] Das et al [8] studied a two- to fourfoldincrease in thermal conductivity enhancement for water-based nanofluids containing Al

2O3or CuO nanoparticles

over a small temperature range 21∘Cndash51∘CThe study of boundary layer flow and heat transfer over

a stretching surface has attracted considerable attention inmany fields of science and technology Few of these appli-cations such as wire and fiber coating materials manufac-tured polymer extrusion food stuff processing drawing ofcopper wires and chemical processing equipment Pioneer-ing work on the dynamics of boundary layer flow overstretching surface was done by Crane [9] who examined thetwo-dimensional Navier-Stokes equations Later on variousaspects of the problem have been investigated by Dutta et al[10] and Chen and Char [11] Khan and Pop [12] presentedthe boundary layer flow of nanofluid past a stretching sheetRecently Hassani et al [13] studied an analytical solutionfor boundary layer flow of a nanofluid past a stretchingsheet Rana and Bhargava [14] analyzed the flow and heattransfer over a nonlinear stretching sheet a numerical studyHamad and Ferdows [15] studied the similarity solution ofboundary layer stagnation-point flow towards a heatedporous stretching sheet saturated with a nanofluid with heatabsorptiongeneration and suctionblowing a Lie groupanalysis

Magnetohydrodynamics (MHD) boundary layer flowover a stretching sheet is important during the last fewdecades due to its numerous applications in industrial man-ufacturing processes such as the aerodynamic extrusion ofplastic sheets liquid film hot rolling wire drawing andglass-fiber and paper production Al-Odat et al [16] analyzedthe thermal boundary layer on an exponentially stretchingcontinuous surface in the presence of magnetic field effectChamkha and Aly [17] have presented theMHD free convec-tion flow of a nanofluid past a vertical plate in the presence ofheat generation or absorption effects Later Aliakbar et al [18]analyzed the influence of thermal radiation on MHD flow ofMaxwellian fluids above stretching sheets Khan et al [19]studied the effects of magnetic field on radiative flow of ananofluid past a stretching sheet Ibrahim and Shankar [20]analyzed MHD boundary layer flow and heat transfer of ananofluid past a permeable stretching sheet with slip condi-tions

However radiation heat transfer has a key impact in hightemperature regime Many technological processes occur athigh temperature and good working knowledge of radiativeheat transfer plays an instrumental role in designing the per-tinent equipment In many practical applications dependingon the surfaces properties and solid geometry the radiativetransport is often comparable with that of convective heattransfer But unfortunately little is known about the effects ofradiation on the boundary layer flow of a radiating fluidIn fact a few difficulties arise in studying radiative fluid flowFirstly when radiative heat transfer takes place the radiation

is absorbedemitted not only at system boundaries but also inthe interior of the system hence prediction of fluid absorp-tion is a difficult task Secondly the absorption coefficient ofthe absorbing emitting fluids is in general strongly depen-dent onwavelengthThirdly presence of radiation term in theenergy equationmakes the equation highly nonlinearThis allleads to computational difficulty Owing to these difficultiesthe effect of radiation on convective flows has been inves-tigated with reasonable simplifications A good literatureon radiative transfer can be seen in well-presented texts bySparrow andCess [21] Ozisik [22] Siegel andHowell [23] andHowell [24] Takhar et al [25] studied radiation effect on thefree convection flow of a gas past a semi-infinite plate Hos-sain and Takhar [26] investigated mixed convection along avertical plate with uniform surface temperature taking radia-tion into account

Thenonadherence of the fluid to a solid boundary knownas velocity slip is a phenomenon that has been observedunder certain circumstances [27] It is a well-known fact thata viscous fluid normally sticks to the boundary But there aremany fluids for example particulate fluids rarefied gas etcwhere theremay be a slip between the fluid and the boundary[28] The effects of slip conditions are very important intechnological applications such as in the polishing of artificialheart valves and the internal cavities [29]The foremost studytaking into account the slip boundary condition over astretching sheet was conducted by Anderson [30] A closedform solution of full Navier-Stokes equations for a hydrody-namic flowover a stretching sheetwas studied by himNext toAndersonWang [31] solved the full Navier-Stokesrsquo equationswith partial slip past a stretching sheet He continues toinvestigate stagnation slip flow and heat transfer on a movingplate [32] Fang et al [33] analyzed the slip condition of aMHD viscous flow over a stretching sheet Hayat et al [34]extended the problem of the previous researchers by incor-porating the thermal slip condition and discussed unsteadymagneto hydrodynamic flow and heat transfer over a perme-able stretching sheet with slip condition also In a similar wayAbu Bakar et al [35] studied the boundary layer flow over astretching sheet with a convective boundary condition andslip effect

But so far no attempt has been made to analyze theeffects of variable thermal conductivity and velocity slip onMHD boundary layer flow of Ethylene-Glycol based Cunanofluids over a stretching sheet with convective boundarycondition Thus the problem is investigated The surfaceexhibits convective heating boundary conditions [36ndash42]Recently Alsaedi et al [43] studied the effects of heat genera-tionabsorption on stagnation point flow of nanofluid over asurface with convective boundary conditions Masood Khanet al [44] investigated the MHD Falkner-Skan flow withmixed convection and convective boundary conditions Anefficient numerical shooting technique with a fourth-orderRunge-Kutta scheme as used to solve the normalized bound-ary layer equations and the effects of material parameters onthe flow field and heat transfer characteristics is discussed indetail

International Journal of Engineering Mathematics 3

x

y

u

g

B0

TfTinfin

uw(x)

Nanofluid

Figure 1 Physical model of the system

2 Mathematical Analysis

A two-dimensional steady laminar boundary layer flow of aviscous incompressible flow of nanofluid past a permeablestretching sheet coinciding with the plane 119910 = 0 is consid-ered The flow is confirmed to 119910 gt 0 The fluid is assumedto be gray absorbing-emitting radiation but nonscatteringmediumThe flow is generated due to stretching of the sheetcaused by the simultaneous application of two equal andopposite forces along the 119909-axis Keeping the origin fixed thesheet is then stretched with velocity 119906

119908(119909) = 119886119909 where 119886 is

constant A uniformmagnetic field is applied in the directiontransverse to the stretching surface The transverse appliedmagnetic field and magnetic Reynolds number are assumedto be very small so that the induced magnetic field isnegligible There is a constant suctioninjection velocity 119881

119908

normal to the sheet The temperature of sheet surface (to bedetermined later) is the result of a convective heating processwhich is characterized by a temperature119879

119891and a heat transfer

coefficient ℎ119891 It is to be mentioned that the hole size (in

nanoscale) of porous sheet is taken to be constantThe fluid isethylene-glycol based nanofluid containing copper (Cu)Thenanofluid is assumed incompressible and flow is assumed tobe laminar A schematic representation of the physical modeland coordinate system is depicted in Figure 1Under the usualassumptions the governing boundary layer equations aregiven by

120597119906

120597119909+120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910=

1

120588119899119891

[120583119899119891

1205972119906

1205971199102+ 119892 (120588120573)

119899119891(119879 minus 119879

infin) minus 120590119861

2

0119906]

(2)

119906120597119879

120597119909+ V

120597119879

120597119910=

1

(120588119862119901)119899119891

120597

120597119910(119896lowast

119899119891

120597119879

120597119910minus 119902119903) (3)

The boundary conditions for the velocity and temperaturefields are

119906 = 119906119908+ 119860

120597119906

120597119910 V = minusV

119908 minus119896

119891

120597119879

120597119910= ℎ119891(119879119891minus 119879infin)

at 119910 = 0

119906 997888rarr 0 119879 997888rarr 119879infin

as 119910 997888rarr infin

(4)

where 119906 and V are the velocity components in the 119909 and 119910

directions respectively119879 is the temperature of the nanofluid119879infinis the ambient fluid temperature 119892 is the acceleration due

to gravity 120590 is the electric conductivity 1198610is the uniform

magnetic field strength and 119902119903is the radiative heat flux A

is the velocity slip factor V119908is the wall mass flux with V

119908lt 0

for suction and V119908

gt 0 for injection respectively 119896119891is the

thermal conductivity of the ordinary fluid and 119879119891is the

temperature of the hot fluidIt should be mentioned that such slip conditions were

recently used in a series of papers [45ndash50] It is worthmentioning at this end that fluid flowwith slip is important inmicroelectromechanical systems (MEMS) The flow in thesesystems deviates significantly from the traditional no slipflow because of the microscale dimensions of these devicesMEMS barometers and commercially compatible flexibleprinted circuit boards are examples for stretchable MEMS[51]

The variable thermal conductivity (Arunachalam andRajappa [52] Chiam [53] and Seddeek and Salem [54]) isconsidered to vary linearly with temperature as given below

119896lowast= 119896 [1+ isin 120579] (5)

where 119896 is thermal conductivity parameterThe effective density of the nanofluid is given by

120588119899119891

= (1 minus 120601) 120588119891+ 120601120588119904 (6)

where 120601 is the solid volume fraction of nanoparticles Ther-mal diffusivity of nanofluid is

120572119899119891

=119896119899119891

(120588119862119901)119899119891

(7)

where the heat capacitance 119862119901of the nanofluid is obtained as

(120588119862119901)119899119891

= (1 minus 120601) (120588119862119901)119891+ 120601 (120588119862

119901)119904 (8)

The thermal conductivity of the nanofluid 119896119899119891

for sphericalnanoparticles can be written as (Maxwell [55])

119896119899119891

119896119891

=(119896119904+ 2119896119891) minus 2120601 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120601 (119896

119891minus 119896119904) (9)

The thermal expansion coefficient of the nanofluid can bedetermined by

(120588120573)119899119891

= (1 minus 120601) (120588120573)119891+ 120601 (120588120573)

119904 (10)

4 International Journal of Engineering Mathematics

Table 1 Thermophysical properties of fluid and nanoparticles [5556]

Physical properties Fluid phase (ethylene-glycol) CuCp (JkgK) 11144 385120588 (kgm3) 2415 8933120573 times 10

minus5 (1K) 65 167119896 (WmK) 0252 401

Also the effective dynamic viscosity of the nanofluid given byBrinkman [56] can be written as

120583119899119891

=120583119891

(1 minus 120601)25

(11)

Here the subscripts 119899119891119891 and 119904 represent the thermophysicalproperties of the nanofluids base fluid and nanosolid parti-cles respectively (Table 1)

By using the Rosseland diffusion approximation (Prasadet al [57]) the radiative heat flux is given by

119902119903= minus

4

3

120590lowast

119896lowast

1205971198794

120597119910 (12)

where 120590lowast is the Stefan-Boltzmann constant and 119896

lowast is themean absorption coefficient It should be noted that by usingthe Rosseland approximation the present analysis is limitedto optically thick fluids Taking the refractive index of the gasmedium to be constant unidirectional radiation flux 119902

119903is

considered and it is assumed that 120597119902119903120597119910 ≫ 120597119902

119903120597119909 This

model is valid for optically thick media in which thermalradiation propagates only a limited distance prior to expe-riencing scattering or absorption The local thermal radia-tion intensity is due to radiation emitting from proximatelocations in the vicinity of which emission and scattering arecomparable to the location of interest The energy transferdepends on conditions only in the area adjacent to theplate regime that is the boundary layer regime Rosselandrsquosmodel yields accurate results for intensive absorption that isoptically thick flows which are optically far from the bound-ary surface Implicit in this approximation is also the exis-tence of wavelength regions where the optical thickness mayexceed a value of five As such the Rosseland model whilelimited compared with other flux models can simulate to areasonable degree of accuracy thermal radiation in problemsranging from thermal radiation transport via gases at lowdensity to thermal radiation simulations associated withnuclear blast waves If the temperature differences within theflow are sufficiently small then (12) can be linearized byexpanding 119879

4 into the Taylor series about 119879infin which after

neglecting higher-order terms takes the form

1198794cong 41198793

infin119879 minus 3119879

4

infin (13)

In view of (12) and (13) (3) becomes

119906120597119879

120597119909+ V

120597119879

120597119910

= 120572119899119891[(1+ isin 120579 +

4

3Nr) 120597

2119879

1205971199102+

isin

(119879119891minus 119879infin)(120597119879

120597119910)

2

]

(14)

where Nr = 16120590lowast1198793

infin3119896119899119891119896lowast is the radiation parameter

To get similarity solutions of (1)ndash(3) subject to the bound-ary conditions (4) we introduce the following similaritytransformations

120578 = (119886

]119891

)

12

119910 119906 = 1198861199091198911015840(120578) V = minus (119886]

119891)12

120579 =119879 minus 119879infin

119879119891minus 119879infin

119872 =1205901198612

0

119886120588119891

120582 =

119892 (120588120573)119891(119879119891minus 119879infin)

120588119891119886119906119908

Nr =4120590lowast1198793

infin

119896lowast119896119899119891

Pr =]119891

120572119891

119878 = minusV119908

radic119886]119891

120575 = 119860radic119886

]119891

Γ =ℎ119891

119896119891

radic]119891

119886

(15)

where ]119891is the kinematic viscosity of the base fluid

In view of (15) (1) (2) and (14) take the followingdimensionless form

119891101584010158401015840(1 minus 120601)

25

[(1 minus 120601 + 120601120588119904

120588119891

)(11989111989110158401015840minus 11989110158402)

+(1 minus 120601 + 120601(120588120573)119904

(120588120573)119891

)120582120579 minus1198721198911015840] = 0

(1 +4

3Nr) 12057910158401015840+ isin 120579120579

10158401015840+ isin 12057910158402

+ Pr(119896119891

119896119899119891

)(1 minus 120601 + 120601

(120588119862119901)119904

(120588119862119901)119891

)1198911205791015840= 0

(16)

where prime denotes the differentiation with respect to 120578The corresponding boundary conditions are

1198911015840(0) = 1 + 120575119891

10158401015840(0) 119891 (0) = 119878

1205791015840(0) = minusΓ (1 minus 120579 (0))

1198911015840(infin) 997888rarr 0 120579 (infin) 997888rarr 0

(17)

International Journal of Engineering Mathematics 5

07

06

05

04

03

02

01

000 2 4 6 8 10 12

Present result

120579(120578)

120578

Abu Bakar et al [35]

Γ = 005 05 10

Figure 2 Comparison of temperature profiles for different Γ

The quantities of practical interest in this study are the skinfriction or the shear stress coefficient119862

119891and the local Nusselt

number Nu119909 which are defined as

119862119891= minus

120583119899119891

1205881198911199062119908

(120597119906

120597119910)

119910=0

Nu119909=

119909119896119899119891

119896119891(119879119908minus 119879infin)(minus

120597119879

120597119910)

119910=0

(18)

Using (6) the skin friction coefficient and local Nusseltnumber can be expressed as

Re12119909

119862119891= minus

1

(1 minus 120601)25

11989110158401015840(0)

Re12119909

Nu119909= minus

119896119899119891

119896119891

1205791015840(0)

(19)

where Re119909= 119906119908119909]119891is the Reynolds number

3 Results and Discussion

The coupled nonlinear differential equations (16) along withthe boundary conditions (17) are solved numerically usingRunge-Kutta fourth order technique along with shootingmethodThe integration length 120578

infinvaries with the parameter

values and it has been suitably chosen each time such that theboundary conditions at the outer edge of the boundary layerare satisfied In order to bring out the salient features of theflow and the heat transfer characteristics in the Cu-EG basednanofluid the numerical values for different values of thegoverning parameters 120601 119872 isin Nr 120575 119878 and Γ are computedand portrayed in Figures 2ndash13

In order to validate our analysis the present results arecompared with the existing results for the case of regular fluidand found an excellent agreement (Figure 2)

Figure 3 shows the temperature profiles for different val-ues of the nanoparticle volume fraction 120601 with water and EG

006

005

004

003

002

001

00000 05 10 15 20 25 30

120579(120578)

120578

Cu-EGCu-water

120601 = 0 005 01 015 02

Figure 3 Temperature profiles for different values of 120601

120601 = 0 005 01 015 02

Cu-EG

120578

0 1 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

Figure 4 Velocity profiles for different values of 120601

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120601 = 0 005 01 015 02

Cu-EG

120578

120579(120578)

Figure 5 Temperature profiles for different values of 120601

based nanofluids It is found that the temperature increaseswith an increase in the nanoparticle volume fraction forboth Cu-water and Cu-EG based fluid And also it is noticedthat EG based nanofluids have more impact in rising thetemperature than that of water-based nanofluids This isdue to the fact that the thermal conductivity increases and

6 International Journal of Engineering Mathematics

Cu-EG

120578

10 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

M = 0 05 10 15 20

Figure 6 Velocity profiles for different values of119872

006

005

004

003

002

001

00000 05 10 15 20 25 30

M = 0 05 10 15 20

Cu-EG

120578

120579(120578)

Figure 7 Temperature profiles for different values of119872

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

Cu-EG

120578

120579(120578)

isin = 0 03 05 07 10

Figure 8 Temperature profiles for different values of isin

then the thermal boundary layer thickness increases with theincrease in volume of nanoparticles Also it is observed thatthe thermal conductivity is more in ethylene-glycol basednanofluid when compared to water-based nanofluid

Figures 4 and 5 represent the influence of the nanoparticlevolume fraction on the velocity and temperature It is foundthat both the nanofluid velocity and the temperature increasewith an increase in nanoparticle volume fraction Thesefigures illustrate in agreement with the physical behavior

015

010

005

0000 1 2 3 4 5

Cu-EG

120578

120579(120578)

Nr = 15 2 3 4 5

Figure 9 Temperature profiles for different values of Nr

10

08

06

04

02

00

f998400 (120578)

120575 = 0 01 03 05 10

Cu-EG

0 1 2 3 4 5

120578

Figure 10 Velocity profiles for different values of 120575

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120575 = 0 01 03 05 10

Cu-EG

120578

120579(120578)

Figure 11 Temperature profiles for different values of 120575

that the thermal conductivity increases and then the thermalboundary layer thickness increases with the increase involume of nanoparticles

Effect of the magnetic parameter 119872 on the velocity andtemperature profiles is shown in Figures 6 and 7 respectivelyIt is observed that the velocity decreases as the magneticparameter increases (Figure 6) The presence of magneticfield normal to the flow in an electrically conducting fluidgives rise to a resistive type of force called Lorentz force

International Journal of Engineering Mathematics 7

S = minus04 minus02 00 02 04

10

08

06

04

02

00

f998400 (120578)

Cu-EG

0 1 2 3 4 5

120578

Figure 12 Velocity profiles for different values of 119878

020

015

010

005

000

120579(120578)

S = minus04 minus02 00 02 04

Cu-EG

0 1 2 3 4 5

120578

Figure 13 Temperature profiles for different values of 119878

04

03

02

01

0000 05 10 15 20 25 30

120578

120579(120578)

Γ = 01 03 05 07 10

Cu-EG

Figure 14 Temperature profiles for different values of Γ

which retards the fluid flow This resistive force controls thenanofluids velocity which is useful in numerous applica-tions such as magnetohydrodynamic power generation andelectromagnetic coating of wires and metal Since the mag-netic parameter is inversely proportional to density so forascending119872 120588 descends thus increasing the temperature ofthe fluid (Figure 7)

Figure 8 illustrates the temperature profiles for variousvalues of thermal conductivity parameter isin It is clear from

our definition that 119896lowast gt 119896 that is the thermal conductivityis higher when isin gt 0 and hence an increase in isin resultsin increase of thermal conductivity thereby raising thetemperature

The influence of effect of radiation parameter Nr on tem-perature is portrayed in Figure 9 The effect of Nr is promi-nently seen throughout the boundary layer It is interestingto observe that the effect of Nr is to increase the temperaturedistribution The radiation parameter (reciprocal of Stephannumber) is the measure of relative importance of thermalradiation transfer to the conduction heat transferThus largervalues of 119873 sound dominance of thermal radiation overconduction Consequently larger values of119873 are indicative oflarger amount of radiative heat energy being poured into thesystem causing rise in 120579(120578)

Figures 10 and 11 demonstrate the variation of the stream-wise velocity and temperature for different values of slipparameter 120575 As the slip parameter increases the slip at thesurface wall increases and a smaller amount of penetrationoccurs due to the stretching surface into the fluid In the noslip condition 120575 approaches zero then 119891

1015840(0) = 1 It is clear

from the figures that the velocity component at the wallreduces with an increase in the slip parameter 120575 and decreasesasymptotically to zero at the edge of the hydrodynamicboundary layer Thus hydrodynamic boundary layer thick-ness decreases as the slip parameter 120575 increases for both theregular and nanofluids and as a result the local velocity alsodecreases This is in agreement with the fact that higher 120575implies an increase in the lubrication and slipperiness of thesurface On the other hand it is evident from Figure 11 thatthe surface slipperiness affects the temperature of the fluidinversely that is an increase with slip parameter tends toincrease temperature of the fluid By escalating 120575 thermalboundary layer thickness enhances So we can interpret thatthe rate of heat transfer decreases with the increase in slipparameter 120575

The effect of suction or injection parameter 119878 on thevelocity and temperature is shown in Figures 12 and 13respectively It is observed that the velocity decreases withincrease in the suctioninjection parameter The physicalexplanation for such behavior is as follows in case of suctionthe heated fluid is pushed towards the wall where thebuoyancy forces can act to retard the fluid due to high influ-ence of the viscosity (Figure 12) The thermal boundary layerthickness for the injection is more pronounced than forsuction and it is quite superior for Cu-EG than for pure water(regular fluid 120601 = 0) Figure 13 explains that for both suc-tioninjection the existence of Cu nanoparticle increases thethermal conductivity rapidly which leads to an increase inthe thickness of the thermal boundary layer Thus the temp-erature of the fluid rises

Figure 14 depicts the temperature profiles for differentvalues of convective parameter Γ It is clearly seen that theincreases in the convective parameter increase the temper-ature profile and thereby reduce the thermal boundary layerthicknessThe variations ofminus11989110158401015840(0) andminus1205791015840(0) which are pro-portional to the local skin-friction coefficient and rate of heattransfers are shown in Tables 2ndash7 From Table 2 it is foundthat both the wall skin friction coefficient and heat transfer

8 International Journal of Engineering Mathematics

Table 2 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601

120601 minus11989110158401015840(0) minus120579

1015840(0)

00 1177071 00955225005 1093940 00951172010 1013062 00947067015 0934425 0094288102 0858128 00938579

Table 3 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and isin

120601 isin minus11989110158401015840(0) minus120579

1015840(0)

00

00 117702 0095551803 117691 0095243905 117684 0095019107 117675 0094777210 11766 00943792

01

00 101307 0094543503 101298 0094201105 101292 0093953307 101285 0093688610 101273 00932570

Table 4 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Nr

120601 Nr minus11989110158401015840(0) minus120579

1015840(0)

00

15 117585 0093119420 117523 0092049930 117396 0090089240 117269 0088329050 117147 00867510

01

10 101157 0092119620 101090 0090941230 100953 0088809740 100821 0086936350 100698 00852933

rate at the surface decrease as 120601 increases From Table 3 it isobserved that the skin friction coefficient at thewall decreaseswith an increase in the parameter isin for both regular fluid andnanofluid Also it is seen that the rate of heat transferdecreases with an increase in the value of isinThis happens dueto the fact that with the increase in 120576 the thermal conductivityincreases which lowers the temperature gradient at the platesurface FromTable 4 it is clear that both the skin friction andheat transfer rate decrease with an increase in the radiationparameter Nr

Table 5 shows the effects of wall skin friction and rate ofheat transfer with suctioninjection parameter 119878 for regularfluid and nanofluid It is evident that increasing suction (119878 gt

0) at thewall tends to have high heat transfer rate but injection(119878 lt 0) decreases heat transfer rate It is also noted that theaddition of nanoparticle produces a remarkable enhancementon heat transfer with respect to that of pure fluid

Table 5 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 119878

120601 119878 minus11989110158401015840(0) minus120579

1015840(0)

00

03 117701 0095522502 113491 0094801801 109395 0093864700 105412 0092619minus01 101538 00909242minus02 097764 00885645minus03 0940748 00852126

01

03 126325 0094510702 120846 0093699401 11555 0092669700 110444 00913403minus01 105532 00895938minus02 100811 00872611minus03 0962756 00841031

Table 6 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 120575

120601 120575 minus11989110158401015840(0) minus120579

1015840(0)

00

00 138291 0095679603 0917088 0095275805 0756782 0095083907 0646713 0094926410 0532992 00947331

01

00 151052 0094723603 096444 0094183705 0786612 0093933607 066708 0093730210 0545629 00934822

Table 7 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Γ

120601 Γ minus11989110158401015840(0) minus120579

1015840(0)

00

01 117701 0095522503 117516 026262605 117355 040335907 117215 052303510 117036 067177

01

01 126325 0094510703 126147 025508105 125999 038578707 125873 049383710 125719 0624363

FromTable 6 we notice that the skin friction coefficient atthe wall decreases with an increase in the slip parameter 120575 forboth regular fluid and nanofluidThis is less pronouncedwithan increase in the value of 120575 That is as expected for the fluidflows at nanoscales the shear stress at the wall decreases withan increase in the slip parameter 120575 It is noticed that in the noslip condition problem the highest wall shear stress occursTable 7 shows the effects of surface skin friction and rate of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Engineering Mathematics

nanofluids is thermal conductivity enhancement a phe-nomenon observed by Masuda et al [5] This phenomenonsuggests the possibility of using nanofluids in advancednuclear systems (Buongiorno and Hu [6]) A benchmarkstudy on the thermal conductivity of nanofluids was made byBuongiorno et al [7] Das et al [8] studied a two- to fourfoldincrease in thermal conductivity enhancement for water-based nanofluids containing Al

2O3or CuO nanoparticles

over a small temperature range 21∘Cndash51∘CThe study of boundary layer flow and heat transfer over

a stretching surface has attracted considerable attention inmany fields of science and technology Few of these appli-cations such as wire and fiber coating materials manufac-tured polymer extrusion food stuff processing drawing ofcopper wires and chemical processing equipment Pioneer-ing work on the dynamics of boundary layer flow overstretching surface was done by Crane [9] who examined thetwo-dimensional Navier-Stokes equations Later on variousaspects of the problem have been investigated by Dutta et al[10] and Chen and Char [11] Khan and Pop [12] presentedthe boundary layer flow of nanofluid past a stretching sheetRecently Hassani et al [13] studied an analytical solutionfor boundary layer flow of a nanofluid past a stretchingsheet Rana and Bhargava [14] analyzed the flow and heattransfer over a nonlinear stretching sheet a numerical studyHamad and Ferdows [15] studied the similarity solution ofboundary layer stagnation-point flow towards a heatedporous stretching sheet saturated with a nanofluid with heatabsorptiongeneration and suctionblowing a Lie groupanalysis

Magnetohydrodynamics (MHD) boundary layer flowover a stretching sheet is important during the last fewdecades due to its numerous applications in industrial man-ufacturing processes such as the aerodynamic extrusion ofplastic sheets liquid film hot rolling wire drawing andglass-fiber and paper production Al-Odat et al [16] analyzedthe thermal boundary layer on an exponentially stretchingcontinuous surface in the presence of magnetic field effectChamkha and Aly [17] have presented theMHD free convec-tion flow of a nanofluid past a vertical plate in the presence ofheat generation or absorption effects Later Aliakbar et al [18]analyzed the influence of thermal radiation on MHD flow ofMaxwellian fluids above stretching sheets Khan et al [19]studied the effects of magnetic field on radiative flow of ananofluid past a stretching sheet Ibrahim and Shankar [20]analyzed MHD boundary layer flow and heat transfer of ananofluid past a permeable stretching sheet with slip condi-tions

However radiation heat transfer has a key impact in hightemperature regime Many technological processes occur athigh temperature and good working knowledge of radiativeheat transfer plays an instrumental role in designing the per-tinent equipment In many practical applications dependingon the surfaces properties and solid geometry the radiativetransport is often comparable with that of convective heattransfer But unfortunately little is known about the effects ofradiation on the boundary layer flow of a radiating fluidIn fact a few difficulties arise in studying radiative fluid flowFirstly when radiative heat transfer takes place the radiation

is absorbedemitted not only at system boundaries but also inthe interior of the system hence prediction of fluid absorp-tion is a difficult task Secondly the absorption coefficient ofthe absorbing emitting fluids is in general strongly depen-dent onwavelengthThirdly presence of radiation term in theenergy equationmakes the equation highly nonlinearThis allleads to computational difficulty Owing to these difficultiesthe effect of radiation on convective flows has been inves-tigated with reasonable simplifications A good literatureon radiative transfer can be seen in well-presented texts bySparrow andCess [21] Ozisik [22] Siegel andHowell [23] andHowell [24] Takhar et al [25] studied radiation effect on thefree convection flow of a gas past a semi-infinite plate Hos-sain and Takhar [26] investigated mixed convection along avertical plate with uniform surface temperature taking radia-tion into account

Thenonadherence of the fluid to a solid boundary knownas velocity slip is a phenomenon that has been observedunder certain circumstances [27] It is a well-known fact thata viscous fluid normally sticks to the boundary But there aremany fluids for example particulate fluids rarefied gas etcwhere theremay be a slip between the fluid and the boundary[28] The effects of slip conditions are very important intechnological applications such as in the polishing of artificialheart valves and the internal cavities [29]The foremost studytaking into account the slip boundary condition over astretching sheet was conducted by Anderson [30] A closedform solution of full Navier-Stokes equations for a hydrody-namic flowover a stretching sheetwas studied by himNext toAndersonWang [31] solved the full Navier-Stokesrsquo equationswith partial slip past a stretching sheet He continues toinvestigate stagnation slip flow and heat transfer on a movingplate [32] Fang et al [33] analyzed the slip condition of aMHD viscous flow over a stretching sheet Hayat et al [34]extended the problem of the previous researchers by incor-porating the thermal slip condition and discussed unsteadymagneto hydrodynamic flow and heat transfer over a perme-able stretching sheet with slip condition also In a similar wayAbu Bakar et al [35] studied the boundary layer flow over astretching sheet with a convective boundary condition andslip effect

But so far no attempt has been made to analyze theeffects of variable thermal conductivity and velocity slip onMHD boundary layer flow of Ethylene-Glycol based Cunanofluids over a stretching sheet with convective boundarycondition Thus the problem is investigated The surfaceexhibits convective heating boundary conditions [36ndash42]Recently Alsaedi et al [43] studied the effects of heat genera-tionabsorption on stagnation point flow of nanofluid over asurface with convective boundary conditions Masood Khanet al [44] investigated the MHD Falkner-Skan flow withmixed convection and convective boundary conditions Anefficient numerical shooting technique with a fourth-orderRunge-Kutta scheme as used to solve the normalized bound-ary layer equations and the effects of material parameters onthe flow field and heat transfer characteristics is discussed indetail

International Journal of Engineering Mathematics 3

x

y

u

g

B0

TfTinfin

uw(x)

Nanofluid

Figure 1 Physical model of the system

2 Mathematical Analysis

A two-dimensional steady laminar boundary layer flow of aviscous incompressible flow of nanofluid past a permeablestretching sheet coinciding with the plane 119910 = 0 is consid-ered The flow is confirmed to 119910 gt 0 The fluid is assumedto be gray absorbing-emitting radiation but nonscatteringmediumThe flow is generated due to stretching of the sheetcaused by the simultaneous application of two equal andopposite forces along the 119909-axis Keeping the origin fixed thesheet is then stretched with velocity 119906

119908(119909) = 119886119909 where 119886 is

constant A uniformmagnetic field is applied in the directiontransverse to the stretching surface The transverse appliedmagnetic field and magnetic Reynolds number are assumedto be very small so that the induced magnetic field isnegligible There is a constant suctioninjection velocity 119881

119908

normal to the sheet The temperature of sheet surface (to bedetermined later) is the result of a convective heating processwhich is characterized by a temperature119879

119891and a heat transfer

coefficient ℎ119891 It is to be mentioned that the hole size (in

nanoscale) of porous sheet is taken to be constantThe fluid isethylene-glycol based nanofluid containing copper (Cu)Thenanofluid is assumed incompressible and flow is assumed tobe laminar A schematic representation of the physical modeland coordinate system is depicted in Figure 1Under the usualassumptions the governing boundary layer equations aregiven by

120597119906

120597119909+120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910=

1

120588119899119891

[120583119899119891

1205972119906

1205971199102+ 119892 (120588120573)

119899119891(119879 minus 119879

infin) minus 120590119861

2

0119906]

(2)

119906120597119879

120597119909+ V

120597119879

120597119910=

1

(120588119862119901)119899119891

120597

120597119910(119896lowast

119899119891

120597119879

120597119910minus 119902119903) (3)

The boundary conditions for the velocity and temperaturefields are

119906 = 119906119908+ 119860

120597119906

120597119910 V = minusV

119908 minus119896

119891

120597119879

120597119910= ℎ119891(119879119891minus 119879infin)

at 119910 = 0

119906 997888rarr 0 119879 997888rarr 119879infin

as 119910 997888rarr infin

(4)

where 119906 and V are the velocity components in the 119909 and 119910

directions respectively119879 is the temperature of the nanofluid119879infinis the ambient fluid temperature 119892 is the acceleration due

to gravity 120590 is the electric conductivity 1198610is the uniform

magnetic field strength and 119902119903is the radiative heat flux A

is the velocity slip factor V119908is the wall mass flux with V

119908lt 0

for suction and V119908

gt 0 for injection respectively 119896119891is the

thermal conductivity of the ordinary fluid and 119879119891is the

temperature of the hot fluidIt should be mentioned that such slip conditions were

recently used in a series of papers [45ndash50] It is worthmentioning at this end that fluid flowwith slip is important inmicroelectromechanical systems (MEMS) The flow in thesesystems deviates significantly from the traditional no slipflow because of the microscale dimensions of these devicesMEMS barometers and commercially compatible flexibleprinted circuit boards are examples for stretchable MEMS[51]

The variable thermal conductivity (Arunachalam andRajappa [52] Chiam [53] and Seddeek and Salem [54]) isconsidered to vary linearly with temperature as given below

119896lowast= 119896 [1+ isin 120579] (5)

where 119896 is thermal conductivity parameterThe effective density of the nanofluid is given by

120588119899119891

= (1 minus 120601) 120588119891+ 120601120588119904 (6)

where 120601 is the solid volume fraction of nanoparticles Ther-mal diffusivity of nanofluid is

120572119899119891

=119896119899119891

(120588119862119901)119899119891

(7)

where the heat capacitance 119862119901of the nanofluid is obtained as

(120588119862119901)119899119891

= (1 minus 120601) (120588119862119901)119891+ 120601 (120588119862

119901)119904 (8)

The thermal conductivity of the nanofluid 119896119899119891

for sphericalnanoparticles can be written as (Maxwell [55])

119896119899119891

119896119891

=(119896119904+ 2119896119891) minus 2120601 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120601 (119896

119891minus 119896119904) (9)

The thermal expansion coefficient of the nanofluid can bedetermined by

(120588120573)119899119891

= (1 minus 120601) (120588120573)119891+ 120601 (120588120573)

119904 (10)

4 International Journal of Engineering Mathematics

Table 1 Thermophysical properties of fluid and nanoparticles [5556]

Physical properties Fluid phase (ethylene-glycol) CuCp (JkgK) 11144 385120588 (kgm3) 2415 8933120573 times 10

minus5 (1K) 65 167119896 (WmK) 0252 401

Also the effective dynamic viscosity of the nanofluid given byBrinkman [56] can be written as

120583119899119891

=120583119891

(1 minus 120601)25

(11)

Here the subscripts 119899119891119891 and 119904 represent the thermophysicalproperties of the nanofluids base fluid and nanosolid parti-cles respectively (Table 1)

By using the Rosseland diffusion approximation (Prasadet al [57]) the radiative heat flux is given by

119902119903= minus

4

3

120590lowast

119896lowast

1205971198794

120597119910 (12)

where 120590lowast is the Stefan-Boltzmann constant and 119896

lowast is themean absorption coefficient It should be noted that by usingthe Rosseland approximation the present analysis is limitedto optically thick fluids Taking the refractive index of the gasmedium to be constant unidirectional radiation flux 119902

119903is

considered and it is assumed that 120597119902119903120597119910 ≫ 120597119902

119903120597119909 This

model is valid for optically thick media in which thermalradiation propagates only a limited distance prior to expe-riencing scattering or absorption The local thermal radia-tion intensity is due to radiation emitting from proximatelocations in the vicinity of which emission and scattering arecomparable to the location of interest The energy transferdepends on conditions only in the area adjacent to theplate regime that is the boundary layer regime Rosselandrsquosmodel yields accurate results for intensive absorption that isoptically thick flows which are optically far from the bound-ary surface Implicit in this approximation is also the exis-tence of wavelength regions where the optical thickness mayexceed a value of five As such the Rosseland model whilelimited compared with other flux models can simulate to areasonable degree of accuracy thermal radiation in problemsranging from thermal radiation transport via gases at lowdensity to thermal radiation simulations associated withnuclear blast waves If the temperature differences within theflow are sufficiently small then (12) can be linearized byexpanding 119879

4 into the Taylor series about 119879infin which after

neglecting higher-order terms takes the form

1198794cong 41198793

infin119879 minus 3119879

4

infin (13)

In view of (12) and (13) (3) becomes

119906120597119879

120597119909+ V

120597119879

120597119910

= 120572119899119891[(1+ isin 120579 +

4

3Nr) 120597

2119879

1205971199102+

isin

(119879119891minus 119879infin)(120597119879

120597119910)

2

]

(14)

where Nr = 16120590lowast1198793

infin3119896119899119891119896lowast is the radiation parameter

To get similarity solutions of (1)ndash(3) subject to the bound-ary conditions (4) we introduce the following similaritytransformations

120578 = (119886

]119891

)

12

119910 119906 = 1198861199091198911015840(120578) V = minus (119886]

119891)12

120579 =119879 minus 119879infin

119879119891minus 119879infin

119872 =1205901198612

0

119886120588119891

120582 =

119892 (120588120573)119891(119879119891minus 119879infin)

120588119891119886119906119908

Nr =4120590lowast1198793

infin

119896lowast119896119899119891

Pr =]119891

120572119891

119878 = minusV119908

radic119886]119891

120575 = 119860radic119886

]119891

Γ =ℎ119891

119896119891

radic]119891

119886

(15)

where ]119891is the kinematic viscosity of the base fluid

In view of (15) (1) (2) and (14) take the followingdimensionless form

119891101584010158401015840(1 minus 120601)

25

[(1 minus 120601 + 120601120588119904

120588119891

)(11989111989110158401015840minus 11989110158402)

+(1 minus 120601 + 120601(120588120573)119904

(120588120573)119891

)120582120579 minus1198721198911015840] = 0

(1 +4

3Nr) 12057910158401015840+ isin 120579120579

10158401015840+ isin 12057910158402

+ Pr(119896119891

119896119899119891

)(1 minus 120601 + 120601

(120588119862119901)119904

(120588119862119901)119891

)1198911205791015840= 0

(16)

where prime denotes the differentiation with respect to 120578The corresponding boundary conditions are

1198911015840(0) = 1 + 120575119891

10158401015840(0) 119891 (0) = 119878

1205791015840(0) = minusΓ (1 minus 120579 (0))

1198911015840(infin) 997888rarr 0 120579 (infin) 997888rarr 0

(17)

International Journal of Engineering Mathematics 5

07

06

05

04

03

02

01

000 2 4 6 8 10 12

Present result

120579(120578)

120578

Abu Bakar et al [35]

Γ = 005 05 10

Figure 2 Comparison of temperature profiles for different Γ

The quantities of practical interest in this study are the skinfriction or the shear stress coefficient119862

119891and the local Nusselt

number Nu119909 which are defined as

119862119891= minus

120583119899119891

1205881198911199062119908

(120597119906

120597119910)

119910=0

Nu119909=

119909119896119899119891

119896119891(119879119908minus 119879infin)(minus

120597119879

120597119910)

119910=0

(18)

Using (6) the skin friction coefficient and local Nusseltnumber can be expressed as

Re12119909

119862119891= minus

1

(1 minus 120601)25

11989110158401015840(0)

Re12119909

Nu119909= minus

119896119899119891

119896119891

1205791015840(0)

(19)

where Re119909= 119906119908119909]119891is the Reynolds number

3 Results and Discussion

The coupled nonlinear differential equations (16) along withthe boundary conditions (17) are solved numerically usingRunge-Kutta fourth order technique along with shootingmethodThe integration length 120578

infinvaries with the parameter

values and it has been suitably chosen each time such that theboundary conditions at the outer edge of the boundary layerare satisfied In order to bring out the salient features of theflow and the heat transfer characteristics in the Cu-EG basednanofluid the numerical values for different values of thegoverning parameters 120601 119872 isin Nr 120575 119878 and Γ are computedand portrayed in Figures 2ndash13

In order to validate our analysis the present results arecompared with the existing results for the case of regular fluidand found an excellent agreement (Figure 2)

Figure 3 shows the temperature profiles for different val-ues of the nanoparticle volume fraction 120601 with water and EG

006

005

004

003

002

001

00000 05 10 15 20 25 30

120579(120578)

120578

Cu-EGCu-water

120601 = 0 005 01 015 02

Figure 3 Temperature profiles for different values of 120601

120601 = 0 005 01 015 02

Cu-EG

120578

0 1 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

Figure 4 Velocity profiles for different values of 120601

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120601 = 0 005 01 015 02

Cu-EG

120578

120579(120578)

Figure 5 Temperature profiles for different values of 120601

based nanofluids It is found that the temperature increaseswith an increase in the nanoparticle volume fraction forboth Cu-water and Cu-EG based fluid And also it is noticedthat EG based nanofluids have more impact in rising thetemperature than that of water-based nanofluids This isdue to the fact that the thermal conductivity increases and

6 International Journal of Engineering Mathematics

Cu-EG

120578

10 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

M = 0 05 10 15 20

Figure 6 Velocity profiles for different values of119872

006

005

004

003

002

001

00000 05 10 15 20 25 30

M = 0 05 10 15 20

Cu-EG

120578

120579(120578)

Figure 7 Temperature profiles for different values of119872

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

Cu-EG

120578

120579(120578)

isin = 0 03 05 07 10

Figure 8 Temperature profiles for different values of isin

then the thermal boundary layer thickness increases with theincrease in volume of nanoparticles Also it is observed thatthe thermal conductivity is more in ethylene-glycol basednanofluid when compared to water-based nanofluid

Figures 4 and 5 represent the influence of the nanoparticlevolume fraction on the velocity and temperature It is foundthat both the nanofluid velocity and the temperature increasewith an increase in nanoparticle volume fraction Thesefigures illustrate in agreement with the physical behavior

015

010

005

0000 1 2 3 4 5

Cu-EG

120578

120579(120578)

Nr = 15 2 3 4 5

Figure 9 Temperature profiles for different values of Nr

10

08

06

04

02

00

f998400 (120578)

120575 = 0 01 03 05 10

Cu-EG

0 1 2 3 4 5

120578

Figure 10 Velocity profiles for different values of 120575

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120575 = 0 01 03 05 10

Cu-EG

120578

120579(120578)

Figure 11 Temperature profiles for different values of 120575

that the thermal conductivity increases and then the thermalboundary layer thickness increases with the increase involume of nanoparticles

Effect of the magnetic parameter 119872 on the velocity andtemperature profiles is shown in Figures 6 and 7 respectivelyIt is observed that the velocity decreases as the magneticparameter increases (Figure 6) The presence of magneticfield normal to the flow in an electrically conducting fluidgives rise to a resistive type of force called Lorentz force

International Journal of Engineering Mathematics 7

S = minus04 minus02 00 02 04

10

08

06

04

02

00

f998400 (120578)

Cu-EG

0 1 2 3 4 5

120578

Figure 12 Velocity profiles for different values of 119878

020

015

010

005

000

120579(120578)

S = minus04 minus02 00 02 04

Cu-EG

0 1 2 3 4 5

120578

Figure 13 Temperature profiles for different values of 119878

04

03

02

01

0000 05 10 15 20 25 30

120578

120579(120578)

Γ = 01 03 05 07 10

Cu-EG

Figure 14 Temperature profiles for different values of Γ

which retards the fluid flow This resistive force controls thenanofluids velocity which is useful in numerous applica-tions such as magnetohydrodynamic power generation andelectromagnetic coating of wires and metal Since the mag-netic parameter is inversely proportional to density so forascending119872 120588 descends thus increasing the temperature ofthe fluid (Figure 7)

Figure 8 illustrates the temperature profiles for variousvalues of thermal conductivity parameter isin It is clear from

our definition that 119896lowast gt 119896 that is the thermal conductivityis higher when isin gt 0 and hence an increase in isin resultsin increase of thermal conductivity thereby raising thetemperature

The influence of effect of radiation parameter Nr on tem-perature is portrayed in Figure 9 The effect of Nr is promi-nently seen throughout the boundary layer It is interestingto observe that the effect of Nr is to increase the temperaturedistribution The radiation parameter (reciprocal of Stephannumber) is the measure of relative importance of thermalradiation transfer to the conduction heat transferThus largervalues of 119873 sound dominance of thermal radiation overconduction Consequently larger values of119873 are indicative oflarger amount of radiative heat energy being poured into thesystem causing rise in 120579(120578)

Figures 10 and 11 demonstrate the variation of the stream-wise velocity and temperature for different values of slipparameter 120575 As the slip parameter increases the slip at thesurface wall increases and a smaller amount of penetrationoccurs due to the stretching surface into the fluid In the noslip condition 120575 approaches zero then 119891

1015840(0) = 1 It is clear

from the figures that the velocity component at the wallreduces with an increase in the slip parameter 120575 and decreasesasymptotically to zero at the edge of the hydrodynamicboundary layer Thus hydrodynamic boundary layer thick-ness decreases as the slip parameter 120575 increases for both theregular and nanofluids and as a result the local velocity alsodecreases This is in agreement with the fact that higher 120575implies an increase in the lubrication and slipperiness of thesurface On the other hand it is evident from Figure 11 thatthe surface slipperiness affects the temperature of the fluidinversely that is an increase with slip parameter tends toincrease temperature of the fluid By escalating 120575 thermalboundary layer thickness enhances So we can interpret thatthe rate of heat transfer decreases with the increase in slipparameter 120575

The effect of suction or injection parameter 119878 on thevelocity and temperature is shown in Figures 12 and 13respectively It is observed that the velocity decreases withincrease in the suctioninjection parameter The physicalexplanation for such behavior is as follows in case of suctionthe heated fluid is pushed towards the wall where thebuoyancy forces can act to retard the fluid due to high influ-ence of the viscosity (Figure 12) The thermal boundary layerthickness for the injection is more pronounced than forsuction and it is quite superior for Cu-EG than for pure water(regular fluid 120601 = 0) Figure 13 explains that for both suc-tioninjection the existence of Cu nanoparticle increases thethermal conductivity rapidly which leads to an increase inthe thickness of the thermal boundary layer Thus the temp-erature of the fluid rises

Figure 14 depicts the temperature profiles for differentvalues of convective parameter Γ It is clearly seen that theincreases in the convective parameter increase the temper-ature profile and thereby reduce the thermal boundary layerthicknessThe variations ofminus11989110158401015840(0) andminus1205791015840(0) which are pro-portional to the local skin-friction coefficient and rate of heattransfers are shown in Tables 2ndash7 From Table 2 it is foundthat both the wall skin friction coefficient and heat transfer

8 International Journal of Engineering Mathematics

Table 2 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601

120601 minus11989110158401015840(0) minus120579

1015840(0)

00 1177071 00955225005 1093940 00951172010 1013062 00947067015 0934425 0094288102 0858128 00938579

Table 3 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and isin

120601 isin minus11989110158401015840(0) minus120579

1015840(0)

00

00 117702 0095551803 117691 0095243905 117684 0095019107 117675 0094777210 11766 00943792

01

00 101307 0094543503 101298 0094201105 101292 0093953307 101285 0093688610 101273 00932570

Table 4 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Nr

120601 Nr minus11989110158401015840(0) minus120579

1015840(0)

00

15 117585 0093119420 117523 0092049930 117396 0090089240 117269 0088329050 117147 00867510

01

10 101157 0092119620 101090 0090941230 100953 0088809740 100821 0086936350 100698 00852933

rate at the surface decrease as 120601 increases From Table 3 it isobserved that the skin friction coefficient at thewall decreaseswith an increase in the parameter isin for both regular fluid andnanofluid Also it is seen that the rate of heat transferdecreases with an increase in the value of isinThis happens dueto the fact that with the increase in 120576 the thermal conductivityincreases which lowers the temperature gradient at the platesurface FromTable 4 it is clear that both the skin friction andheat transfer rate decrease with an increase in the radiationparameter Nr

Table 5 shows the effects of wall skin friction and rate ofheat transfer with suctioninjection parameter 119878 for regularfluid and nanofluid It is evident that increasing suction (119878 gt

0) at thewall tends to have high heat transfer rate but injection(119878 lt 0) decreases heat transfer rate It is also noted that theaddition of nanoparticle produces a remarkable enhancementon heat transfer with respect to that of pure fluid

Table 5 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 119878

120601 119878 minus11989110158401015840(0) minus120579

1015840(0)

00

03 117701 0095522502 113491 0094801801 109395 0093864700 105412 0092619minus01 101538 00909242minus02 097764 00885645minus03 0940748 00852126

01

03 126325 0094510702 120846 0093699401 11555 0092669700 110444 00913403minus01 105532 00895938minus02 100811 00872611minus03 0962756 00841031

Table 6 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 120575

120601 120575 minus11989110158401015840(0) minus120579

1015840(0)

00

00 138291 0095679603 0917088 0095275805 0756782 0095083907 0646713 0094926410 0532992 00947331

01

00 151052 0094723603 096444 0094183705 0786612 0093933607 066708 0093730210 0545629 00934822

Table 7 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Γ

120601 Γ minus11989110158401015840(0) minus120579

1015840(0)

00

01 117701 0095522503 117516 026262605 117355 040335907 117215 052303510 117036 067177

01

01 126325 0094510703 126147 025508105 125999 038578707 125873 049383710 125719 0624363

FromTable 6 we notice that the skin friction coefficient atthe wall decreases with an increase in the slip parameter 120575 forboth regular fluid and nanofluidThis is less pronouncedwithan increase in the value of 120575 That is as expected for the fluidflows at nanoscales the shear stress at the wall decreases withan increase in the slip parameter 120575 It is noticed that in the noslip condition problem the highest wall shear stress occursTable 7 shows the effects of surface skin friction and rate of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 3

x

y

u

g

B0

TfTinfin

uw(x)

Nanofluid

Figure 1 Physical model of the system

2 Mathematical Analysis

A two-dimensional steady laminar boundary layer flow of aviscous incompressible flow of nanofluid past a permeablestretching sheet coinciding with the plane 119910 = 0 is consid-ered The flow is confirmed to 119910 gt 0 The fluid is assumedto be gray absorbing-emitting radiation but nonscatteringmediumThe flow is generated due to stretching of the sheetcaused by the simultaneous application of two equal andopposite forces along the 119909-axis Keeping the origin fixed thesheet is then stretched with velocity 119906

119908(119909) = 119886119909 where 119886 is

constant A uniformmagnetic field is applied in the directiontransverse to the stretching surface The transverse appliedmagnetic field and magnetic Reynolds number are assumedto be very small so that the induced magnetic field isnegligible There is a constant suctioninjection velocity 119881

119908

normal to the sheet The temperature of sheet surface (to bedetermined later) is the result of a convective heating processwhich is characterized by a temperature119879

119891and a heat transfer

coefficient ℎ119891 It is to be mentioned that the hole size (in

nanoscale) of porous sheet is taken to be constantThe fluid isethylene-glycol based nanofluid containing copper (Cu)Thenanofluid is assumed incompressible and flow is assumed tobe laminar A schematic representation of the physical modeland coordinate system is depicted in Figure 1Under the usualassumptions the governing boundary layer equations aregiven by

120597119906

120597119909+120597V120597119910

= 0 (1)

119906120597119906

120597119909+ V

120597119906

120597119910=

1

120588119899119891

[120583119899119891

1205972119906

1205971199102+ 119892 (120588120573)

119899119891(119879 minus 119879

infin) minus 120590119861

2

0119906]

(2)

119906120597119879

120597119909+ V

120597119879

120597119910=

1

(120588119862119901)119899119891

120597

120597119910(119896lowast

119899119891

120597119879

120597119910minus 119902119903) (3)

The boundary conditions for the velocity and temperaturefields are

119906 = 119906119908+ 119860

120597119906

120597119910 V = minusV

119908 minus119896

119891

120597119879

120597119910= ℎ119891(119879119891minus 119879infin)

at 119910 = 0

119906 997888rarr 0 119879 997888rarr 119879infin

as 119910 997888rarr infin

(4)

where 119906 and V are the velocity components in the 119909 and 119910

directions respectively119879 is the temperature of the nanofluid119879infinis the ambient fluid temperature 119892 is the acceleration due

to gravity 120590 is the electric conductivity 1198610is the uniform

magnetic field strength and 119902119903is the radiative heat flux A

is the velocity slip factor V119908is the wall mass flux with V

119908lt 0

for suction and V119908

gt 0 for injection respectively 119896119891is the

thermal conductivity of the ordinary fluid and 119879119891is the

temperature of the hot fluidIt should be mentioned that such slip conditions were

recently used in a series of papers [45ndash50] It is worthmentioning at this end that fluid flowwith slip is important inmicroelectromechanical systems (MEMS) The flow in thesesystems deviates significantly from the traditional no slipflow because of the microscale dimensions of these devicesMEMS barometers and commercially compatible flexibleprinted circuit boards are examples for stretchable MEMS[51]

The variable thermal conductivity (Arunachalam andRajappa [52] Chiam [53] and Seddeek and Salem [54]) isconsidered to vary linearly with temperature as given below

119896lowast= 119896 [1+ isin 120579] (5)

where 119896 is thermal conductivity parameterThe effective density of the nanofluid is given by

120588119899119891

= (1 minus 120601) 120588119891+ 120601120588119904 (6)

where 120601 is the solid volume fraction of nanoparticles Ther-mal diffusivity of nanofluid is

120572119899119891

=119896119899119891

(120588119862119901)119899119891

(7)

where the heat capacitance 119862119901of the nanofluid is obtained as

(120588119862119901)119899119891

= (1 minus 120601) (120588119862119901)119891+ 120601 (120588119862

119901)119904 (8)

The thermal conductivity of the nanofluid 119896119899119891

for sphericalnanoparticles can be written as (Maxwell [55])

119896119899119891

119896119891

=(119896119904+ 2119896119891) minus 2120601 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120601 (119896

119891minus 119896119904) (9)

The thermal expansion coefficient of the nanofluid can bedetermined by

(120588120573)119899119891

= (1 minus 120601) (120588120573)119891+ 120601 (120588120573)

119904 (10)

4 International Journal of Engineering Mathematics

Table 1 Thermophysical properties of fluid and nanoparticles [5556]

Physical properties Fluid phase (ethylene-glycol) CuCp (JkgK) 11144 385120588 (kgm3) 2415 8933120573 times 10

minus5 (1K) 65 167119896 (WmK) 0252 401

Also the effective dynamic viscosity of the nanofluid given byBrinkman [56] can be written as

120583119899119891

=120583119891

(1 minus 120601)25

(11)

Here the subscripts 119899119891119891 and 119904 represent the thermophysicalproperties of the nanofluids base fluid and nanosolid parti-cles respectively (Table 1)

By using the Rosseland diffusion approximation (Prasadet al [57]) the radiative heat flux is given by

119902119903= minus

4

3

120590lowast

119896lowast

1205971198794

120597119910 (12)

where 120590lowast is the Stefan-Boltzmann constant and 119896

lowast is themean absorption coefficient It should be noted that by usingthe Rosseland approximation the present analysis is limitedto optically thick fluids Taking the refractive index of the gasmedium to be constant unidirectional radiation flux 119902

119903is

considered and it is assumed that 120597119902119903120597119910 ≫ 120597119902

119903120597119909 This

model is valid for optically thick media in which thermalradiation propagates only a limited distance prior to expe-riencing scattering or absorption The local thermal radia-tion intensity is due to radiation emitting from proximatelocations in the vicinity of which emission and scattering arecomparable to the location of interest The energy transferdepends on conditions only in the area adjacent to theplate regime that is the boundary layer regime Rosselandrsquosmodel yields accurate results for intensive absorption that isoptically thick flows which are optically far from the bound-ary surface Implicit in this approximation is also the exis-tence of wavelength regions where the optical thickness mayexceed a value of five As such the Rosseland model whilelimited compared with other flux models can simulate to areasonable degree of accuracy thermal radiation in problemsranging from thermal radiation transport via gases at lowdensity to thermal radiation simulations associated withnuclear blast waves If the temperature differences within theflow are sufficiently small then (12) can be linearized byexpanding 119879

4 into the Taylor series about 119879infin which after

neglecting higher-order terms takes the form

1198794cong 41198793

infin119879 minus 3119879

4

infin (13)

In view of (12) and (13) (3) becomes

119906120597119879

120597119909+ V

120597119879

120597119910

= 120572119899119891[(1+ isin 120579 +

4

3Nr) 120597

2119879

1205971199102+

isin

(119879119891minus 119879infin)(120597119879

120597119910)

2

]

(14)

where Nr = 16120590lowast1198793

infin3119896119899119891119896lowast is the radiation parameter

To get similarity solutions of (1)ndash(3) subject to the bound-ary conditions (4) we introduce the following similaritytransformations

120578 = (119886

]119891

)

12

119910 119906 = 1198861199091198911015840(120578) V = minus (119886]

119891)12

120579 =119879 minus 119879infin

119879119891minus 119879infin

119872 =1205901198612

0

119886120588119891

120582 =

119892 (120588120573)119891(119879119891minus 119879infin)

120588119891119886119906119908

Nr =4120590lowast1198793

infin

119896lowast119896119899119891

Pr =]119891

120572119891

119878 = minusV119908

radic119886]119891

120575 = 119860radic119886

]119891

Γ =ℎ119891

119896119891

radic]119891

119886

(15)

where ]119891is the kinematic viscosity of the base fluid

In view of (15) (1) (2) and (14) take the followingdimensionless form

119891101584010158401015840(1 minus 120601)

25

[(1 minus 120601 + 120601120588119904

120588119891

)(11989111989110158401015840minus 11989110158402)

+(1 minus 120601 + 120601(120588120573)119904

(120588120573)119891

)120582120579 minus1198721198911015840] = 0

(1 +4

3Nr) 12057910158401015840+ isin 120579120579

10158401015840+ isin 12057910158402

+ Pr(119896119891

119896119899119891

)(1 minus 120601 + 120601

(120588119862119901)119904

(120588119862119901)119891

)1198911205791015840= 0

(16)

where prime denotes the differentiation with respect to 120578The corresponding boundary conditions are

1198911015840(0) = 1 + 120575119891

10158401015840(0) 119891 (0) = 119878

1205791015840(0) = minusΓ (1 minus 120579 (0))

1198911015840(infin) 997888rarr 0 120579 (infin) 997888rarr 0

(17)

International Journal of Engineering Mathematics 5

07

06

05

04

03

02

01

000 2 4 6 8 10 12

Present result

120579(120578)

120578

Abu Bakar et al [35]

Γ = 005 05 10

Figure 2 Comparison of temperature profiles for different Γ

The quantities of practical interest in this study are the skinfriction or the shear stress coefficient119862

119891and the local Nusselt

number Nu119909 which are defined as

119862119891= minus

120583119899119891

1205881198911199062119908

(120597119906

120597119910)

119910=0

Nu119909=

119909119896119899119891

119896119891(119879119908minus 119879infin)(minus

120597119879

120597119910)

119910=0

(18)

Using (6) the skin friction coefficient and local Nusseltnumber can be expressed as

Re12119909

119862119891= minus

1

(1 minus 120601)25

11989110158401015840(0)

Re12119909

Nu119909= minus

119896119899119891

119896119891

1205791015840(0)

(19)

where Re119909= 119906119908119909]119891is the Reynolds number

3 Results and Discussion

The coupled nonlinear differential equations (16) along withthe boundary conditions (17) are solved numerically usingRunge-Kutta fourth order technique along with shootingmethodThe integration length 120578

infinvaries with the parameter

values and it has been suitably chosen each time such that theboundary conditions at the outer edge of the boundary layerare satisfied In order to bring out the salient features of theflow and the heat transfer characteristics in the Cu-EG basednanofluid the numerical values for different values of thegoverning parameters 120601 119872 isin Nr 120575 119878 and Γ are computedand portrayed in Figures 2ndash13

In order to validate our analysis the present results arecompared with the existing results for the case of regular fluidand found an excellent agreement (Figure 2)

Figure 3 shows the temperature profiles for different val-ues of the nanoparticle volume fraction 120601 with water and EG

006

005

004

003

002

001

00000 05 10 15 20 25 30

120579(120578)

120578

Cu-EGCu-water

120601 = 0 005 01 015 02

Figure 3 Temperature profiles for different values of 120601

120601 = 0 005 01 015 02

Cu-EG

120578

0 1 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

Figure 4 Velocity profiles for different values of 120601

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120601 = 0 005 01 015 02

Cu-EG

120578

120579(120578)

Figure 5 Temperature profiles for different values of 120601

based nanofluids It is found that the temperature increaseswith an increase in the nanoparticle volume fraction forboth Cu-water and Cu-EG based fluid And also it is noticedthat EG based nanofluids have more impact in rising thetemperature than that of water-based nanofluids This isdue to the fact that the thermal conductivity increases and

6 International Journal of Engineering Mathematics

Cu-EG

120578

10 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

M = 0 05 10 15 20

Figure 6 Velocity profiles for different values of119872

006

005

004

003

002

001

00000 05 10 15 20 25 30

M = 0 05 10 15 20

Cu-EG

120578

120579(120578)

Figure 7 Temperature profiles for different values of119872

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

Cu-EG

120578

120579(120578)

isin = 0 03 05 07 10

Figure 8 Temperature profiles for different values of isin

then the thermal boundary layer thickness increases with theincrease in volume of nanoparticles Also it is observed thatthe thermal conductivity is more in ethylene-glycol basednanofluid when compared to water-based nanofluid

Figures 4 and 5 represent the influence of the nanoparticlevolume fraction on the velocity and temperature It is foundthat both the nanofluid velocity and the temperature increasewith an increase in nanoparticle volume fraction Thesefigures illustrate in agreement with the physical behavior

015

010

005

0000 1 2 3 4 5

Cu-EG

120578

120579(120578)

Nr = 15 2 3 4 5

Figure 9 Temperature profiles for different values of Nr

10

08

06

04

02

00

f998400 (120578)

120575 = 0 01 03 05 10

Cu-EG

0 1 2 3 4 5

120578

Figure 10 Velocity profiles for different values of 120575

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120575 = 0 01 03 05 10

Cu-EG

120578

120579(120578)

Figure 11 Temperature profiles for different values of 120575

that the thermal conductivity increases and then the thermalboundary layer thickness increases with the increase involume of nanoparticles

Effect of the magnetic parameter 119872 on the velocity andtemperature profiles is shown in Figures 6 and 7 respectivelyIt is observed that the velocity decreases as the magneticparameter increases (Figure 6) The presence of magneticfield normal to the flow in an electrically conducting fluidgives rise to a resistive type of force called Lorentz force

International Journal of Engineering Mathematics 7

S = minus04 minus02 00 02 04

10

08

06

04

02

00

f998400 (120578)

Cu-EG

0 1 2 3 4 5

120578

Figure 12 Velocity profiles for different values of 119878

020

015

010

005

000

120579(120578)

S = minus04 minus02 00 02 04

Cu-EG

0 1 2 3 4 5

120578

Figure 13 Temperature profiles for different values of 119878

04

03

02

01

0000 05 10 15 20 25 30

120578

120579(120578)

Γ = 01 03 05 07 10

Cu-EG

Figure 14 Temperature profiles for different values of Γ

which retards the fluid flow This resistive force controls thenanofluids velocity which is useful in numerous applica-tions such as magnetohydrodynamic power generation andelectromagnetic coating of wires and metal Since the mag-netic parameter is inversely proportional to density so forascending119872 120588 descends thus increasing the temperature ofthe fluid (Figure 7)

Figure 8 illustrates the temperature profiles for variousvalues of thermal conductivity parameter isin It is clear from

our definition that 119896lowast gt 119896 that is the thermal conductivityis higher when isin gt 0 and hence an increase in isin resultsin increase of thermal conductivity thereby raising thetemperature

The influence of effect of radiation parameter Nr on tem-perature is portrayed in Figure 9 The effect of Nr is promi-nently seen throughout the boundary layer It is interestingto observe that the effect of Nr is to increase the temperaturedistribution The radiation parameter (reciprocal of Stephannumber) is the measure of relative importance of thermalradiation transfer to the conduction heat transferThus largervalues of 119873 sound dominance of thermal radiation overconduction Consequently larger values of119873 are indicative oflarger amount of radiative heat energy being poured into thesystem causing rise in 120579(120578)

Figures 10 and 11 demonstrate the variation of the stream-wise velocity and temperature for different values of slipparameter 120575 As the slip parameter increases the slip at thesurface wall increases and a smaller amount of penetrationoccurs due to the stretching surface into the fluid In the noslip condition 120575 approaches zero then 119891

1015840(0) = 1 It is clear

from the figures that the velocity component at the wallreduces with an increase in the slip parameter 120575 and decreasesasymptotically to zero at the edge of the hydrodynamicboundary layer Thus hydrodynamic boundary layer thick-ness decreases as the slip parameter 120575 increases for both theregular and nanofluids and as a result the local velocity alsodecreases This is in agreement with the fact that higher 120575implies an increase in the lubrication and slipperiness of thesurface On the other hand it is evident from Figure 11 thatthe surface slipperiness affects the temperature of the fluidinversely that is an increase with slip parameter tends toincrease temperature of the fluid By escalating 120575 thermalboundary layer thickness enhances So we can interpret thatthe rate of heat transfer decreases with the increase in slipparameter 120575

The effect of suction or injection parameter 119878 on thevelocity and temperature is shown in Figures 12 and 13respectively It is observed that the velocity decreases withincrease in the suctioninjection parameter The physicalexplanation for such behavior is as follows in case of suctionthe heated fluid is pushed towards the wall where thebuoyancy forces can act to retard the fluid due to high influ-ence of the viscosity (Figure 12) The thermal boundary layerthickness for the injection is more pronounced than forsuction and it is quite superior for Cu-EG than for pure water(regular fluid 120601 = 0) Figure 13 explains that for both suc-tioninjection the existence of Cu nanoparticle increases thethermal conductivity rapidly which leads to an increase inthe thickness of the thermal boundary layer Thus the temp-erature of the fluid rises

Figure 14 depicts the temperature profiles for differentvalues of convective parameter Γ It is clearly seen that theincreases in the convective parameter increase the temper-ature profile and thereby reduce the thermal boundary layerthicknessThe variations ofminus11989110158401015840(0) andminus1205791015840(0) which are pro-portional to the local skin-friction coefficient and rate of heattransfers are shown in Tables 2ndash7 From Table 2 it is foundthat both the wall skin friction coefficient and heat transfer

8 International Journal of Engineering Mathematics

Table 2 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601

120601 minus11989110158401015840(0) minus120579

1015840(0)

00 1177071 00955225005 1093940 00951172010 1013062 00947067015 0934425 0094288102 0858128 00938579

Table 3 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and isin

120601 isin minus11989110158401015840(0) minus120579

1015840(0)

00

00 117702 0095551803 117691 0095243905 117684 0095019107 117675 0094777210 11766 00943792

01

00 101307 0094543503 101298 0094201105 101292 0093953307 101285 0093688610 101273 00932570

Table 4 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Nr

120601 Nr minus11989110158401015840(0) minus120579

1015840(0)

00

15 117585 0093119420 117523 0092049930 117396 0090089240 117269 0088329050 117147 00867510

01

10 101157 0092119620 101090 0090941230 100953 0088809740 100821 0086936350 100698 00852933

rate at the surface decrease as 120601 increases From Table 3 it isobserved that the skin friction coefficient at thewall decreaseswith an increase in the parameter isin for both regular fluid andnanofluid Also it is seen that the rate of heat transferdecreases with an increase in the value of isinThis happens dueto the fact that with the increase in 120576 the thermal conductivityincreases which lowers the temperature gradient at the platesurface FromTable 4 it is clear that both the skin friction andheat transfer rate decrease with an increase in the radiationparameter Nr

Table 5 shows the effects of wall skin friction and rate ofheat transfer with suctioninjection parameter 119878 for regularfluid and nanofluid It is evident that increasing suction (119878 gt

0) at thewall tends to have high heat transfer rate but injection(119878 lt 0) decreases heat transfer rate It is also noted that theaddition of nanoparticle produces a remarkable enhancementon heat transfer with respect to that of pure fluid

Table 5 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 119878

120601 119878 minus11989110158401015840(0) minus120579

1015840(0)

00

03 117701 0095522502 113491 0094801801 109395 0093864700 105412 0092619minus01 101538 00909242minus02 097764 00885645minus03 0940748 00852126

01

03 126325 0094510702 120846 0093699401 11555 0092669700 110444 00913403minus01 105532 00895938minus02 100811 00872611minus03 0962756 00841031

Table 6 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 120575

120601 120575 minus11989110158401015840(0) minus120579

1015840(0)

00

00 138291 0095679603 0917088 0095275805 0756782 0095083907 0646713 0094926410 0532992 00947331

01

00 151052 0094723603 096444 0094183705 0786612 0093933607 066708 0093730210 0545629 00934822

Table 7 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Γ

120601 Γ minus11989110158401015840(0) minus120579

1015840(0)

00

01 117701 0095522503 117516 026262605 117355 040335907 117215 052303510 117036 067177

01

01 126325 0094510703 126147 025508105 125999 038578707 125873 049383710 125719 0624363

FromTable 6 we notice that the skin friction coefficient atthe wall decreases with an increase in the slip parameter 120575 forboth regular fluid and nanofluidThis is less pronouncedwithan increase in the value of 120575 That is as expected for the fluidflows at nanoscales the shear stress at the wall decreases withan increase in the slip parameter 120575 It is noticed that in the noslip condition problem the highest wall shear stress occursTable 7 shows the effects of surface skin friction and rate of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

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Mathematical Problems in Engineering

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Engineering Mathematics

Table 1 Thermophysical properties of fluid and nanoparticles [5556]

Physical properties Fluid phase (ethylene-glycol) CuCp (JkgK) 11144 385120588 (kgm3) 2415 8933120573 times 10

minus5 (1K) 65 167119896 (WmK) 0252 401

Also the effective dynamic viscosity of the nanofluid given byBrinkman [56] can be written as

120583119899119891

=120583119891

(1 minus 120601)25

(11)

Here the subscripts 119899119891119891 and 119904 represent the thermophysicalproperties of the nanofluids base fluid and nanosolid parti-cles respectively (Table 1)

By using the Rosseland diffusion approximation (Prasadet al [57]) the radiative heat flux is given by

119902119903= minus

4

3

120590lowast

119896lowast

1205971198794

120597119910 (12)

where 120590lowast is the Stefan-Boltzmann constant and 119896

lowast is themean absorption coefficient It should be noted that by usingthe Rosseland approximation the present analysis is limitedto optically thick fluids Taking the refractive index of the gasmedium to be constant unidirectional radiation flux 119902

119903is

considered and it is assumed that 120597119902119903120597119910 ≫ 120597119902

119903120597119909 This

model is valid for optically thick media in which thermalradiation propagates only a limited distance prior to expe-riencing scattering or absorption The local thermal radia-tion intensity is due to radiation emitting from proximatelocations in the vicinity of which emission and scattering arecomparable to the location of interest The energy transferdepends on conditions only in the area adjacent to theplate regime that is the boundary layer regime Rosselandrsquosmodel yields accurate results for intensive absorption that isoptically thick flows which are optically far from the bound-ary surface Implicit in this approximation is also the exis-tence of wavelength regions where the optical thickness mayexceed a value of five As such the Rosseland model whilelimited compared with other flux models can simulate to areasonable degree of accuracy thermal radiation in problemsranging from thermal radiation transport via gases at lowdensity to thermal radiation simulations associated withnuclear blast waves If the temperature differences within theflow are sufficiently small then (12) can be linearized byexpanding 119879

4 into the Taylor series about 119879infin which after

neglecting higher-order terms takes the form

1198794cong 41198793

infin119879 minus 3119879

4

infin (13)

In view of (12) and (13) (3) becomes

119906120597119879

120597119909+ V

120597119879

120597119910

= 120572119899119891[(1+ isin 120579 +

4

3Nr) 120597

2119879

1205971199102+

isin

(119879119891minus 119879infin)(120597119879

120597119910)

2

]

(14)

where Nr = 16120590lowast1198793

infin3119896119899119891119896lowast is the radiation parameter

To get similarity solutions of (1)ndash(3) subject to the bound-ary conditions (4) we introduce the following similaritytransformations

120578 = (119886

]119891

)

12

119910 119906 = 1198861199091198911015840(120578) V = minus (119886]

119891)12

120579 =119879 minus 119879infin

119879119891minus 119879infin

119872 =1205901198612

0

119886120588119891

120582 =

119892 (120588120573)119891(119879119891minus 119879infin)

120588119891119886119906119908

Nr =4120590lowast1198793

infin

119896lowast119896119899119891

Pr =]119891

120572119891

119878 = minusV119908

radic119886]119891

120575 = 119860radic119886

]119891

Γ =ℎ119891

119896119891

radic]119891

119886

(15)

where ]119891is the kinematic viscosity of the base fluid

In view of (15) (1) (2) and (14) take the followingdimensionless form

119891101584010158401015840(1 minus 120601)

25

[(1 minus 120601 + 120601120588119904

120588119891

)(11989111989110158401015840minus 11989110158402)

+(1 minus 120601 + 120601(120588120573)119904

(120588120573)119891

)120582120579 minus1198721198911015840] = 0

(1 +4

3Nr) 12057910158401015840+ isin 120579120579

10158401015840+ isin 12057910158402

+ Pr(119896119891

119896119899119891

)(1 minus 120601 + 120601

(120588119862119901)119904

(120588119862119901)119891

)1198911205791015840= 0

(16)

where prime denotes the differentiation with respect to 120578The corresponding boundary conditions are

1198911015840(0) = 1 + 120575119891

10158401015840(0) 119891 (0) = 119878

1205791015840(0) = minusΓ (1 minus 120579 (0))

1198911015840(infin) 997888rarr 0 120579 (infin) 997888rarr 0

(17)

International Journal of Engineering Mathematics 5

07

06

05

04

03

02

01

000 2 4 6 8 10 12

Present result

120579(120578)

120578

Abu Bakar et al [35]

Γ = 005 05 10

Figure 2 Comparison of temperature profiles for different Γ

The quantities of practical interest in this study are the skinfriction or the shear stress coefficient119862

119891and the local Nusselt

number Nu119909 which are defined as

119862119891= minus

120583119899119891

1205881198911199062119908

(120597119906

120597119910)

119910=0

Nu119909=

119909119896119899119891

119896119891(119879119908minus 119879infin)(minus

120597119879

120597119910)

119910=0

(18)

Using (6) the skin friction coefficient and local Nusseltnumber can be expressed as

Re12119909

119862119891= minus

1

(1 minus 120601)25

11989110158401015840(0)

Re12119909

Nu119909= minus

119896119899119891

119896119891

1205791015840(0)

(19)

where Re119909= 119906119908119909]119891is the Reynolds number

3 Results and Discussion

The coupled nonlinear differential equations (16) along withthe boundary conditions (17) are solved numerically usingRunge-Kutta fourth order technique along with shootingmethodThe integration length 120578

infinvaries with the parameter

values and it has been suitably chosen each time such that theboundary conditions at the outer edge of the boundary layerare satisfied In order to bring out the salient features of theflow and the heat transfer characteristics in the Cu-EG basednanofluid the numerical values for different values of thegoverning parameters 120601 119872 isin Nr 120575 119878 and Γ are computedand portrayed in Figures 2ndash13

In order to validate our analysis the present results arecompared with the existing results for the case of regular fluidand found an excellent agreement (Figure 2)

Figure 3 shows the temperature profiles for different val-ues of the nanoparticle volume fraction 120601 with water and EG

006

005

004

003

002

001

00000 05 10 15 20 25 30

120579(120578)

120578

Cu-EGCu-water

120601 = 0 005 01 015 02

Figure 3 Temperature profiles for different values of 120601

120601 = 0 005 01 015 02

Cu-EG

120578

0 1 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

Figure 4 Velocity profiles for different values of 120601

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120601 = 0 005 01 015 02

Cu-EG

120578

120579(120578)

Figure 5 Temperature profiles for different values of 120601

based nanofluids It is found that the temperature increaseswith an increase in the nanoparticle volume fraction forboth Cu-water and Cu-EG based fluid And also it is noticedthat EG based nanofluids have more impact in rising thetemperature than that of water-based nanofluids This isdue to the fact that the thermal conductivity increases and

6 International Journal of Engineering Mathematics

Cu-EG

120578

10 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

M = 0 05 10 15 20

Figure 6 Velocity profiles for different values of119872

006

005

004

003

002

001

00000 05 10 15 20 25 30

M = 0 05 10 15 20

Cu-EG

120578

120579(120578)

Figure 7 Temperature profiles for different values of119872

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

Cu-EG

120578

120579(120578)

isin = 0 03 05 07 10

Figure 8 Temperature profiles for different values of isin

then the thermal boundary layer thickness increases with theincrease in volume of nanoparticles Also it is observed thatthe thermal conductivity is more in ethylene-glycol basednanofluid when compared to water-based nanofluid

Figures 4 and 5 represent the influence of the nanoparticlevolume fraction on the velocity and temperature It is foundthat both the nanofluid velocity and the temperature increasewith an increase in nanoparticle volume fraction Thesefigures illustrate in agreement with the physical behavior

015

010

005

0000 1 2 3 4 5

Cu-EG

120578

120579(120578)

Nr = 15 2 3 4 5

Figure 9 Temperature profiles for different values of Nr

10

08

06

04

02

00

f998400 (120578)

120575 = 0 01 03 05 10

Cu-EG

0 1 2 3 4 5

120578

Figure 10 Velocity profiles for different values of 120575

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120575 = 0 01 03 05 10

Cu-EG

120578

120579(120578)

Figure 11 Temperature profiles for different values of 120575

that the thermal conductivity increases and then the thermalboundary layer thickness increases with the increase involume of nanoparticles

Effect of the magnetic parameter 119872 on the velocity andtemperature profiles is shown in Figures 6 and 7 respectivelyIt is observed that the velocity decreases as the magneticparameter increases (Figure 6) The presence of magneticfield normal to the flow in an electrically conducting fluidgives rise to a resistive type of force called Lorentz force

International Journal of Engineering Mathematics 7

S = minus04 minus02 00 02 04

10

08

06

04

02

00

f998400 (120578)

Cu-EG

0 1 2 3 4 5

120578

Figure 12 Velocity profiles for different values of 119878

020

015

010

005

000

120579(120578)

S = minus04 minus02 00 02 04

Cu-EG

0 1 2 3 4 5

120578

Figure 13 Temperature profiles for different values of 119878

04

03

02

01

0000 05 10 15 20 25 30

120578

120579(120578)

Γ = 01 03 05 07 10

Cu-EG

Figure 14 Temperature profiles for different values of Γ

which retards the fluid flow This resistive force controls thenanofluids velocity which is useful in numerous applica-tions such as magnetohydrodynamic power generation andelectromagnetic coating of wires and metal Since the mag-netic parameter is inversely proportional to density so forascending119872 120588 descends thus increasing the temperature ofthe fluid (Figure 7)

Figure 8 illustrates the temperature profiles for variousvalues of thermal conductivity parameter isin It is clear from

our definition that 119896lowast gt 119896 that is the thermal conductivityis higher when isin gt 0 and hence an increase in isin resultsin increase of thermal conductivity thereby raising thetemperature

The influence of effect of radiation parameter Nr on tem-perature is portrayed in Figure 9 The effect of Nr is promi-nently seen throughout the boundary layer It is interestingto observe that the effect of Nr is to increase the temperaturedistribution The radiation parameter (reciprocal of Stephannumber) is the measure of relative importance of thermalradiation transfer to the conduction heat transferThus largervalues of 119873 sound dominance of thermal radiation overconduction Consequently larger values of119873 are indicative oflarger amount of radiative heat energy being poured into thesystem causing rise in 120579(120578)

Figures 10 and 11 demonstrate the variation of the stream-wise velocity and temperature for different values of slipparameter 120575 As the slip parameter increases the slip at thesurface wall increases and a smaller amount of penetrationoccurs due to the stretching surface into the fluid In the noslip condition 120575 approaches zero then 119891

1015840(0) = 1 It is clear

from the figures that the velocity component at the wallreduces with an increase in the slip parameter 120575 and decreasesasymptotically to zero at the edge of the hydrodynamicboundary layer Thus hydrodynamic boundary layer thick-ness decreases as the slip parameter 120575 increases for both theregular and nanofluids and as a result the local velocity alsodecreases This is in agreement with the fact that higher 120575implies an increase in the lubrication and slipperiness of thesurface On the other hand it is evident from Figure 11 thatthe surface slipperiness affects the temperature of the fluidinversely that is an increase with slip parameter tends toincrease temperature of the fluid By escalating 120575 thermalboundary layer thickness enhances So we can interpret thatthe rate of heat transfer decreases with the increase in slipparameter 120575

The effect of suction or injection parameter 119878 on thevelocity and temperature is shown in Figures 12 and 13respectively It is observed that the velocity decreases withincrease in the suctioninjection parameter The physicalexplanation for such behavior is as follows in case of suctionthe heated fluid is pushed towards the wall where thebuoyancy forces can act to retard the fluid due to high influ-ence of the viscosity (Figure 12) The thermal boundary layerthickness for the injection is more pronounced than forsuction and it is quite superior for Cu-EG than for pure water(regular fluid 120601 = 0) Figure 13 explains that for both suc-tioninjection the existence of Cu nanoparticle increases thethermal conductivity rapidly which leads to an increase inthe thickness of the thermal boundary layer Thus the temp-erature of the fluid rises

Figure 14 depicts the temperature profiles for differentvalues of convective parameter Γ It is clearly seen that theincreases in the convective parameter increase the temper-ature profile and thereby reduce the thermal boundary layerthicknessThe variations ofminus11989110158401015840(0) andminus1205791015840(0) which are pro-portional to the local skin-friction coefficient and rate of heattransfers are shown in Tables 2ndash7 From Table 2 it is foundthat both the wall skin friction coefficient and heat transfer

8 International Journal of Engineering Mathematics

Table 2 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601

120601 minus11989110158401015840(0) minus120579

1015840(0)

00 1177071 00955225005 1093940 00951172010 1013062 00947067015 0934425 0094288102 0858128 00938579

Table 3 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and isin

120601 isin minus11989110158401015840(0) minus120579

1015840(0)

00

00 117702 0095551803 117691 0095243905 117684 0095019107 117675 0094777210 11766 00943792

01

00 101307 0094543503 101298 0094201105 101292 0093953307 101285 0093688610 101273 00932570

Table 4 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Nr

120601 Nr minus11989110158401015840(0) minus120579

1015840(0)

00

15 117585 0093119420 117523 0092049930 117396 0090089240 117269 0088329050 117147 00867510

01

10 101157 0092119620 101090 0090941230 100953 0088809740 100821 0086936350 100698 00852933

rate at the surface decrease as 120601 increases From Table 3 it isobserved that the skin friction coefficient at thewall decreaseswith an increase in the parameter isin for both regular fluid andnanofluid Also it is seen that the rate of heat transferdecreases with an increase in the value of isinThis happens dueto the fact that with the increase in 120576 the thermal conductivityincreases which lowers the temperature gradient at the platesurface FromTable 4 it is clear that both the skin friction andheat transfer rate decrease with an increase in the radiationparameter Nr

Table 5 shows the effects of wall skin friction and rate ofheat transfer with suctioninjection parameter 119878 for regularfluid and nanofluid It is evident that increasing suction (119878 gt

0) at thewall tends to have high heat transfer rate but injection(119878 lt 0) decreases heat transfer rate It is also noted that theaddition of nanoparticle produces a remarkable enhancementon heat transfer with respect to that of pure fluid

Table 5 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 119878

120601 119878 minus11989110158401015840(0) minus120579

1015840(0)

00

03 117701 0095522502 113491 0094801801 109395 0093864700 105412 0092619minus01 101538 00909242minus02 097764 00885645minus03 0940748 00852126

01

03 126325 0094510702 120846 0093699401 11555 0092669700 110444 00913403minus01 105532 00895938minus02 100811 00872611minus03 0962756 00841031

Table 6 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 120575

120601 120575 minus11989110158401015840(0) minus120579

1015840(0)

00

00 138291 0095679603 0917088 0095275805 0756782 0095083907 0646713 0094926410 0532992 00947331

01

00 151052 0094723603 096444 0094183705 0786612 0093933607 066708 0093730210 0545629 00934822

Table 7 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Γ

120601 Γ minus11989110158401015840(0) minus120579

1015840(0)

00

01 117701 0095522503 117516 026262605 117355 040335907 117215 052303510 117036 067177

01

01 126325 0094510703 126147 025508105 125999 038578707 125873 049383710 125719 0624363

FromTable 6 we notice that the skin friction coefficient atthe wall decreases with an increase in the slip parameter 120575 forboth regular fluid and nanofluidThis is less pronouncedwithan increase in the value of 120575 That is as expected for the fluidflows at nanoscales the shear stress at the wall decreases withan increase in the slip parameter 120575 It is noticed that in the noslip condition problem the highest wall shear stress occursTable 7 shows the effects of surface skin friction and rate of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 5

07

06

05

04

03

02

01

000 2 4 6 8 10 12

Present result

120579(120578)

120578

Abu Bakar et al [35]

Γ = 005 05 10

Figure 2 Comparison of temperature profiles for different Γ

The quantities of practical interest in this study are the skinfriction or the shear stress coefficient119862

119891and the local Nusselt

number Nu119909 which are defined as

119862119891= minus

120583119899119891

1205881198911199062119908

(120597119906

120597119910)

119910=0

Nu119909=

119909119896119899119891

119896119891(119879119908minus 119879infin)(minus

120597119879

120597119910)

119910=0

(18)

Using (6) the skin friction coefficient and local Nusseltnumber can be expressed as

Re12119909

119862119891= minus

1

(1 minus 120601)25

11989110158401015840(0)

Re12119909

Nu119909= minus

119896119899119891

119896119891

1205791015840(0)

(19)

where Re119909= 119906119908119909]119891is the Reynolds number

3 Results and Discussion

The coupled nonlinear differential equations (16) along withthe boundary conditions (17) are solved numerically usingRunge-Kutta fourth order technique along with shootingmethodThe integration length 120578

infinvaries with the parameter

values and it has been suitably chosen each time such that theboundary conditions at the outer edge of the boundary layerare satisfied In order to bring out the salient features of theflow and the heat transfer characteristics in the Cu-EG basednanofluid the numerical values for different values of thegoverning parameters 120601 119872 isin Nr 120575 119878 and Γ are computedand portrayed in Figures 2ndash13

In order to validate our analysis the present results arecompared with the existing results for the case of regular fluidand found an excellent agreement (Figure 2)

Figure 3 shows the temperature profiles for different val-ues of the nanoparticle volume fraction 120601 with water and EG

006

005

004

003

002

001

00000 05 10 15 20 25 30

120579(120578)

120578

Cu-EGCu-water

120601 = 0 005 01 015 02

Figure 3 Temperature profiles for different values of 120601

120601 = 0 005 01 015 02

Cu-EG

120578

0 1 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

Figure 4 Velocity profiles for different values of 120601

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120601 = 0 005 01 015 02

Cu-EG

120578

120579(120578)

Figure 5 Temperature profiles for different values of 120601

based nanofluids It is found that the temperature increaseswith an increase in the nanoparticle volume fraction forboth Cu-water and Cu-EG based fluid And also it is noticedthat EG based nanofluids have more impact in rising thetemperature than that of water-based nanofluids This isdue to the fact that the thermal conductivity increases and

6 International Journal of Engineering Mathematics

Cu-EG

120578

10 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

M = 0 05 10 15 20

Figure 6 Velocity profiles for different values of119872

006

005

004

003

002

001

00000 05 10 15 20 25 30

M = 0 05 10 15 20

Cu-EG

120578

120579(120578)

Figure 7 Temperature profiles for different values of119872

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

Cu-EG

120578

120579(120578)

isin = 0 03 05 07 10

Figure 8 Temperature profiles for different values of isin

then the thermal boundary layer thickness increases with theincrease in volume of nanoparticles Also it is observed thatthe thermal conductivity is more in ethylene-glycol basednanofluid when compared to water-based nanofluid

Figures 4 and 5 represent the influence of the nanoparticlevolume fraction on the velocity and temperature It is foundthat both the nanofluid velocity and the temperature increasewith an increase in nanoparticle volume fraction Thesefigures illustrate in agreement with the physical behavior

015

010

005

0000 1 2 3 4 5

Cu-EG

120578

120579(120578)

Nr = 15 2 3 4 5

Figure 9 Temperature profiles for different values of Nr

10

08

06

04

02

00

f998400 (120578)

120575 = 0 01 03 05 10

Cu-EG

0 1 2 3 4 5

120578

Figure 10 Velocity profiles for different values of 120575

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120575 = 0 01 03 05 10

Cu-EG

120578

120579(120578)

Figure 11 Temperature profiles for different values of 120575

that the thermal conductivity increases and then the thermalboundary layer thickness increases with the increase involume of nanoparticles

Effect of the magnetic parameter 119872 on the velocity andtemperature profiles is shown in Figures 6 and 7 respectivelyIt is observed that the velocity decreases as the magneticparameter increases (Figure 6) The presence of magneticfield normal to the flow in an electrically conducting fluidgives rise to a resistive type of force called Lorentz force

International Journal of Engineering Mathematics 7

S = minus04 minus02 00 02 04

10

08

06

04

02

00

f998400 (120578)

Cu-EG

0 1 2 3 4 5

120578

Figure 12 Velocity profiles for different values of 119878

020

015

010

005

000

120579(120578)

S = minus04 minus02 00 02 04

Cu-EG

0 1 2 3 4 5

120578

Figure 13 Temperature profiles for different values of 119878

04

03

02

01

0000 05 10 15 20 25 30

120578

120579(120578)

Γ = 01 03 05 07 10

Cu-EG

Figure 14 Temperature profiles for different values of Γ

which retards the fluid flow This resistive force controls thenanofluids velocity which is useful in numerous applica-tions such as magnetohydrodynamic power generation andelectromagnetic coating of wires and metal Since the mag-netic parameter is inversely proportional to density so forascending119872 120588 descends thus increasing the temperature ofthe fluid (Figure 7)

Figure 8 illustrates the temperature profiles for variousvalues of thermal conductivity parameter isin It is clear from

our definition that 119896lowast gt 119896 that is the thermal conductivityis higher when isin gt 0 and hence an increase in isin resultsin increase of thermal conductivity thereby raising thetemperature

The influence of effect of radiation parameter Nr on tem-perature is portrayed in Figure 9 The effect of Nr is promi-nently seen throughout the boundary layer It is interestingto observe that the effect of Nr is to increase the temperaturedistribution The radiation parameter (reciprocal of Stephannumber) is the measure of relative importance of thermalradiation transfer to the conduction heat transferThus largervalues of 119873 sound dominance of thermal radiation overconduction Consequently larger values of119873 are indicative oflarger amount of radiative heat energy being poured into thesystem causing rise in 120579(120578)

Figures 10 and 11 demonstrate the variation of the stream-wise velocity and temperature for different values of slipparameter 120575 As the slip parameter increases the slip at thesurface wall increases and a smaller amount of penetrationoccurs due to the stretching surface into the fluid In the noslip condition 120575 approaches zero then 119891

1015840(0) = 1 It is clear

from the figures that the velocity component at the wallreduces with an increase in the slip parameter 120575 and decreasesasymptotically to zero at the edge of the hydrodynamicboundary layer Thus hydrodynamic boundary layer thick-ness decreases as the slip parameter 120575 increases for both theregular and nanofluids and as a result the local velocity alsodecreases This is in agreement with the fact that higher 120575implies an increase in the lubrication and slipperiness of thesurface On the other hand it is evident from Figure 11 thatthe surface slipperiness affects the temperature of the fluidinversely that is an increase with slip parameter tends toincrease temperature of the fluid By escalating 120575 thermalboundary layer thickness enhances So we can interpret thatthe rate of heat transfer decreases with the increase in slipparameter 120575

The effect of suction or injection parameter 119878 on thevelocity and temperature is shown in Figures 12 and 13respectively It is observed that the velocity decreases withincrease in the suctioninjection parameter The physicalexplanation for such behavior is as follows in case of suctionthe heated fluid is pushed towards the wall where thebuoyancy forces can act to retard the fluid due to high influ-ence of the viscosity (Figure 12) The thermal boundary layerthickness for the injection is more pronounced than forsuction and it is quite superior for Cu-EG than for pure water(regular fluid 120601 = 0) Figure 13 explains that for both suc-tioninjection the existence of Cu nanoparticle increases thethermal conductivity rapidly which leads to an increase inthe thickness of the thermal boundary layer Thus the temp-erature of the fluid rises

Figure 14 depicts the temperature profiles for differentvalues of convective parameter Γ It is clearly seen that theincreases in the convective parameter increase the temper-ature profile and thereby reduce the thermal boundary layerthicknessThe variations ofminus11989110158401015840(0) andminus1205791015840(0) which are pro-portional to the local skin-friction coefficient and rate of heattransfers are shown in Tables 2ndash7 From Table 2 it is foundthat both the wall skin friction coefficient and heat transfer

8 International Journal of Engineering Mathematics

Table 2 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601

120601 minus11989110158401015840(0) minus120579

1015840(0)

00 1177071 00955225005 1093940 00951172010 1013062 00947067015 0934425 0094288102 0858128 00938579

Table 3 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and isin

120601 isin minus11989110158401015840(0) minus120579

1015840(0)

00

00 117702 0095551803 117691 0095243905 117684 0095019107 117675 0094777210 11766 00943792

01

00 101307 0094543503 101298 0094201105 101292 0093953307 101285 0093688610 101273 00932570

Table 4 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Nr

120601 Nr minus11989110158401015840(0) minus120579

1015840(0)

00

15 117585 0093119420 117523 0092049930 117396 0090089240 117269 0088329050 117147 00867510

01

10 101157 0092119620 101090 0090941230 100953 0088809740 100821 0086936350 100698 00852933

rate at the surface decrease as 120601 increases From Table 3 it isobserved that the skin friction coefficient at thewall decreaseswith an increase in the parameter isin for both regular fluid andnanofluid Also it is seen that the rate of heat transferdecreases with an increase in the value of isinThis happens dueto the fact that with the increase in 120576 the thermal conductivityincreases which lowers the temperature gradient at the platesurface FromTable 4 it is clear that both the skin friction andheat transfer rate decrease with an increase in the radiationparameter Nr

Table 5 shows the effects of wall skin friction and rate ofheat transfer with suctioninjection parameter 119878 for regularfluid and nanofluid It is evident that increasing suction (119878 gt

0) at thewall tends to have high heat transfer rate but injection(119878 lt 0) decreases heat transfer rate It is also noted that theaddition of nanoparticle produces a remarkable enhancementon heat transfer with respect to that of pure fluid

Table 5 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 119878

120601 119878 minus11989110158401015840(0) minus120579

1015840(0)

00

03 117701 0095522502 113491 0094801801 109395 0093864700 105412 0092619minus01 101538 00909242minus02 097764 00885645minus03 0940748 00852126

01

03 126325 0094510702 120846 0093699401 11555 0092669700 110444 00913403minus01 105532 00895938minus02 100811 00872611minus03 0962756 00841031

Table 6 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 120575

120601 120575 minus11989110158401015840(0) minus120579

1015840(0)

00

00 138291 0095679603 0917088 0095275805 0756782 0095083907 0646713 0094926410 0532992 00947331

01

00 151052 0094723603 096444 0094183705 0786612 0093933607 066708 0093730210 0545629 00934822

Table 7 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Γ

120601 Γ minus11989110158401015840(0) minus120579

1015840(0)

00

01 117701 0095522503 117516 026262605 117355 040335907 117215 052303510 117036 067177

01

01 126325 0094510703 126147 025508105 125999 038578707 125873 049383710 125719 0624363

FromTable 6 we notice that the skin friction coefficient atthe wall decreases with an increase in the slip parameter 120575 forboth regular fluid and nanofluidThis is less pronouncedwithan increase in the value of 120575 That is as expected for the fluidflows at nanoscales the shear stress at the wall decreases withan increase in the slip parameter 120575 It is noticed that in the noslip condition problem the highest wall shear stress occursTable 7 shows the effects of surface skin friction and rate of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Engineering Mathematics

Cu-EG

120578

10 2 3 4 5

10

08

06

04

02

00

f998400 (120578)

M = 0 05 10 15 20

Figure 6 Velocity profiles for different values of119872

006

005

004

003

002

001

00000 05 10 15 20 25 30

M = 0 05 10 15 20

Cu-EG

120578

120579(120578)

Figure 7 Temperature profiles for different values of119872

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

Cu-EG

120578

120579(120578)

isin = 0 03 05 07 10

Figure 8 Temperature profiles for different values of isin

then the thermal boundary layer thickness increases with theincrease in volume of nanoparticles Also it is observed thatthe thermal conductivity is more in ethylene-glycol basednanofluid when compared to water-based nanofluid

Figures 4 and 5 represent the influence of the nanoparticlevolume fraction on the velocity and temperature It is foundthat both the nanofluid velocity and the temperature increasewith an increase in nanoparticle volume fraction Thesefigures illustrate in agreement with the physical behavior

015

010

005

0000 1 2 3 4 5

Cu-EG

120578

120579(120578)

Nr = 15 2 3 4 5

Figure 9 Temperature profiles for different values of Nr

10

08

06

04

02

00

f998400 (120578)

120575 = 0 01 03 05 10

Cu-EG

0 1 2 3 4 5

120578

Figure 10 Velocity profiles for different values of 120575

007

006

005

004

003

002

001

00000 05 10 15 20 25 30

120575 = 0 01 03 05 10

Cu-EG

120578

120579(120578)

Figure 11 Temperature profiles for different values of 120575

that the thermal conductivity increases and then the thermalboundary layer thickness increases with the increase involume of nanoparticles

Effect of the magnetic parameter 119872 on the velocity andtemperature profiles is shown in Figures 6 and 7 respectivelyIt is observed that the velocity decreases as the magneticparameter increases (Figure 6) The presence of magneticfield normal to the flow in an electrically conducting fluidgives rise to a resistive type of force called Lorentz force

International Journal of Engineering Mathematics 7

S = minus04 minus02 00 02 04

10

08

06

04

02

00

f998400 (120578)

Cu-EG

0 1 2 3 4 5

120578

Figure 12 Velocity profiles for different values of 119878

020

015

010

005

000

120579(120578)

S = minus04 minus02 00 02 04

Cu-EG

0 1 2 3 4 5

120578

Figure 13 Temperature profiles for different values of 119878

04

03

02

01

0000 05 10 15 20 25 30

120578

120579(120578)

Γ = 01 03 05 07 10

Cu-EG

Figure 14 Temperature profiles for different values of Γ

which retards the fluid flow This resistive force controls thenanofluids velocity which is useful in numerous applica-tions such as magnetohydrodynamic power generation andelectromagnetic coating of wires and metal Since the mag-netic parameter is inversely proportional to density so forascending119872 120588 descends thus increasing the temperature ofthe fluid (Figure 7)

Figure 8 illustrates the temperature profiles for variousvalues of thermal conductivity parameter isin It is clear from

our definition that 119896lowast gt 119896 that is the thermal conductivityis higher when isin gt 0 and hence an increase in isin resultsin increase of thermal conductivity thereby raising thetemperature

The influence of effect of radiation parameter Nr on tem-perature is portrayed in Figure 9 The effect of Nr is promi-nently seen throughout the boundary layer It is interestingto observe that the effect of Nr is to increase the temperaturedistribution The radiation parameter (reciprocal of Stephannumber) is the measure of relative importance of thermalradiation transfer to the conduction heat transferThus largervalues of 119873 sound dominance of thermal radiation overconduction Consequently larger values of119873 are indicative oflarger amount of radiative heat energy being poured into thesystem causing rise in 120579(120578)

Figures 10 and 11 demonstrate the variation of the stream-wise velocity and temperature for different values of slipparameter 120575 As the slip parameter increases the slip at thesurface wall increases and a smaller amount of penetrationoccurs due to the stretching surface into the fluid In the noslip condition 120575 approaches zero then 119891

1015840(0) = 1 It is clear

from the figures that the velocity component at the wallreduces with an increase in the slip parameter 120575 and decreasesasymptotically to zero at the edge of the hydrodynamicboundary layer Thus hydrodynamic boundary layer thick-ness decreases as the slip parameter 120575 increases for both theregular and nanofluids and as a result the local velocity alsodecreases This is in agreement with the fact that higher 120575implies an increase in the lubrication and slipperiness of thesurface On the other hand it is evident from Figure 11 thatthe surface slipperiness affects the temperature of the fluidinversely that is an increase with slip parameter tends toincrease temperature of the fluid By escalating 120575 thermalboundary layer thickness enhances So we can interpret thatthe rate of heat transfer decreases with the increase in slipparameter 120575

The effect of suction or injection parameter 119878 on thevelocity and temperature is shown in Figures 12 and 13respectively It is observed that the velocity decreases withincrease in the suctioninjection parameter The physicalexplanation for such behavior is as follows in case of suctionthe heated fluid is pushed towards the wall where thebuoyancy forces can act to retard the fluid due to high influ-ence of the viscosity (Figure 12) The thermal boundary layerthickness for the injection is more pronounced than forsuction and it is quite superior for Cu-EG than for pure water(regular fluid 120601 = 0) Figure 13 explains that for both suc-tioninjection the existence of Cu nanoparticle increases thethermal conductivity rapidly which leads to an increase inthe thickness of the thermal boundary layer Thus the temp-erature of the fluid rises

Figure 14 depicts the temperature profiles for differentvalues of convective parameter Γ It is clearly seen that theincreases in the convective parameter increase the temper-ature profile and thereby reduce the thermal boundary layerthicknessThe variations ofminus11989110158401015840(0) andminus1205791015840(0) which are pro-portional to the local skin-friction coefficient and rate of heattransfers are shown in Tables 2ndash7 From Table 2 it is foundthat both the wall skin friction coefficient and heat transfer

8 International Journal of Engineering Mathematics

Table 2 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601

120601 minus11989110158401015840(0) minus120579

1015840(0)

00 1177071 00955225005 1093940 00951172010 1013062 00947067015 0934425 0094288102 0858128 00938579

Table 3 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and isin

120601 isin minus11989110158401015840(0) minus120579

1015840(0)

00

00 117702 0095551803 117691 0095243905 117684 0095019107 117675 0094777210 11766 00943792

01

00 101307 0094543503 101298 0094201105 101292 0093953307 101285 0093688610 101273 00932570

Table 4 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Nr

120601 Nr minus11989110158401015840(0) minus120579

1015840(0)

00

15 117585 0093119420 117523 0092049930 117396 0090089240 117269 0088329050 117147 00867510

01

10 101157 0092119620 101090 0090941230 100953 0088809740 100821 0086936350 100698 00852933

rate at the surface decrease as 120601 increases From Table 3 it isobserved that the skin friction coefficient at thewall decreaseswith an increase in the parameter isin for both regular fluid andnanofluid Also it is seen that the rate of heat transferdecreases with an increase in the value of isinThis happens dueto the fact that with the increase in 120576 the thermal conductivityincreases which lowers the temperature gradient at the platesurface FromTable 4 it is clear that both the skin friction andheat transfer rate decrease with an increase in the radiationparameter Nr

Table 5 shows the effects of wall skin friction and rate ofheat transfer with suctioninjection parameter 119878 for regularfluid and nanofluid It is evident that increasing suction (119878 gt

0) at thewall tends to have high heat transfer rate but injection(119878 lt 0) decreases heat transfer rate It is also noted that theaddition of nanoparticle produces a remarkable enhancementon heat transfer with respect to that of pure fluid

Table 5 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 119878

120601 119878 minus11989110158401015840(0) minus120579

1015840(0)

00

03 117701 0095522502 113491 0094801801 109395 0093864700 105412 0092619minus01 101538 00909242minus02 097764 00885645minus03 0940748 00852126

01

03 126325 0094510702 120846 0093699401 11555 0092669700 110444 00913403minus01 105532 00895938minus02 100811 00872611minus03 0962756 00841031

Table 6 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 120575

120601 120575 minus11989110158401015840(0) minus120579

1015840(0)

00

00 138291 0095679603 0917088 0095275805 0756782 0095083907 0646713 0094926410 0532992 00947331

01

00 151052 0094723603 096444 0094183705 0786612 0093933607 066708 0093730210 0545629 00934822

Table 7 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Γ

120601 Γ minus11989110158401015840(0) minus120579

1015840(0)

00

01 117701 0095522503 117516 026262605 117355 040335907 117215 052303510 117036 067177

01

01 126325 0094510703 126147 025508105 125999 038578707 125873 049383710 125719 0624363

FromTable 6 we notice that the skin friction coefficient atthe wall decreases with an increase in the slip parameter 120575 forboth regular fluid and nanofluidThis is less pronouncedwithan increase in the value of 120575 That is as expected for the fluidflows at nanoscales the shear stress at the wall decreases withan increase in the slip parameter 120575 It is noticed that in the noslip condition problem the highest wall shear stress occursTable 7 shows the effects of surface skin friction and rate of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 7

S = minus04 minus02 00 02 04

10

08

06

04

02

00

f998400 (120578)

Cu-EG

0 1 2 3 4 5

120578

Figure 12 Velocity profiles for different values of 119878

020

015

010

005

000

120579(120578)

S = minus04 minus02 00 02 04

Cu-EG

0 1 2 3 4 5

120578

Figure 13 Temperature profiles for different values of 119878

04

03

02

01

0000 05 10 15 20 25 30

120578

120579(120578)

Γ = 01 03 05 07 10

Cu-EG

Figure 14 Temperature profiles for different values of Γ

which retards the fluid flow This resistive force controls thenanofluids velocity which is useful in numerous applica-tions such as magnetohydrodynamic power generation andelectromagnetic coating of wires and metal Since the mag-netic parameter is inversely proportional to density so forascending119872 120588 descends thus increasing the temperature ofthe fluid (Figure 7)

Figure 8 illustrates the temperature profiles for variousvalues of thermal conductivity parameter isin It is clear from

our definition that 119896lowast gt 119896 that is the thermal conductivityis higher when isin gt 0 and hence an increase in isin resultsin increase of thermal conductivity thereby raising thetemperature

The influence of effect of radiation parameter Nr on tem-perature is portrayed in Figure 9 The effect of Nr is promi-nently seen throughout the boundary layer It is interestingto observe that the effect of Nr is to increase the temperaturedistribution The radiation parameter (reciprocal of Stephannumber) is the measure of relative importance of thermalradiation transfer to the conduction heat transferThus largervalues of 119873 sound dominance of thermal radiation overconduction Consequently larger values of119873 are indicative oflarger amount of radiative heat energy being poured into thesystem causing rise in 120579(120578)

Figures 10 and 11 demonstrate the variation of the stream-wise velocity and temperature for different values of slipparameter 120575 As the slip parameter increases the slip at thesurface wall increases and a smaller amount of penetrationoccurs due to the stretching surface into the fluid In the noslip condition 120575 approaches zero then 119891

1015840(0) = 1 It is clear

from the figures that the velocity component at the wallreduces with an increase in the slip parameter 120575 and decreasesasymptotically to zero at the edge of the hydrodynamicboundary layer Thus hydrodynamic boundary layer thick-ness decreases as the slip parameter 120575 increases for both theregular and nanofluids and as a result the local velocity alsodecreases This is in agreement with the fact that higher 120575implies an increase in the lubrication and slipperiness of thesurface On the other hand it is evident from Figure 11 thatthe surface slipperiness affects the temperature of the fluidinversely that is an increase with slip parameter tends toincrease temperature of the fluid By escalating 120575 thermalboundary layer thickness enhances So we can interpret thatthe rate of heat transfer decreases with the increase in slipparameter 120575

The effect of suction or injection parameter 119878 on thevelocity and temperature is shown in Figures 12 and 13respectively It is observed that the velocity decreases withincrease in the suctioninjection parameter The physicalexplanation for such behavior is as follows in case of suctionthe heated fluid is pushed towards the wall where thebuoyancy forces can act to retard the fluid due to high influ-ence of the viscosity (Figure 12) The thermal boundary layerthickness for the injection is more pronounced than forsuction and it is quite superior for Cu-EG than for pure water(regular fluid 120601 = 0) Figure 13 explains that for both suc-tioninjection the existence of Cu nanoparticle increases thethermal conductivity rapidly which leads to an increase inthe thickness of the thermal boundary layer Thus the temp-erature of the fluid rises

Figure 14 depicts the temperature profiles for differentvalues of convective parameter Γ It is clearly seen that theincreases in the convective parameter increase the temper-ature profile and thereby reduce the thermal boundary layerthicknessThe variations ofminus11989110158401015840(0) andminus1205791015840(0) which are pro-portional to the local skin-friction coefficient and rate of heattransfers are shown in Tables 2ndash7 From Table 2 it is foundthat both the wall skin friction coefficient and heat transfer

8 International Journal of Engineering Mathematics

Table 2 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601

120601 minus11989110158401015840(0) minus120579

1015840(0)

00 1177071 00955225005 1093940 00951172010 1013062 00947067015 0934425 0094288102 0858128 00938579

Table 3 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and isin

120601 isin minus11989110158401015840(0) minus120579

1015840(0)

00

00 117702 0095551803 117691 0095243905 117684 0095019107 117675 0094777210 11766 00943792

01

00 101307 0094543503 101298 0094201105 101292 0093953307 101285 0093688610 101273 00932570

Table 4 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Nr

120601 Nr minus11989110158401015840(0) minus120579

1015840(0)

00

15 117585 0093119420 117523 0092049930 117396 0090089240 117269 0088329050 117147 00867510

01

10 101157 0092119620 101090 0090941230 100953 0088809740 100821 0086936350 100698 00852933

rate at the surface decrease as 120601 increases From Table 3 it isobserved that the skin friction coefficient at thewall decreaseswith an increase in the parameter isin for both regular fluid andnanofluid Also it is seen that the rate of heat transferdecreases with an increase in the value of isinThis happens dueto the fact that with the increase in 120576 the thermal conductivityincreases which lowers the temperature gradient at the platesurface FromTable 4 it is clear that both the skin friction andheat transfer rate decrease with an increase in the radiationparameter Nr

Table 5 shows the effects of wall skin friction and rate ofheat transfer with suctioninjection parameter 119878 for regularfluid and nanofluid It is evident that increasing suction (119878 gt

0) at thewall tends to have high heat transfer rate but injection(119878 lt 0) decreases heat transfer rate It is also noted that theaddition of nanoparticle produces a remarkable enhancementon heat transfer with respect to that of pure fluid

Table 5 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 119878

120601 119878 minus11989110158401015840(0) minus120579

1015840(0)

00

03 117701 0095522502 113491 0094801801 109395 0093864700 105412 0092619minus01 101538 00909242minus02 097764 00885645minus03 0940748 00852126

01

03 126325 0094510702 120846 0093699401 11555 0092669700 110444 00913403minus01 105532 00895938minus02 100811 00872611minus03 0962756 00841031

Table 6 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 120575

120601 120575 minus11989110158401015840(0) minus120579

1015840(0)

00

00 138291 0095679603 0917088 0095275805 0756782 0095083907 0646713 0094926410 0532992 00947331

01

00 151052 0094723603 096444 0094183705 0786612 0093933607 066708 0093730210 0545629 00934822

Table 7 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Γ

120601 Γ minus11989110158401015840(0) minus120579

1015840(0)

00

01 117701 0095522503 117516 026262605 117355 040335907 117215 052303510 117036 067177

01

01 126325 0094510703 126147 025508105 125999 038578707 125873 049383710 125719 0624363

FromTable 6 we notice that the skin friction coefficient atthe wall decreases with an increase in the slip parameter 120575 forboth regular fluid and nanofluidThis is less pronouncedwithan increase in the value of 120575 That is as expected for the fluidflows at nanoscales the shear stress at the wall decreases withan increase in the slip parameter 120575 It is noticed that in the noslip condition problem the highest wall shear stress occursTable 7 shows the effects of surface skin friction and rate of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Engineering Mathematics

Table 2 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601

120601 minus11989110158401015840(0) minus120579

1015840(0)

00 1177071 00955225005 1093940 00951172010 1013062 00947067015 0934425 0094288102 0858128 00938579

Table 3 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and isin

120601 isin minus11989110158401015840(0) minus120579

1015840(0)

00

00 117702 0095551803 117691 0095243905 117684 0095019107 117675 0094777210 11766 00943792

01

00 101307 0094543503 101298 0094201105 101292 0093953307 101285 0093688610 101273 00932570

Table 4 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Nr

120601 Nr minus11989110158401015840(0) minus120579

1015840(0)

00

15 117585 0093119420 117523 0092049930 117396 0090089240 117269 0088329050 117147 00867510

01

10 101157 0092119620 101090 0090941230 100953 0088809740 100821 0086936350 100698 00852933

rate at the surface decrease as 120601 increases From Table 3 it isobserved that the skin friction coefficient at thewall decreaseswith an increase in the parameter isin for both regular fluid andnanofluid Also it is seen that the rate of heat transferdecreases with an increase in the value of isinThis happens dueto the fact that with the increase in 120576 the thermal conductivityincreases which lowers the temperature gradient at the platesurface FromTable 4 it is clear that both the skin friction andheat transfer rate decrease with an increase in the radiationparameter Nr

Table 5 shows the effects of wall skin friction and rate ofheat transfer with suctioninjection parameter 119878 for regularfluid and nanofluid It is evident that increasing suction (119878 gt

0) at thewall tends to have high heat transfer rate but injection(119878 lt 0) decreases heat transfer rate It is also noted that theaddition of nanoparticle produces a remarkable enhancementon heat transfer with respect to that of pure fluid

Table 5 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 119878

120601 119878 minus11989110158401015840(0) minus120579

1015840(0)

00

03 117701 0095522502 113491 0094801801 109395 0093864700 105412 0092619minus01 101538 00909242minus02 097764 00885645minus03 0940748 00852126

01

03 126325 0094510702 120846 0093699401 11555 0092669700 110444 00913403minus01 105532 00895938minus02 100811 00872611minus03 0962756 00841031

Table 6 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and 120575

120601 120575 minus11989110158401015840(0) minus120579

1015840(0)

00

00 138291 0095679603 0917088 0095275805 0756782 0095083907 0646713 0094926410 0532992 00947331

01

00 151052 0094723603 096444 0094183705 0786612 0093933607 066708 0093730210 0545629 00934822

Table 7 Values of minus11989110158401015840(0) and minus1205791015840(0) for various values of 120601 and Γ

120601 Γ minus11989110158401015840(0) minus120579

1015840(0)

00

01 117701 0095522503 117516 026262605 117355 040335907 117215 052303510 117036 067177

01

01 126325 0094510703 126147 025508105 125999 038578707 125873 049383710 125719 0624363

FromTable 6 we notice that the skin friction coefficient atthe wall decreases with an increase in the slip parameter 120575 forboth regular fluid and nanofluidThis is less pronouncedwithan increase in the value of 120575 That is as expected for the fluidflows at nanoscales the shear stress at the wall decreases withan increase in the slip parameter 120575 It is noticed that in the noslip condition problem the highest wall shear stress occursTable 7 shows the effects of surface skin friction and rate of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Engineering Mathematics 9

heat transfer with parameter Γ for regular fluid and nanofluidIt is evident that the surface skin friction decreases but theheat transfer rate decreases with an increase in convectionparameter Γ

Conflict of Interests

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

Acknowledgments

This work is carried out by the financial support of UniversityGrants Commission under Major Research Project (UGC-F no 41-7942012(SR)) for which the first author is highlygrateful The authors are highly grateful to the reviewers fortheir constructive comments which helped to enrich thequality of the paper

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids withnanoparticlerdquo in Developments and Applications of Non-Newt-onian Flows D A Siginer and P HWang Eds vol 66 pp 99ndash105 ASME New York NY USA 1995

[2] M Prodanovi S Ryoo and R A Rahmani ldquoEffects of magneticfield on the motion of multiphase fluids containing paramag-netic nanoparticles in porous mediardquo in Proceedings of the SPEImproved Oil Recovery Symposium Tulsa Okla USA 2010

[3] C T Nguyen G Roy C Gauthier and N Galanis ldquoHeattransfer enhancement using Al

2O3-water nanofluid for an elec-

tronic liquid cooling systemrdquo AppliedThermal Engineering vol27 no 8-9 pp 1501ndash1506 2007

[4] A Akbarinia M Abdolzadeh and R Laur ldquoCritical inves-tigation of heat transfer enhancement using nanofluids inmicrochannels with slip and non-slip flow regimesrdquo AppliedThermal Engineering vol 31 no 4 pp 556ndash565 2011

[5] H Masuda A Ebata K Teramae and N Hishinuma ldquoAlter-ation of thermal conductivity and viscosity of liquid by dispers-ing ultra-fine particlesrdquo Netsu Bussei vol 7 pp 227ndash233 1993

[6] J Buongiorno and W Hu ldquoNanofluid coolants for advancednuclear power plantsrdquo in Proceedings of the ICAPP Paper no5705 Seoul Republic of Korea May 2005

[7] J Buongiorno D C Venerus N Prabhat et al ldquoA benchmarkstudy on the thermal conductivity of nanofluidsrdquo JournalApplied Physics vol 106 no 9 Article ID 094312 2009

[8] S K Das N Putra P Thiesen and W Roetzel ldquoTemperaturedependence of thermal conductivity enhancement for nanoflu-idsrdquo Journal of Heat Transfer vol 125 no 4 pp 567ndash574 2003

[9] L J Crane ldquoFlow past a stretching platerdquo Zeitschrift fur ange-wandteMathematik und Physik vol 21 no 4 pp 645ndash647 1970

[10] B K Dutta P Roy and A S Gupta ldquoTemperature field in flowover a stretching sheet with uniform heat fluxrdquo InternationalCommunications in Heat and Mass Transfer vol 12 no 1 pp89ndash94 1985

[11] C K Chen and M I Char ldquoHeat transfer of a continuousstretching surface with suction or blowingrdquo Journal of Mathe-matical Analysis and Applications vol 135 no 2 pp 568ndash5801988

[12] W A Khan and I Pop ldquoBoundary-layer flow of a nanofluid pasta stretching sheetrdquo International Journal of Heat and MassTransfer vol 53 no 11-12 pp 2477ndash2483 2010

[13] MHassaniMM Tabar HNemati G Domairry and FNoorildquoAn analytical solution for boundary layer flow of a nanofluidpast a stretching sheetrdquo International Journal of Thermal Sci-ences vol 50 no 11 pp 2256ndash2263 2011

[14] P Rana and R Bhargava ldquoFlow and heat transfer of a nanofluidover a nonlinearly stretching sheet a numerical studyrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 212ndash226 2012

[15] M AHamad andM Ferdows ldquoSimilarity solution of boundarylayer stagnation-point flow towards a heated porous stretch-ing sheet saturated with a nanofluid with heat absorptiongeneration and suctionblowing a Lie group analysisrdquoCommu-nications in Nonlinear Science and Numerical Simulation vol 17no 1 pp 132ndash140 2012

[16] M Q Al-Odat R A Damesh and T A Al-Azab ldquoThermalboundary layer on an exponentially stretching continuoussurface in the presence of magnetic field effectrdquo InternationalJournal of Applied Mechanics and Engineering vol 11 no 2 pp289ndash299 2006

[17] A J Chamkha and A M Aly ldquoMHD free convection flow of ananofluid past a vertical plate in the presence of heat generationor absorption effectsrdquo Chemical Engineering Communicationsvol 198 no 3 pp 425ndash441 2011

[18] V Aliakbar A Alizadeh-Pahlavan and K Sadeghy ldquoThe influ-ence of thermal radiation on MHD flow of Maxwellian fluidsabove stretching sheetsrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 14 no 3 pp 779ndash794 2009

[19] M S Khan M M Alam and M Ferdows ldquoEffects of magneticfield on radiative flow of a nanofluid past a stretching sheetrdquoProcedia Engineering vol 56 pp 316ndash322 2013

[20] W Ibrahim and B Shankar ldquoMHD boundary layer flow andheat transfer of a nanofluid past a permeable stretching sheetwith velocity thermal and solutal slip boundary conditionsrdquoComputers amp Fluids vol 75 pp 1ndash10 2013

[21] M E Sparrow and D R Cess Radiation Heat Transfer BrooksCole Belmont Calif USA 1970

[22] N M Ozisik Radiative Transfer and Interaction with Conduc-tion and Convection John Wiley amp Sons New York NY USA1973

[23] R Siegel and J R Howell Thermal Radiation Heat TransferHemisphere New York NY USA 2nd edition 1992

[24] J R Howell ldquoRadiative transfer in porous mediardquo inHandbookof Porous Media K Vafai Ed pp 663ndash698 CRC Press NewYork NY USA 2000

[25] H S Takhar R S R Gorla and VM Soundalgekar ldquoRadiationeffects onMHD free convection flowof a gas past a semi-infinitevertical platerdquo International Journal of Numerical Methods forHeat and Fluid Flow vol 6 no 2 pp 77ndash83 1996

[26] M A Hossain and H S Takhar ldquoRadiation effect on mixedconvection along a vertical plate with uniform surface temper-aturerdquoHeat and Mass Transfer vol 31 no 4 pp 243ndash248 1996

[27] A Yoshimura and R K Prudrsquohomme ldquoWall slip corrections forCouette and parallel disk viscometersrdquo Journal of Rheology vol32 no 1 pp 53ndash67 1988

[28] V P Shidlovskiy Introduction to the Dynamics of Rarefied GasesAmerican Elsevier Publishing New York NY USA 1967

[29] SMansur andA Ishak ldquoThemagnetohydrodynamic boundarylayer flow of a nanofluid past a stretchingshrinking sheet with

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Engineering Mathematics

slip boundary conditionsrdquo Journal of Applied Mathematics vol2014 Article ID 907152 7 pages 2014

[30] H I Andersson ldquoSlip flow past a stretching surfacerdquo ActaMechanica vol 158 no 1-2 pp 121ndash125 2002

[31] C Y Wang ldquoFlow due to a stretching boundary with partialslipmdashan exact solution of the Navier-Stokes equationsrdquo Chemi-cal Engineering Science vol 57 no 17 pp 3745ndash3747 2002

[32] C Y Wang ldquoStagnation slip flow and heat transfer on a movingplaterdquo Chemical Engineering Science vol 61 no 23 pp 7668ndash7672 2006

[33] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[34] T Hayat M Qasim and S Mesloub ldquoMHD flow and heattransfer over permeable stretching sheet with slip conditionsrdquoInternational Journal for Numerical Methods in Fluids vol 66no 8 pp 963ndash975 2011

[35] A Abu Bakar M K A W Wan Zaimi R Hamid A B Bidinand A Ishak ldquoBoundary layer flow over a stretching sheet witha convective boundary condition and slip effectrdquoWorld AppliedSciences Journal vol 17 pp 49ndash53 2012

[36] O DMakinde and A Aziz ldquoBoundary layer flow of a nanofluidpast a stretching sheet with a convective boundary conditionrdquoInternational Journal ofThermal Sciences vol 50 no 7 pp 1326ndash1332 2011

[37] O D Makinde and A Aziz ldquoMHD mixed convection from avertical plate embedded in a porous medium with a convectiveboundary conditionrdquo International Journal of Thermal Sciencesvol 49 no 9 pp 1813ndash1820 2010

[38] O D Makinde ldquoSimilarity solution of hydromagnetic heat andmass transfer over a vertical plate with a convective surfaceboundary conditionrdquo International Journal of Physical Sciencesvol 5 no 6 pp 700ndash710 2010

[39] ODMakinde ldquoOnMHDheat andmass transfer over amovingvertical plate with a convective surface boundary conditionrdquoCanadian Journal of Chemical Engineering vol 88 no 6 pp983ndash990 2010

[40] S V Subhashini N Samuel and I Pop ldquoDouble-diffusive con-vection from a permeable vertical surface under convectiveboundary conditionrdquo International Communications in Heatand Mass Transfer vol 38 no 9 pp 1183ndash1188 2011

[41] A Ishak ldquoSimilarity solutions for flow and heat transfer overa permeable surface with convective boundary conditionrdquoApplied Mathematics and Computation vol 217 no 2 pp 837ndash842 2010

[42] T Hayat Z Iqbal M Mustafa and S Obaidat ldquoBoundarylayer flow of an Oldroyd-B fluid with convective boundaryconditionsrdquo Heat Transfer Asian Research vol 40 no 8 pp744ndash755 2011

[43] A Alsaedi M Awais and T Hayat ldquoEffects of heat generationabsorption on stagnation point flow of nanofluid over a surfacewith convective boundary conditionsrdquo Communications inNonlinear Science and Numerical Simulation vol 17 no 11 pp4210ndash4223 2012

[44] MKhan RAli andA Shahzad ldquoMHDFalkner-Skan flowwithmixed convection and convective boundary conditionsrdquoWalailak Journal of Science and Technology vol 10 no 5 pp517ndash529 2013

[45] C Y Wang ldquoAnalysis of viscous flow due to a stretching sheetwith surface slip and suctionrdquo Nonlinear Analysis Real WorldApplications vol 10 no 1 pp 375ndash380 2009

[46] T Fang J Zhang and S Yao ldquoSlip MHD viscous flow over astretching sheetmdashan exact solutionrdquo Communications in Non-linear Science andNumerical Simulation vol 14 no 11 pp 3731ndash3737 2009

[47] S Mukhopadhyay and H I Andersson ldquoEffects of slip and heattransfer analysis of flow over an unsteady stretching surfacerdquoHeat and Mass Transfer vol 45 no 11 pp 1447ndash1452 2009

[48] K Bhattacharyya S Mukhopadhyay and G C Layek ldquoSlipeffects on boundary layer stagnation-point flow and heat trans-fer towards a shrinking sheetrdquo International Journal of Heat andMass Transfer vol 54 no 1ndash3 pp 308ndash313 2011

[49] F Aman A Ishak and I Pop ldquoMixed convection boundarylayer flow near stagnation-point on vertical surface with sliprdquoAppliedMathematics andMechanics English Edition vol 32 no12 pp 1599ndash1606 2011

[50] T Fang S Yao J Zhang and A Aziz ldquoViscous flow over ashrinking sheet with a second order slip flow modelrdquo Commu-nications inNonlinear Science andNumerical Simulation vol 15no 7 pp 1831ndash1842 2010

[51] L P Jentoft Y Tenzer D Vogt J Liu R J Wood and R DHowe ldquoFlexible stretchable tactile arrays from MEMS barom-etersrdquo in Proceedings of the 16th International Conference onAdvanced Robotics (ICAR rsquo13) November 2013

[52] M Arunachalam and N R Rajappa ldquoForced convection inliquid metals with variable thermal conductivity and capacityrdquoActa Mechanica vol 31 no 1-2 pp 25ndash31 1978

[53] T C Chiam ldquoHeat transfer in a fluid with variable thermalconductivity over a linearly stretching sheetrdquo Acta Mechanicavol 129 no 1-2 pp 63ndash72 1998

[54] M A Seddeek and A M Salem ldquoLaminar mixed convectionadjacent to vertical continuously stretching sheets with variableviscosity and variable thermal diffusivityrdquo Heat and MassTransferWaerme- und Stoffuebertragung vol 41 no 12 pp1048ndash1055 2005

[55] J Maxwell A Treatise on Electricity and Magnetism OxfordUniversity Press Cambridge UK 2nd edition 1904

[56] H C Brinkman ldquoThe viscosity of concentrated suspensionsand solutionsrdquoThe Journal of Chemical Physics vol 20 pp 571ndash581 1952

[57] V R Prasad N B Reddy R Muthucumaraswamy and B VasuldquoFinite difference analysis of radiative free convection flow pastan impulsively started vertical plate with variable heat andmassfluxrdquo Journal of Applied FluidMechanics vol 4 no 1 pp 59ndash682011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

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International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of