research article similarity solution of marangoni...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 634746 8 pageshttpdxdoiorg1011552013634746

Research ArticleSimilarity Solution of Marangoni Convection BoundaryLayer Flow over a Flat Surface in a Nanofluid

Norihan Md Arifin1 Roslinda Nazar2 and Ioan Pop3

1 Institute for Mathematical Research and Department of Mathematics Universiti Putra Malaysia 43400 SerdangSelangor Malaysia

2 School of Mathematical Sciences Faculty of Science amp Technology Universiti Kebangsaan Malaysia 43600 BangiSelangor Malaysia

3 Department of Mathematics Babes-Bolyai University 400084 Cluj-Napoca Romania

Correspondence should be addressed to Norihan Md Arifin norihanarifinyahoocom

Received 21 June 2013 Accepted 10 December 2013

Academic Editor Mohamed Fathy El-Amin

Copyright copy 2013 Norihan Md Arifin 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

The problem of steady Marangoni boundary layer flow and heat transfer over a flat plate in a nanofluid is studied using differenttypes of nanoparticles The general governing partial differential equations are transformed into a set of two nonlinear ordinarydifferential equations using unique similarity transformation Numerical solutions of the similarity equations are obtained using theRunge-Kutta-Fehlberg (RKF)methodThree different types of nanoparticles are considered namely Cu Al

2O3 and TiO

2 by using

water as a base fluid with Prandtl number Pr = 62 The effects of the nanoparticle volume fraction 120601 and the constant exponentmon the flow and heat transfer characteristics are obtained and discussed

1 Introduction

A nanofluid is a colloidal mixture of nanosized particles(lt100 nm) in a base fluid It is known that nanofluid cantremendously enhance the heat transfer characteristics of theoriginal (base) fluid One such characteristic of nanofluid isthe anomalous high thermal conductivity at very low concen-tration of nanoparticles and the considerable enhancementof convective heat transfer Thus nanofluids have manyapplications in industry such as coolants lubricants heatexchangers and microchannel heat sinks Nanoparticles aremade of various materials such as oxide ceramics andnitride ceramics The objective of nanofluids is to achievethe best possible thermal properties with the least possible(lt1) volume fraction of nanoparticles in the base fluid[1] There have been many studies in the literature to bet-ter understand the mechanism behind the enhanced heattransfer characteristics An excellent collection of paperson this topic can be found in the book by Das et al [2]and in several review papers ([3ndash8]) There are also severalexperimental studies to better understand the mechanism

of heat transfer enhancement for natural convection heattransfer in nanofluids ([1 9ndash12])

Marangoni flow induced by surface tension along a liquidsurface causes undesirable effects in crystal growth melts inthe same manner as buoyancy-induced natural convection[13] These undesirable effects also occur in space-basedcrystal growth experiments since Marangoni flow is involvedin microgravity as well as in earth gravity An excellentview of the Marangoni effect from the perspective of allthree possible interfaces as motion inducing agents has beendone by Tadmor [14] It is worth mentioning that thereare two existing models for Marangoni boundary layer thathave been studied namely model for nonisobaricMarangoniboundary layer as discussed by Golia and Viviani [15] andmodel for Marangoni boundary layer over a flat plate studiedby Christopher and Wang [13] Marangoni boundary layerstudied by Golia and Viviani [15] has been extended by Popet al [16] where they included the concentration equationChamkha et al [17] studied the same model with Golia andViviani [15] in which they considered the gravity effectsHamid et al [18] extended the problem of the thermosolutal

2 Journal of Applied Mathematics

Marangoni forced convection boundary layer flow by Popet al [16] when the wall is permeable Very recently Matel al [19] discussed the radiation effects on the problem ofMarangoni boundary layer with permeable surface On theother hand nanofluid equationsmodel as proposed by Tiwariand Das [20] has been used by Arifin et al [21] for theMarangoni boundary layer problembyGolia andViviani [15]They found that the numerical results also indicate that forboth a regular fluid (120593 = 0) and a nanofluid (120593 = 0) dualsolutions exist when 120573 lt 05 These dual solutions were notdiscussed by Golia and Viviani [15] This problem has beenextended byRemeli et al [22] to the problemwith suction andinjection effects Mat et al [23] also extended the problem ofMarangoni boundary layer in a nanofluid by Arifin et al [21]to the radiation effect

It is worth mentioning that Christopher and Wang [13]considered the Marangoni boundary layer over a flat platewhere the term119906

119890(119909) which is the velocity of the external flow

in Golia and Viviani [15] has been neglected The similaritysolutions of the Christopher and Wang [13] problem arealso different from the Golia and Viviani [15] problem Theproblem of Christopher and Wang [13] has been extendedby several researchers such as Al-Mudhaf and Chamkha [24]where they have presented the similarity solutions for MHDMarangoni convection in the presence of heat generation orabsorption effects and Magyari and Chamkha [25] reportedthe exact analytical solutions of thermosolutal Marangoniflows in the presence of temperature-dependent volumetricheat sourcesinks as well as of a first-order chemical reactionRecently Hamid et al [26] studied the two-dimensionalMarangoni convection flow past a flat plate in the presenceof thermal radiation suction and injection effects FurtherMHD thermosolutal Marangoni convection boundary layerover a flat surface considering the effects of the thermal dif-fusion and diffusion-thermo with fluid suction and injectionhas been examined by Hamid et al [27]

It should be highlighted that the present paper presentsa similarity solution for the steady Marangoni convectionboundary layer flow over a static semi-infinite flat plate dueto an imposed temperature gradient in a nanofluid whichextends the problem by Christopher and Wang [13] to thecase of nanofluid The nanofluid equations model proposedby Tiwari and Das [20] has been used This model has beenvery successfully used in several papers [21 28ndash32] Thuswe wish to highlight that this present study is original andall the results are new To the best of our knowledge thepresent problem has not been considered before The studyof nanofluid is still at its early stage and it seems difficult tohave a precise idea on the way the use of nanoparticles acts inheat transfer A clear picture on the boundary layer flows ofnanofluid is yet to emerge

2 Problem Formulation

We consider the steady two-dimensional boundary layer flowpast a semi-infinite flat plate in a water-based nanofluidcontaining different types of nanoparticles namely copper(Cu) alumina (Al

2O3) and titania (TiO

2) with Marangoni

effectsThe nanofluid is assumed incompressible and the flowis assumed to be laminar It is also assumed that the base fluid(ie water) and the nanoparticles are in thermal equilibriumand no slip occurs between them The thermophysical prop-erties of the nanofluids are given in Table 1 (see Oztop andAbu-Nada [29]) Further we consider a Cartesian coordinatesystem (119909 119910) where 119909 and 119910 are the coordinates measuredalong the plate and normal to it respectively and the flowtakes place at 119910 ge 0 It is also assumed that the temperatureof the plate is 119879

119908(119909) and that of the ambient nanofluid is 119879

infin

Following [15ndash17 25 33 34] the surface tension 120590 is assumedto vary linearly with temperature as follow

120590 = 1205900[1 minus 120574 (119879 minus 119879

0)] (1)

where 1205900and 119879

0are the characteristics surface tension and

temperature respectively and we assume that 1198790

equiv 119879infin

Equation (1) is a commonly made assumption [34] For mostliquids the surface tension decreases with temperature thatis 120574 is a positive fluid property

The steady boundary layer equations for a nanofluid inthe coordinates 119909 and 119910 are ([13 20])

120597119906

120597119909

+

120597V120597119910

= 0 (2)

119906

120597119906

120597119909

+ V120597119906

120597119910

=

120583nf120588nf

120597

2119906

120597119910

2 (3)

119906

120597119879

120597119909

+ V120597119879

120597119910

= 120572nf120597

2119879

120597119910

2(4)

subject to the boundary conditions

V = 0 119879 = 1198790+ 119860119909

119898+1 120583nf

120597119906

120597119910

=

120597120590

120597119879

120597119879

120597119909

at119910 = 0

119906 = 0 119879 = 119879infin

as 119910 997888rarr infin

(5)

Here 119906 and V are the velocity components along the 119909- and119910-axes respectively 119879 is the temperature of the nanofluidmis the constant exponent of the temperature120572nf is the thermaldiffusivity of the nanofluid 120588nf is the effective density of thenanofluid 119896nf is the effective thermal conductivity of thenanofluid and 120583nf is the effective viscosity of the nanofluidwhich are given by

120572nf =119896nf

(120588119862119901)

nf

120588nf = (1 minus 120601) 120588119891+ 120601120588119904

120583nf =120583119891

(1 minus 120601)

25 (120588119862

119901)

nf= (1 minus 120601) (120588119862

119901)

119891+ 120601(120588119862

119901)

119904

119896nf119896119891

=

(119896119904+ 2119896119891) minus 2120601 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120601 (119896

119891minus 119896119904)

(6)

where 120601 is the nanoparticle volume fraction 120588119891

is thereference density of the fluid fraction 120588

119904is the reference

Journal of Applied Mathematics 3

velocity of the solid fraction 120583119891is the viscosity of the fluid

fraction 119896119891is the thermal conductivity of the fluid 119896

119904is the

thermal conductivity of the solid and (120588119862119901)nf is the heat

capacity of the nanofluidWe look now for a similarity solution of (2)ndash(4) subject

to the boundary conditions (5) of the following form

120595 = 1198621119909

(2+119898)3119891 (120578) 120579 (120578) =

119879 minus 119879infin

119860119909

1+119898

120578 = 1198622119909

(119898minus1)3119910

(7)

where120595 is the stream function which is defined as 119906 = 120597120595120597119910

and V = minus120597120595120597119909 Further1198981198601198621 and119862

2are constants with

119860 1198621 and 119862

2given by

119860 =

Δ119879

119871

119898+1 119862

1=

3radic

1205900120574119860120583119891

120588

2

119891

1198622=

3radic

1205900120574119860120588119891

120583

2

119891

(8)

with 119871 being the length of the surface and Δ119879 being theconstant characteristic temperature Substituting (7) into (2)and (3) we get the following ordinary differential equations

1

(1 minus 120601)

25

(1 minus 120601 + 120601120588119904120588119891)

119891

101584010158401015840+

2 + 119898

3

119891119891

10158401015840

minus

1 + 2119898

3

119891

10158402= 0

1

Pr119896nf119896119891

(1 minus 120601 + 120601(120588119862119901)

119904(120588119862119901)

119891)

120579

10158401015840+

2 + 119898

3

119891120579

minus (1 + 119898)119891

1015840120579 = 0

(9)

and the boundary conditions (5) become

119891 (0) = 0

1

(1 minus 120601)

25119891

10158401015840(0) = minus1 120579 (0) = 1

119891

1015840(infin) = 0 120579 (infin) = 0

(10)

We can now determine the surface velocity 119906(119909 0) = 119906119908(119909)

as

119906119908(119909) =

3radic

(1205900120574119860)

2

120588119891120583119891

119909

(1+2119898)3119891

1015840(0)

(11)

A quantity of interest is the local Nusselt number Nu119909which

is defined as

Nu119909=

119909119902119908(119909)

119896119891 [

119879 (119909 0) minus 119879 (119909infin)]

(12)

where 119902119908(119909) is the heat flux from the surface of the plate and

it is given by

119902119908(119909) = minus119896nf(

120597119879

120597119910

)

119910=0

(13)

Using (7) (12) and (13) we get

Nu119909= minus

119896nf119896119891

1198622119909

(2+119898)3120579

1015840(0) (14)

The average Nusselt number Nu119871based on the average tem-

perature difference between the temperature of the surfaceand the temperature far from the surface (ambient fluid) isgiven by

Nu119871= minus

6 + 3119898

5 + 4119898

119896nf119896119891

Ma13119871

Prminus131205791015840 (0) (15)

where Ma119871is the Marangoni based on 119871 and is defined as

Ma119871=

120590119879119860119871

2+119898

120583119891120572119891

=

120590119879Δ119879119871

120583119891120572119891

(16)

Also the total mass flow in the boundary layer per unitwidth is given by

= int

infin

0

120588119891119906 119889119910 =

3radic1205900120574120588119891120583119891119909

(2+119898)3119891 (infin) (17)

3 Results and Discussion

The nonlinear ordinary differential equation (9) subject tothe boundary conditions (10) forms a two-point boundaryvalue problem (BVP) and is solved numerically using theRunge-Kutta-Fehlberg fourth-fifth-order (RKF45) methodusing Maple 12- and the algorithm RKF45 in Maple hasbeen well tested for its accuracy and robustness [35] In thismethod it is most important to choose the appropriate finitevalue of the edge of boundary layer 120578 rarr infin (say 120578

infin) that is

between 4 and 10 which is in accordance with the standardpractice in the boundary layer analysis We begin with someinitial guess value of 120578

infinand solve (9) subject to the boundary

conditions (10) with some particular set of parameters toobtain the surface velocity1198911015840(0) and the temperature gradientminus120579

1015840(0) The solution process is repeated until further changes

(increment) in 120578infin

would not lead to any changes in thevalues of 1198911015840(0) and minus120579

1015840(0) or in other words the results are

independent of the value of 120578infinThe initial step size employed

is ℎ = Δ120578 = 01 Following Oztop and Abu-Nada [29] weconsidered the range of nanoparticles volume fraction 120601 as0 le 120601 le 02 The Prandtl number of the base fluid (water) iskept constant at 62 Further it should also be pointed out thatthe thermophysical properties of fluid and nanoparticles (CuAl2O3 and TiO

2) used in this study are given in Table 1 It is

worth mentioning that the present study reduces to that ofa classical viscous (regular) fluid studied by Christopher andWang [13] when 120601 = 0

Figures 1 and 2 show the distribution of the dimensionlessvelocity 119891

1015840(120578) and temperature 120579(120578) profiles for the three

types of the nanoparticles considered when the solid volumefraction of the nanofluid parameter 120601 = 01 and119898 = 0 (whichcorresponds to a linear variation of the surface temperaturewith the distance 119909 measured along the flat plate) whileFigures 3 and 4 display the variation with 120601 of the reduced


Table 1 Thermophysical properties of fluid and nanoparticles [29]

Physical properties Fluid phase (water) Cu Al2O3 TiO2

119862119901(JkgK) 4179 385 765 6862

120588 (kgm3) 9971 8933 3970 4250119896 (WmK) 0613 400 40 89538

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

120578

Cu TiO2 Al2O3

f998400 (120578)

Figure 1 Dimensionless velocity profiles 1198911015840(120578) for different types ofnanoparticles when 120601 = 01 and119898 = 0

surface velocity 1198911015840(0) and reduced temperature gradientminus120579

1015840(0) respectively It is seen from Figure 1 that the velocity

profiles in Figure 1 for Al2O3and TiO

2are almost identical

while the profile for Cu is smaller This is consistent with thevariation of the reduced surface velocity 119891

1015840(0) as shown in

Figure 3 In Figure 2 it is shown that the temperature profileis the highest for higher thermal diffusivity nanoparticle (Cu)On the other hand the thermal boundary layer thicknessas shown in Figure 2 decreases with a decrease in thermaldiffusivity which in turn gives rise to the minus1205791015840(0) as illustratedin Figure 4 Figures 3 and 4 display the surface velocity1198911015840(0)and the surface temperature gradient minus1205791015840(0) respectivelyfor different types of nanoparticles (Cu Al

2O3 and TiO

2)

when 119898 = 0 One can see that the surface velocity 1198911015840(0)and the surface temperature gradient minus1205791015840(0) decrease as 120601increases for all three nanoparticles (Cu Al

2O3 TiO

2) It

should be noticed that the entire values of minus1205791015840(0) are alwayspositive that is the heat is transferred from hot surface to thecold surface In Figure 4 we are looking at the variation oftemperature gradient with the nanoparticle volume fraction120601 It is observed that the reduced value of thermal diffu-sivity leads to higher temperature gradients and thereforehigher enhancements in heat transfer Nanoparticles with lowthermal diffusivity TiO

2 have better enhancement on heat

transfer compared to Cu and Al2O3 Further Figure 5 shows

the variations of the reduced surface velocity 119891

1015840(0) with 119898

120579(120578

)

120578

TiO2 Al2O3 Cu

1

08

06

04

02

0

0 1 2 3

Figure 2 Dimensionless temperature profiles 120579(120578) for differenttypes of nanoparticles when 120601 = 01 and119898 = 0

Cu TiO2 Al2O3

14

13

12

11

10

09

08

07

0 005 010 015 020

120601

f998400 (0)

Figure 3 Variation of surface velocity 119891

1015840(0) with 120601 for different

types of nanoparticles when Pr = 62 and119898 = 0

where 119898 = 0 refers to a linear variation of the surfacetemperature with the distance 119909measured along the flat plateand119898 = 1 is a quadratic variation of the surface temperaturewith 119909 while 119898 = minus05 refers to a temperature variationrelative to the square root of 119909 It should also be noticed thatfor 120601 = 0 (regular fluid) we reproduced the variations ofsurface velocity obtained by Christopher and Wang [13] asillustrated by dashed lines in the figure

Journal of Applied Mathematics 5minus120579998400 (0)

120601

Cu Al2O3 TiO2

32

30

28

26

24

22

20

18

16

14

0 005 010 015 020

Figure 4 Variation of minus1205791015840(0) with 120601 for different types of nanopar-ticles when Pr = 62 and119898 = 0

f998400 (0)

120601 = 01

120601 = 00

m

6

5

4

3

2

1

0

minus1 0 1 2

Al2O3 TiO2 Cu



types of nanoparticles when Pr = 62 120601 = 0 (regular fluid) and120601 = 01

Figures 6 to 11 show the dimensionless velocity 1198911015840(120578) andtemperature 120579(120578) profiles for different values of 120601 in the range0 le 120601 le 02when119898 = 0with different types of nanoparticlesnamely Cu Al

2O3 and TiO

2 respectively It is worth men-

tioning that nanoparticle volume fraction is a key parameterfor studying the effect of nanoparticles on flow fields and

120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

f998400 (120578)

Figure 6Dimensionless velocity profiles1198911015840(120578) for Cu nanoparticleswith119898 = 0 and various values of 120601

120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3

Figure 7 Dimensionless temperature profiles 120579(120578) for Cu nanopar-ticles with119898 = 0 and various values of 120601

temperature distributions More fluid is heated for highervalues of nanoparticle volume fraction Flow strength alsoincreases with increasing of nanoparticle volume fractionAs the nanoparticle volume fraction increases movements ofparticles become irregular and random due to increasing ofenergy exchange rates in the fluid (see [29]) It is observedfrom these figures that for any type of nanoparticles as thenanoparticle volume fraction 120601 increases both the surfacevelocities and the temperature gradients decrease which is


120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

f998400 (120578)

Figure 8 Dimensionless velocity profiles 1198911015840(120578) for Al2O3nanopar-

ticles with119898 = 0 and various values of 120601

120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3

Figure 9 Dimensionless temperature profiles 120579(120578) for Al2O3

nanoparticles with119898 = 0 and various values of 120601

again in agreement with Figures 3 and 4 It should also benoticed again that for 120601 = 0 (regular fluid) we reproducedthe velocity and temperature profiles obtained byChristopherand Wang [13]

4 Conclusion

Wehave theoretically and numerically studied the problem ofsteady two-dimensional laminar Marangoni-driven bound-ary layer flow in nanofluids It is worth mentioning that the

f998400 (120578)

120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0

1 2 3 4 5 6 7 8

Figure 10 Dimensionless temperature profiles 120579(120578) for TiO2

nanoparticles with119898 = 0 and various values of 120601120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3

Figure 11 Dimensionless velocity profiles 1198911015840(120578) for TiO2nanopar-


novelty of the present paper is to study numerically the heattransfer in a liquid layer driven by Marangoni flow with vari-ous types of nanoparticles (Cu Al

2O3 and TiO

2) in the base

fluid which has not been considered before The nonlinearordinary differential equation (9) subject to the boundaryconditions (10) forms a two-point boundary value problem(BVP) and is solved numerically using the Runge-Kutta-Fehlberg fourth-fifth-order (RKF45) method using Maple 12


and the algorithm RKF45 in Maple has been well tested forits accuracy and robustness Similarity solutions are obtainedfor the surface velocity 119891

1015840(0) and the surface temperature

gradient minus120579

1015840(0) as well as the velocity and temperature

profiles for some values of the governing parameters namelythe solid volume fraction of the nanofluid 120601 (0 le 120601 le 02)the constant exponent 119898 and the Prandtl number Pr It wasfound that nanoparticles with low thermal diffusivity (TiO

2)

have better enhancement on heat transfer compared to Al2O3

and Cu

Acknowledgments

The authors gratefully acknowledge the financial supportreceived in the form of a FRGS Research Grant from theMinistry of Higher Education Malaysia and a Research Uni-versity Grant (AP-2013-009) from the Universiti KebangsaanMalaysiaThey also wish to express their sincere thanks to thereviewers for the valuable comments and suggestions

References

[1] L Godson B Raja DM Lal and SWongwises ldquoEnhancementof heat transfer using nanofluidsmdashan overviewrdquo Renewable andSustainable Energy Reviews vol 14 no 2 pp 629ndash641 2010

[2] S K Das S U S Choi W Yu and T Pradeep NanofluidsScience and Technology Wiley Hoboken NJ USA 2007

[3] J Buongiorno ldquoConvective transport in nanofluidsrdquo Journal ofHeat Transfer vol 128 no 3 pp 240ndash250 2006

[4] WDaungthongsuk and SWongwises ldquoA critical review of con-vective heat transfer of nanofluidsrdquo Renewable and SustainableEnergy Reviews vol 11 no 5 pp 797ndash817 2007

[5] V Trisaksri and S Wongwises ldquoCritical review of heat transfercharacteristics of nanofluidsrdquoRenewable and Sustainable EnergyReviews vol 11 no 3 pp 512ndash523 2007

[6] X-Q Wang and A S Mujumdar ldquoHeat transfer characteristicsof nanofluids a reviewrdquo International Journal of Thermal Sci-ences vol 46 no 1 pp 1ndash19 2007

[7] X-QWang andA SMujumdar ldquoA review on nanofluidsmdashpartI theoretical and numerical investigationsrdquo Brazilian Journal ofChemical Engineering vol 25 no 4 pp 613ndash630 2008

[8] S Kakac and A Pramuanjaroenkij ldquoReview of convective heattransfer enhancement with nanofluidsrdquo International Journal ofHeat and Mass Transfer vol 52 no 13-14 pp 3187ndash3196 2009

[9] N Putra W Roetzel and S K Das ldquoNatural convection ofnano-fluidsrdquo Heat and Mass Transfer vol 39 no 8-9 pp 775ndash784 2003

[10] D Wen and Y Ding ldquoFormulation of nanofluids for naturalconvective heat transfer applicationsrdquo International Journal ofHeat and Fluid Flow vol 26 no 6 pp 855ndash864 2005

[11] K S Hwang J-H Lee and S P Jang ldquoBuoyancy-drivenheat transfer of water-based Al

2O3nanofluids in a rectangular

cavityrdquo International Journal of Heat and Mass Transfer vol 50no 19-20 pp 4003ndash4010 2007

[12] X-QWang andA SMujumdar ldquoA review on nanofluidsmdashpartII experiments and applicationsrdquo Brazilian Journal of ChemicalEngineering vol 25 no 4 pp 631ndash648 2008

[13] D M Christopher and B Wang ldquoPrandtl number effects forMarangoni convection over a flat surfacerdquo International Journalof Thermal Sciences vol 40 no 6 pp 564ndash570 2001

[14] R Tadmor ldquoMarangoni flow revisitedrdquo Journal of Colloid andInterface Science vol 332 no 2 pp 451ndash454 2009

[15] C Golia and A Viviani ldquoNon isobaric boundary layers relatedto Marangoni flowsrdquo Meccanica vol 21 no 4 pp 200ndash2041986

[16] I Pop A Postelnicu andTGrosan ldquoThermosolutalMarangoniforced convection boundary layersrdquo Meccanica vol 36 no 5pp 555ndash571 2001

[17] A J Chamkha I Pop andH S Takhar ldquoMarangonimixed con-vection boundary layer flowrdquoMeccanica vol 41 no 2 pp 219ndash232 2006

[18] R A Hamid N M Arifin R Nazar F M Ali and I PopldquoDual Solutions on thermosolutalMarangoni forced convectionboundary layer with suction and injectionrdquo MathematicalProblems in Engineering vol 2011 Article ID 875754 19 pages2011

[19] N Mat N M Arifin R Nazar F Ismail and I Pop ldquoRadiationeffects on Marangoni convection boundary layer over a perme-able surfacerdquoMeccanica vol 48 pp 83ndash89 2013

[20] R K Tiwari and M K Das ldquoHeat transfer augmentation in atwo-sided lid-driven differentially heated square cavity utilizingnanofluidsrdquo International Journal of Heat and Mass Transfervol 50 no 9-10 pp 2002ndash2018 2007

[21] N M Arifin R Nazar and I Pop ldquoNon-Isobaric Marangoniboundary layer flow for Cu Al

2O3and TiO

2nanoparticles in a

water based fluidrdquoMeccanica vol 46 no 4 pp 833ndash843 2011[22] A Remeli NMArifin RNazar F Ismail and I Pop ldquoMarang-

oni-driven boundary layer flow in a nanofluid with suction andinjectionrdquo World Applied Sciences Journal vol 17 pp 21ndash262012

[23] N A A Mat N M Arifin R Nazar and F Ismail ldquoRadiationeffect onMarangoni convection boundary layer flow of a nano-fluidrdquoMathematical Sciences vol 6 article 21 2012

[24] A Al-Mudhaf and A J Chamkha ldquoSimilarity solutions forMHD thermosolutal Marangoni convection over a flat surfacein the presence of heat generation or absorption effectsrdquo Heatand Mass Transfer vol 42 no 2 pp 112ndash121 2005

[25] E Magyari and A J Chamkha ldquoExact analytical results forthe thermosolutal MHDMarangoni boundary layersrdquo Interna-tional Journal of Thermal Sciences vol 47 no 7 pp 848ndash8572008

[26] R A Hamid N M Arifin R Nazar and F M Ali ldquoRadiationeffects onMarangoni convection over a flat surface with suctionand injectionrdquoMalaysian Journal of Mathematical Sciences vol5 no 1 pp 13ndash25 2011

[27] R AHamidWMK AW Zaimi NMArifin N A A Bakarand B Bidin ldquoThermal-diffusion and diffusion-thermo effectson MHD thermosolutal Marangoni convection boundary layerflow over a permeable surfacerdquo Journal of Applied Mathematicsvol 2012 Article ID 750939 14 pages 2012

[28] E Abu-Nada ldquoApplication of nanofluids for heat transferenhancement of separated flows encountered in a backwardfacing steprdquo International Journal of Heat and Fluid Flow vol29 no 1 pp 242ndash249 2008

[29] H F Oztop and E Abu-Nada ldquoNumerical study of natural con-vection in partially heated rectangular enclosures filled withnanofluidsrdquo International Journal of Heat and Fluid Flow vol29 no 5 pp 1326ndash1336 2008

[30] E Abu-Nada and H F Oztop ldquoEffects of inclination angle onnatural convection in enclosures filled with Cu-water nano-fluidrdquo International Journal of Heat and Fluid Flow vol 30 no4 pp 669ndash678 2009


[31] M Muthtamilselvan P Kandaswamy and J Lee ldquoHeat transferenhancement of copper-water nanofluids in a lid-driven enclo-surerdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 6 pp 1501ndash1510 2010

[32] N Bachok A Ishak R Nazar and N Senu ldquoStagnation-pointflow over a permeable stretchingshrinking sheet in a copper-water nanofluidrdquo Boundary Value Problems vol 2013 article 392013

[33] C Golia and A Viviani ldquoMarangoni buoyant boundary layersrdquoAerotecnica Missili e Spazio vol 64 pp 29ndash35 1985

[34] B S Dandapat B Santra and H I Andersson ldquoThermo-capillarity in a liquid film on an unsteady stretching surfacerdquoInternational Journal of Heat and Mass Transfer vol 46 no 16pp 3009ndash3015 2003

[35] A Aziz ldquoA similarity solution for laminar thermal boundarylayer over a flat plate with a convective surface boundary con-ditionrdquo Communications in Nonlinear Science and NumericalSimulation vol 14 no 4 pp 1064ndash1068 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


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Journal of


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OptimizationJournal of


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International Journal of


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Journal of


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



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Marangoni forced convection boundary layer flow by Popet al [16] when the wall is permeable Very recently Matel al [19] discussed the radiation effects on the problem ofMarangoni boundary layer with permeable surface On theother hand nanofluid equationsmodel as proposed by Tiwariand Das [20] has been used by Arifin et al [21] for theMarangoni boundary layer problembyGolia andViviani [15]They found that the numerical results also indicate that forboth a regular fluid (120593 = 0) and a nanofluid (120593 = 0) dualsolutions exist when 120573 lt 05 These dual solutions were notdiscussed by Golia and Viviani [15] This problem has beenextended byRemeli et al [22] to the problemwith suction andinjection effects Mat et al [23] also extended the problem ofMarangoni boundary layer in a nanofluid by Arifin et al [21]to the radiation effect

It is worth mentioning that Christopher and Wang [13]considered the Marangoni boundary layer over a flat platewhere the term119906

119890(119909) which is the velocity of the external flow

in Golia and Viviani [15] has been neglected The similaritysolutions of the Christopher and Wang [13] problem arealso different from the Golia and Viviani [15] problem Theproblem of Christopher and Wang [13] has been extendedby several researchers such as Al-Mudhaf and Chamkha [24]where they have presented the similarity solutions for MHDMarangoni convection in the presence of heat generation orabsorption effects and Magyari and Chamkha [25] reportedthe exact analytical solutions of thermosolutal Marangoniflows in the presence of temperature-dependent volumetricheat sourcesinks as well as of a first-order chemical reactionRecently Hamid et al [26] studied the two-dimensionalMarangoni convection flow past a flat plate in the presenceof thermal radiation suction and injection effects FurtherMHD thermosolutal Marangoni convection boundary layerover a flat surface considering the effects of the thermal dif-fusion and diffusion-thermo with fluid suction and injectionhas been examined by Hamid et al [27]

It should be highlighted that the present paper presentsa similarity solution for the steady Marangoni convectionboundary layer flow over a static semi-infinite flat plate dueto an imposed temperature gradient in a nanofluid whichextends the problem by Christopher and Wang [13] to thecase of nanofluid The nanofluid equations model proposedby Tiwari and Das [20] has been used This model has beenvery successfully used in several papers [21 28ndash32] Thuswe wish to highlight that this present study is original andall the results are new To the best of our knowledge thepresent problem has not been considered before The studyof nanofluid is still at its early stage and it seems difficult tohave a precise idea on the way the use of nanoparticles acts inheat transfer A clear picture on the boundary layer flows ofnanofluid is yet to emerge

2 Problem Formulation

We consider the steady two-dimensional boundary layer flowpast a semi-infinite flat plate in a water-based nanofluidcontaining different types of nanoparticles namely copper(Cu) alumina (Al

2O3) and titania (TiO

2) with Marangoni

effectsThe nanofluid is assumed incompressible and the flowis assumed to be laminar It is also assumed that the base fluid(ie water) and the nanoparticles are in thermal equilibriumand no slip occurs between them The thermophysical prop-erties of the nanofluids are given in Table 1 (see Oztop andAbu-Nada [29]) Further we consider a Cartesian coordinatesystem (119909 119910) where 119909 and 119910 are the coordinates measuredalong the plate and normal to it respectively and the flowtakes place at 119910 ge 0 It is also assumed that the temperatureof the plate is 119879

119908(119909) and that of the ambient nanofluid is 119879

infin

Following [15ndash17 25 33 34] the surface tension 120590 is assumedto vary linearly with temperature as follow

120590 = 1205900[1 minus 120574 (119879 minus 119879

0)] (1)

where 1205900and 119879

0are the characteristics surface tension and

temperature respectively and we assume that 1198790

equiv 119879infin

Equation (1) is a commonly made assumption [34] For mostliquids the surface tension decreases with temperature thatis 120574 is a positive fluid property

The steady boundary layer equations for a nanofluid inthe coordinates 119909 and 119910 are ([13 20])

120597119906

120597119909

+

120597V120597119910

= 0 (2)

119906

120597119906

120597119909

+ V120597119906

120597119910

=

120583nf120588nf

120597

2119906

120597119910

2 (3)

119906

120597119879

120597119909

+ V120597119879

120597119910

= 120572nf120597

2119879

120597119910

2(4)

subject to the boundary conditions

V = 0 119879 = 1198790+ 119860119909

119898+1 120583nf

120597119906

120597119910

=

120597120590

120597119879

120597119879

120597119909

at119910 = 0

119906 = 0 119879 = 119879infin

as 119910 997888rarr infin

(5)

Here 119906 and V are the velocity components along the 119909- and119910-axes respectively 119879 is the temperature of the nanofluidmis the constant exponent of the temperature120572nf is the thermaldiffusivity of the nanofluid 120588nf is the effective density of thenanofluid 119896nf is the effective thermal conductivity of thenanofluid and 120583nf is the effective viscosity of the nanofluidwhich are given by

120572nf =119896nf

(120588119862119901)

nf

120588nf = (1 minus 120601) 120588119891+ 120601120588119904

120583nf =120583119891

(1 minus 120601)

25 (120588119862

119901)

nf= (1 minus 120601) (120588119862

119901)

119891+ 120601(120588119862

119901)

119904

119896nf119896119891

=

(119896119904+ 2119896119891) minus 2120601 (119896

119891minus 119896119904)

(119896119904+ 2119896119891) + 120601 (119896

119891minus 119896119904)

(6)

where 120601 is the nanoparticle volume fraction 120588119891

is thereference density of the fluid fraction 120588

119904is the reference




119904is the




120595 = 1198621119909

(2+119898)3119891 (120578) 120579 (120578) =


119860119909

1+119898

120578 = 1198622119909

(119898minus1)3119910

(7)



2are constants with

119860 1198621 and 119862

2given by

119860 =

Δ119879

119871

119898+1 119862

1=

3radic

1205900120574119860120583119891

120588

2

119891

1198622=

3radic

1205900120574119860120588119891

120583

2

119891

(8)


1

(1 minus 120601)

25

(1 minus 120601 + 120601120588119904120588119891)

119891

101584010158401015840+

2 + 119898

3

119891119891

10158401015840

minus

1 + 2119898

3

119891

10158402= 0

1

Pr119896nf119896119891

(1 minus 120601 + 120601(120588119862119901)

119904(120588119862119901)

119891)

120579

10158401015840+

2 + 119898

3

119891120579

minus (1 + 119898)119891

1015840120579 = 0

(9)


119891 (0) = 0

1

(1 minus 120601)

25119891

10158401015840(0) = minus1 120579 (0) = 1

119891

1015840(infin) = 0 120579 (infin) = 0

(10)


as

119906119908(119909) =

3radic

(1205900120574119860)

2

120588119891120583119891

119909

(1+2119898)3119891

1015840(0)

(11)


is defined as

Nu119909=

119909119902119908(119909)

119896119891 [

119879 (119909 0) minus 119879 (119909infin)]

(12)


it is given by

119902119908(119909) = minus119896nf(

120597119879

120597119910

)

119910=0

(13)


Nu119909= minus

119896nf119896119891

1198622119909

(2+119898)3120579

1015840(0) (14)



Nu119871= minus

6 + 3119898

5 + 4119898

119896nf119896119891

Ma13119871

Prminus131205791015840 (0) (15)


Ma119871=

120590119879119860119871

2+119898

120583119891120572119891

=

120590119879Δ119879119871

120583119891120572119891

(16)


= int

infin

0

120588119891119906 119889119910 =

3radic1205900120574120588119891120583119891119909

(2+119898)3119891 (infin) (17)



infin) that is


















119862119901(JkgK) 4179 385 765 6862

120588 (kgm3) 9971 8933 3970 4250119896 (WmK) 0613 400 40 89538

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

120578

Cu TiO2 Al2O3

f998400 (120578)









2O3 and TiO

2)


2O3 TiO

2) It





1015840(0) with 119898

120579(120578

)

120578

TiO2 Al2O3 Cu

1

08

06

04

02

0

0 1 2 3


Cu TiO2 Al2O3

14

13

12

11

10

09

08

07

0 005 010 015 020

120601

f998400 (0)






120601

Cu Al2O3 TiO2

32

30

28

26

24

22

20

18

16

14

0 005 010 015 020


f998400 (0)

120601 = 01

120601 = 00

m

6

5

4

3

2

1

0

minus1 0 1 2

Al2O3 TiO2 Cu





2O3 and TiO



120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

f998400 (120578)


120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

f998400 (120578)



120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




4 Conclusion


f998400 (120578)

120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0

1 2 3 4 5 6 7 8



)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




2O3 and TiO

2) in the base








2)


and Cu

Acknowledgments


References
























2O3and TiO

2nanoparticles in a
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119904is the




120595 = 1198621119909

(2+119898)3119891 (120578) 120579 (120578) =


119860119909

1+119898

120578 = 1198622119909

(119898minus1)3119910

(7)



2are constants with

119860 1198621 and 119862

2given by

119860 =

Δ119879

119871

119898+1 119862

1=

3radic

1205900120574119860120583119891

120588

2

119891

1198622=

3radic

1205900120574119860120588119891

120583

2

119891

(8)


1

(1 minus 120601)

25

(1 minus 120601 + 120601120588119904120588119891)

119891

101584010158401015840+

2 + 119898

3

119891119891

10158401015840

minus

1 + 2119898

3

119891

10158402= 0

1

Pr119896nf119896119891

(1 minus 120601 + 120601(120588119862119901)

119904(120588119862119901)

119891)

120579

10158401015840+

2 + 119898

3

119891120579

minus (1 + 119898)119891

1015840120579 = 0

(9)


119891 (0) = 0

1

(1 minus 120601)

25119891

10158401015840(0) = minus1 120579 (0) = 1

119891

1015840(infin) = 0 120579 (infin) = 0

(10)


as

119906119908(119909) =

3radic

(1205900120574119860)

2

120588119891120583119891

119909

(1+2119898)3119891

1015840(0)

(11)


is defined as

Nu119909=

119909119902119908(119909)

119896119891 [

119879 (119909 0) minus 119879 (119909infin)]

(12)


it is given by

119902119908(119909) = minus119896nf(

120597119879

120597119910

)

119910=0

(13)


Nu119909= minus

119896nf119896119891

1198622119909

(2+119898)3120579

1015840(0) (14)



Nu119871= minus

6 + 3119898

5 + 4119898

119896nf119896119891

Ma13119871

Prminus131205791015840 (0) (15)


Ma119871=

120590119879119860119871

2+119898

120583119891120572119891

=

120590119879Δ119879119871

120583119891120572119891

(16)


= int

infin

0

120588119891119906 119889119910 =

3radic1205900120574120588119891120583119891119909

(2+119898)3119891 (infin) (17)



infin) that is


















119862119901(JkgK) 4179 385 765 6862

120588 (kgm3) 9971 8933 3970 4250119896 (WmK) 0613 400 40 89538

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

120578

Cu TiO2 Al2O3

f998400 (120578)









2O3 and TiO

2)


2O3 TiO

2) It





1015840(0) with 119898

120579(120578

)

120578

TiO2 Al2O3 Cu

1

08

06

04

02

0

0 1 2 3


Cu TiO2 Al2O3

14

13

12

11

10

09

08

07

0 005 010 015 020

120601

f998400 (0)






120601

Cu Al2O3 TiO2

32

30

28

26

24

22

20

18

16

14

0 005 010 015 020


f998400 (0)

120601 = 01

120601 = 00

m

6

5

4

3

2

1

0

minus1 0 1 2

Al2O3 TiO2 Cu





2O3 and TiO



120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

f998400 (120578)


120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

f998400 (120578)



120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




4 Conclusion


f998400 (120578)

120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0

1 2 3 4 5 6 7 8



)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




2O3 and TiO

2) in the base








2)


and Cu

Acknowledgments


References
























2O3and TiO

2nanoparticles in a
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119862119901(JkgK) 4179 385 765 6862

120588 (kgm3) 9971 8933 3970 4250119896 (WmK) 0613 400 40 89538

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

120578

Cu TiO2 Al2O3

f998400 (120578)









2O3 and TiO

2)


2O3 TiO

2) It





1015840(0) with 119898

120579(120578

)

120578

TiO2 Al2O3 Cu

1

08

06

04

02

0

0 1 2 3


Cu TiO2 Al2O3

14

13

12

11

10

09

08

07

0 005 010 015 020

120601

f998400 (0)






120601

Cu Al2O3 TiO2

32

30

28

26

24

22

20

18

16

14

0 005 010 015 020


f998400 (0)

120601 = 01

120601 = 00

m

6

5

4

3

2

1

0

minus1 0 1 2

Al2O3 TiO2 Cu





2O3 and TiO



120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

f998400 (120578)


120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

f998400 (120578)



120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




4 Conclusion


f998400 (120578)

120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0

1 2 3 4 5 6 7 8



)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




2O3 and TiO

2) in the base








2)


and Cu

Acknowledgments


References
























2O3and TiO

2nanoparticles in a
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120601

Cu Al2O3 TiO2

32

30

28

26

24

22

20

18

16

14

0 005 010 015 020


f998400 (0)

120601 = 01

120601 = 00

m

6

5

4

3

2

1

0

minus1 0 1 2

Al2O3 TiO2 Cu





2O3 and TiO



120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0 1 2 3 4 5 6 7 8

f998400 (120578)


120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

f998400 (120578)



120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




4 Conclusion


f998400 (120578)

120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0

1 2 3 4 5 6 7 8



)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




2O3 and TiO

2) in the base








2)


and Cu

Acknowledgments


References
























2O3and TiO

2nanoparticles in a
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0 1 2 3 4 5 6 7 8

f998400 (120578)



120579(120578

)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




4 Conclusion


f998400 (120578)

120601 = 0 005 01 015 02

120578

12

1

08

06

04

02

0

0

1 2 3 4 5 6 7 8



)

120601 = 0 005 01 015 02

120578

1

08

06

04

02

0

0 1 2 3




2O3 and TiO

2) in the base








2)


and Cu

Acknowledgments


References
























2O3and TiO

2nanoparticles in a
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















2)


and Cu

Acknowledgments


References
























2O3and TiO

2nanoparticles in a
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article similarity solution of marangoni...

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