research article the effect of heat transfer on ... - hindawi

Hindawi Publishing CorporationInternational Journal of Engineering MathematicsVolume 2013 Article ID 581507 6 pageshttpdxdoiorg1011552013581507

Research ArticleThe Effect of Heat Transfer on MHD Marangoni Boundary LayerFlow Past a Flat Plate in Nanofluid

D R V S R K Sastry1 A S N Murti2 and T Poorna Kantha2

1 Department of Mathematics Aditya Engineering College Surampalem Ardhra Pradesh 533437 India2Deptartment of Engineering Mathematics GITAM University Visakhapatnam Ardhra Pradesh 530023 India

Correspondence should be addressed to D R V S R K Sastry sastry devyahoocoin

Received 30 March 2013 Revised 8 May 2013 Accepted 8 May 2013

Academic Editor Yurong Liu

Copyright copy 2013 D R V S R K Sastry 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 heat transfer on the Marangoni convection boundary layer flow in an electrically conducting nanofluid is studiedSimilarity transformations are used to transform the set of governing partial differential equations of the flow into a set of nonlinearordinary differential equations Numerical solutions of the similarity equations are then solved through the MATLAB ldquobvp4crdquofunction Different nanoparticles like Cu Al

2O3 and TiO

2are taken into consideration with water as base fluid The velocity and

temperature profiles are shown in graphs Also the effects of the Prandtl number and solid volume fraction on heat transfer arediscussed

1 Introduction

The convection induced by the variations of the surfacetension gradients is known as theMarangoni convectionThisconvection has received great consideration in view of itsapplication in the fields of welding and crystal growth Alsothis convection is necessary to stabilize the soap films anddrying silicon wafers During the study of the existence ofthe steady dissipative layers which occur along the liquid-liquid or liquid-gas interfaces Napolitano [1] first calledthe boundary layer as the Marangoni boundary layer Manyresearchers such as Okano et al [2] Christopher and Wang[3] Pop et al [4] and Magyari and Chamkha [5] haveinvestigated theMarangoni convection in various geometriesAl-Mudhaf andChamkha [6] obtained the similarity solutionfor the MHD thermosolutal Marangoni convection over aflat surface in the presence of heat generation or absorptionwith fluid suction and injection Chen [7] investigated theflow and the heat transfer characteristics on the forcedconvection in a power law liquid film under an appliedMarangoni convection over a stretching sheet In recent yearsthe study on convective transport of nanofluids has becomeone of the popular topics of interest Nanotechnology takes

important part for the development of high performancecompact and cost-effective liquid cooling systemsMoreovernanofluids have effective applications inmany industries suchas electronics transportation biomedical and many moreNanotechnology has been an ongoing topic of discussionin public health as some of the researchers claimed thatnanoparticles could present possible dangers in health andenvironment Jang and Choi [8] have introduced nanosizedparticle in a base fluid which is also termed nanofluidfor the first time Arifin et al [9] have examined theinfluence of nanoparticles on the Marangoni boundary layerflow using a model proposed by Tiwari and Das [10] Anextendedworkwas done by Buongiorno [11] Daungthongsukand Wongwises [12] Trisaksri and Wongwises [13] Wangand Mujumdar [14] and Kakac and Pramuanjaroenkij [15]Recently Hamid et al [16] studied the radiation effectson the Marangoni boundary layer flow past a flat platein nanofluid In the present paper we study a numericalsolution of MHD heat transfer problem in nanofluid withnanoparticles Cu Al

2O3 and TiO

2 We also observed the

effects of the Prandtl number and solid volume fractionon the Nusselt number The results are shown graphi-cally

2 International Journal of Engineering Mathematics

2 Mathematical Formulation

Consider a steady two-dimensional Marangoni boundarylayer flow past a permeable flat plate in a water-basednanofluid containing different types of nano particles like Cu(Copper) Al

2O3(Aluminium Oxide) and TiO

2(Titanium

dioxide) Assume that the fluid is incompressible and theflow is laminar Also it is assumed that the base fluid andthe particles are in thermal equilibrium and no slip occursbetween them The thermophysical properties of nanoparti-cles are given in the Table 1 Further we consider a Cartesiancoordinate system (119909 119910) where 119909 and 119910 are the coordinatesmeasured along the plate and normal to it respectively andthe flow takes place at 119910 ge 0 Assume that the temperature ofthe plate is 119879

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

infin

We further assume that the surface tension 120590 is to varylinearly with temperature as

120590 = 1205900[1 minus 120574 (119879 minus 119879

infin)] (1)

where 1205900is the surface tension at the interface and 120574 is the

rate of change of surface tension with temperature (a positivefluid property) It is also assumed that a uniform magneticfield 119867

0is imposed in the direction normal to the surface

(Figure 1) Then the steady state boundary layer equationsfor a nanofluid in the Cartesian coordinates are given by

120597119906

120597119909+120597V

120597119910= 0 (2)

119906120597119906

120597119909+ V120597119906

120597119910=120583nf120588nf

1205972119906

1205971199102minus120590

120588nf1198672

0119906 (3)

119906120597119879

120597119909+ V120597119879

120597119910= 120572nf

1205972119879

1205971199102 (4)

together with the boundary conditions

V = 0 119879 = 119879infin+ 1198861199092

120583nf120597119906

120597119910=120597120590

120597119879

120597119879

120597119909at 119910 = 0

119906 = 0 119879 = 119879infin

as 119910 997888rarr infin

(5)

Here 119906 and V are the components of velocity along the 119909- and119910-axes respectively 119879 is the temperature 120572nf is the thermaldiffusivity 120588nf is the effective density 119896nf is the effectivethermal conductivity and 120583nf is the effective viscosity ofthe nanofluid Moreover 119886 is the coefficient of temperaturegradient Consider the following

120572nf =119896nf

(120588119862119901)nf

120588nf = (1 minus 120601) 120588119891 + 120601120588119904

120583nf =120583119891

(1 minus 120601)25

Table 1Thermophysical properties of pure water and nanoparticles(Oztop and Abu-Nada [17])

Physical property Pure water Cu Al2O3 TiO2

120588 (kgm3) 9971 8933 3970 4250119862119901(JkgK) 4179 385 765 6862

119896 (WmK) 0613 401 40 89538

Interface Transversemagnetic field (1198670)

119884

119883

Δ119879 Δ120590

Figure 1 Schematic diagram of the problem

(120588119862119901)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)

119886 =Δ119879

1198712

(6)

where 120601 is the solid volume fraction of the nanofluid 120588119891is

the reference density of the fluid fraction 120588119904is the reference

density 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 nanofluid 119871 is the length of the surface andΔ119879 is the constant characteristic temperature

A similarity solution of (2)ndash(5) is obtained by introducingan independent variable 120578 and a dependent variable 119891 interms of the stream function 120595 as

120595 = 1198621119909119891 (120578) 120578 = 119862

2119910 (7)

with 119906 = 120597120595120597119910 and V = minus120597120595120597119909The constants 119862

1and 119862

2are given by

1198621= (

1205900120574119886120583119891

1205882119891

)

13

1198622= (

1205900120574119886120588119891

1205832119891

)

13

(8)

Further the dimensionless temperature 120579 is given by

120579 (120578) =119879 minus 119879infin

1198861199092 (9)

International Journal of Engineering Mathematics 3

Substituting (6) (7) (8) and (9) into (3) and (4) we obtain aset of nonlinear ordinary differential equations

119891101584010158401015840= (1 minus 120601)

25

[(1 minus 120601) + 120601120588119904

120588119891

] (11989110158402minus 11989111989110158401015840)

+119872(119875119903)13

(119896119891)13

(119862119901119891)minus13

(120588119891)minus23

1198911015840

12057910158401015840=[(1 minus 120601) + 120601 (120588

119904119862119901119904120588119891119862119901119891)]

119896nf119896119891(21198911015840120579 minus 119891120579

1015840) 119875119903

(10)

and the boundary conditions become

119891 (0) = 0 120579 (0) = 11

(1 minus 120601)2511989110158401015840(0) = minus2

1198911015840(infin) = 0 120579 (infin) = 0

(11)

where the magnetic field parameter119872 = 120590131198672

0(120574119886)23

Also one can define the surface velocity and the localNusselt number respectively as

119906119908(119909) = 3radic

(1205900120574119886)2

120588119891120583119891

1199091198911015840(0) (12)

Nu119909=

119909119902119908(119909)

119896119891 [119879 (119909 0) minus 119879 (119909infin)]

(13)

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

is given by

119902119908(119909) = minus119896nf(

120597119879

120597119910)119910=0

(14)

Using the above nondimension quantities one can obtain thelocal Nusselt number as

Nu119909= minus

119896nf119896119891

11986221205791015840(0) (15)

Based on the average temperature difference between thetemperature of the surface and the ambient fluid temperaturewe define

Nu119871= minus

119896nf119896119891

(Ma119871

Pr)13

1205791015840(0) (16)

where Ma119871is the Marangoni based on L and is defined as

Ma119871=(120597120590120597119879) (Δ119879) 119871

120572119891120583119891

(17)

3 Results and Discussion

Numerical solutions were obtained for the effect of thePrandtl number and solid volume fraction on the Marangoniheat transfer in a nanofluid In this paper we considered

0 05 1 15 2 25 3 35 40

02

04

06

08

1 Pr = 62

119872 = 0 100 200 500

120578

120601 = 01

Cu-water

119891(120578

)

Figure 2 Velocity profile for Cu nanoparticles for various119872

Pr = 62120601 = 01

119872 = 0 100 200 500

0 05 1 15 2 25 3 35 4120578

0

02

12

04

06

08

1

119891(120578

)

Al2O3-water

Figure 3 Velocity profile for Al2O3nanoparticles for various119872

three different nanoparticles whose thermophysical proper-ties were given in Table 1 The nonlinear ordinary differentialequations (10) subject to the boundary conditions (11) weresolved numerically using the MATLAB ldquobvp4crdquo routine Weconsidered the range of nanoparticles volume fraction 120601 as0 le 0 le 03 and the Prandtl number Pr as 2 le Pr le 8 (for thebase fluid (water) Pr = 62) The influences of the magneticfield parameter (119872) the nanoparticles volume fraction (120601)on velocity and temperature and also the influence of thePrandtl number (Pr) and solid volume fraction (120601) on theNusselt number are presented in graphs

Figures 2 3 and 4 display the velocity profiles andFigures 5 6 and 7 display the temperature profiles of Cu-water Al

2O3-water andTiO

2-water respectively for different

values of magnetic field parameter 119872 It is observed fromthe figures that the velocity in the boundary layer decreasesand temperature increases as the Magnetic field parameterincreases this is due to the resistive force called the Lorentzforce which is produced by the inducedmagnetic fieldwithinthe boundary layer


Pr = 62120601 = 01

119872 = 0 100 200 500

0

02

12

04

06

08

1

119891(120578

)

0 05 1 15 2 25 3 35 4120578

TiO2-water

Figure 4 Velocity profile for TiO2nanoparticles for various119872

0 02 04 06 08 1 12 14 16 18 20

01

02

03

04

05

06

07

08

09

1Cu-water

Pr = 62120601 = 01

119872 = 0 100 200 500

120578

120579(120578)

Figure 5 Temperature profile for Cu nanoparticles for various119872

Pr = 62120601 = 01

119872 = 0 100 200 500

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

Al2O3-water

Figure 6 Temperature profile for Al2O3nanoparticles for various

119872

Pr = 62120601 = 01

119872 = 0 100 200 500

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

TiO2-water

Figure 7 Temperature profile for TiO2nanoparticles for various119872

0 05 1 15 2 25 3 35 4 45 50

05

1

15

Cu

119872 = 10

Pr = 62

120601 = 0 01 02 03119891(120578

)

120578

Al2O3

TiO2

Figure 8 Velocity profile for different 120601

Figure 8 depicts the influence of volume fraction onthe velocity profile of the nanofluid particles It is observednear the wall that velocity decreases with an increase in thevolume fraction 120601 Also it is observed that the velocity ofTiO2nanoparticles is higher than that of Cu nanoparticles

From Figure 9 it is clear that an increase in the valueof volume fraction enhances the temperature profile andCu nanoparticles exhibit more temperature than that ofthe other nanoparticles It is also known from Figure 10that temperature decreases with an increase in the Prandtlnumber This is because of a decrease in thermal diffusivitywith an increase in the Prandtl number (Pr)

Figures 11 and 12 depict the influence of the Prandtlnumber and volume fraction on heat transfer respectivelyIt is observed that the Nusselt number increases with an


119872 = 10

Pr = 62

120601 = 0 01 02 03

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

CuAl2O3

TiO2

Figure 9 Temperature profile for different 120601

Pr = 2 4 6

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

0 05 1 15 2 25 3120578

119872 = 10

120601 = 01

CuAl2O3

TiO2

Figure 10 Temperature profile for different Pr

increase in Prandtl number and decreases with an increasein the volume fraction

From Table 2 it is observed that skin friction decreaseswith the increase in the volume fraction

4 Conclusion

In the present paper we studied the effect of heat transferon the Marangoni boundary layer flow past a flat plate innanofluids in presence of transverse magnetic field With thesimilarity transformation the governing equations of motiontogether with boundary conditions were transformed to a set

2 3 4 5 616

2

24

28

32

36

Pr

119872 = 10

120601 = 01

minus1205791(0)

Cu Al2O3 TiO2

Figure 11 Heat transfer effect against the Prandtl number

0 005 01 015 02

25

3

35

4

45

5

Cu Al2O3 TiO2

119872 = 10

Pr = 62

minus1205791(0)

120601

Figure 12 Heat transfer effect against the volume fraction

Table 2The numeric values of skin friction and theNusselt numberfor various values of 120601

120601 11989110158401015840(0) minus1205791015840(0)

0 2 46842958601 153686694 34036131502 11448668 258644151

of nonlinear ordinary differential equations The numericalsolutions are then obtained for these equations by the helpof MATLAB ldquobvp4crdquo programming tool Different types ofnanoparticles like Cu Al

2O3 and TiO

2were taken into

consideration with H2O as base fluidThe effects of magnetic

field parameter 119872 solid volume fraction of the nanofluid120601 on the velocity and temperature fields for different nanoparticles and the Prandtl number Pr on temperature fieldwere plotted and analyzed Also the effects of the Prandtlnumber Pr and solid volume fraction 120601 on local the Nusseltnumber for the different nanoparticles were discussed for afixed value of magnetic field parameter 119872 It is found thatthe inclusion of the magnetic field parameter on the flow


increased the temperature and decreased the velocity fieldsin all types of nanofluids A similar profile was observed onthe inclusion of solid volume fraction of the nanoparticlesIt was noted that presence of the Prandtl number reduced thetemperature field Also it was observed that for a fixed Prandtlnumber and other parameters the rate of heat transfer ismore in TiO

2-H2O

References

[1] L G Napolitano ldquoMicrogravity fluid dynamicsrdquo in Proceedingsof the 2nd Levitch Conference Washington DC USA 1978

[2] Y Okano M Itoh and A Hirata ldquoNatural and marangoniconvections in a two-dimensional rectangular open boatrdquoJournal of Chemical Engineering of Japan vol 22 no 3 pp 275ndash281 1989

[3] 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

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

[5] E Magyari and A J Chamkha ldquoExact analytical solutions forthermosolutal Marangoni convection in the presence of heatand mass generation or consumptionrdquoHeat and Mass Transfervol 43 no 9 pp 965ndash974 2007

[6] 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

[7] C H Chen ldquoMarangoni effects on forced convection of power-law liquids in a thin film over a stretching surfacerdquo PhysicsLetters A vol 370 no 1 pp 51ndash57 2007

[8] S P Jang and S U S Choi ldquoRole of Brownian motion in theenhanced thermal conductivity of nanofluidsrdquo Applied PhysicsLetters vol 84 no 21 pp 4316ndash4318 2004

[9] NM Arifin R Nazar and I Pop ldquoMarangoni driven boundarylayer flow past a flat plate in nanofluid with suctioninjectionrdquoin Proceedings of the International Conference on MathematicalScience (ICMS rsquo10) pp 94ndash99 Bolu Turkey November 2010

[10] 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

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

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

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

[14] 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

[15] 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

[16] R A Hamid N M Arifin R M Nazar and I Pop ldquoRadiationeffects on Marangoni boundary layer flow past a flat plate in

nanofluidrdquo in Proceedings of the International MultiConferenceof Engineers and Computer Scientists 2011 (IMECS rsquo11) vol 3 pp1260ndash1263 Hong Kong March 2011

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

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


2 Mathematical Formulation

Consider a steady two-dimensional Marangoni boundarylayer flow past a permeable flat plate in a water-basednanofluid containing different types of nano particles like Cu(Copper) Al

2O3(Aluminium Oxide) and TiO

2(Titanium

dioxide) Assume that the fluid is incompressible and theflow is laminar Also it is assumed that the base fluid andthe particles are in thermal equilibrium and no slip occursbetween them The thermophysical properties of nanoparti-cles are given in the Table 1 Further we consider a Cartesiancoordinate system (119909 119910) where 119909 and 119910 are the coordinatesmeasured along the plate and normal to it respectively andthe flow takes place at 119910 ge 0 Assume that the temperature ofthe plate is 119879

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

infin

We further assume that the surface tension 120590 is to varylinearly with temperature as

120590 = 1205900[1 minus 120574 (119879 minus 119879

infin)] (1)

where 1205900is the surface tension at the interface and 120574 is the

rate of change of surface tension with temperature (a positivefluid property) It is also assumed that a uniform magneticfield 119867

0is imposed in the direction normal to the surface

(Figure 1) Then the steady state boundary layer equationsfor a nanofluid in the Cartesian coordinates are given by

120597119906

120597119909+120597V

120597119910= 0 (2)

119906120597119906

120597119909+ V120597119906

120597119910=120583nf120588nf

1205972119906

1205971199102minus120590

120588nf1198672

0119906 (3)

119906120597119879

120597119909+ V120597119879

120597119910= 120572nf

1205972119879

1205971199102 (4)

together with the boundary conditions

V = 0 119879 = 119879infin+ 1198861199092

120583nf120597119906

120597119910=120597120590

120597119879

120597119879

120597119909at 119910 = 0

119906 = 0 119879 = 119879infin

as 119910 997888rarr infin

(5)

Here 119906 and V are the components of velocity along the 119909- and119910-axes respectively 119879 is the temperature 120572nf is the thermaldiffusivity 120588nf is the effective density 119896nf is the effectivethermal conductivity and 120583nf is the effective viscosity ofthe nanofluid Moreover 119886 is the coefficient of temperaturegradient Consider the following

120572nf =119896nf

(120588119862119901)nf

120588nf = (1 minus 120601) 120588119891 + 120601120588119904

120583nf =120583119891

(1 minus 120601)25

Table 1Thermophysical properties of pure water and nanoparticles(Oztop and Abu-Nada [17])

Physical property Pure water Cu Al2O3 TiO2

120588 (kgm3) 9971 8933 3970 4250119862119901(JkgK) 4179 385 765 6862

119896 (WmK) 0613 401 40 89538

Interface Transversemagnetic field (1198670)

119884

119883

Δ119879 Δ120590

Figure 1 Schematic diagram of the problem

(120588119862119901)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)

119886 =Δ119879

1198712

(6)

where 120601 is the solid volume fraction of the nanofluid 120588119891is

the reference density of the fluid fraction 120588119904is the reference

density 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 nanofluid 119871 is the length of the surface andΔ119879 is the constant characteristic temperature

A similarity solution of (2)ndash(5) is obtained by introducingan independent variable 120578 and a dependent variable 119891 interms of the stream function 120595 as

120595 = 1198621119909119891 (120578) 120578 = 119862

2119910 (7)

with 119906 = 120597120595120597119910 and V = minus120597120595120597119909The constants 119862

1and 119862

2are given by

1198621= (

1205900120574119886120583119891

1205882119891

)

13

1198622= (

1205900120574119886120588119891

1205832119891

)

13

(8)

Further the dimensionless temperature 120579 is given by

120579 (120578) =119879 minus 119879infin

1198861199092 (9)



119891101584010158401015840= (1 minus 120601)

25

[(1 minus 120601) + 120601120588119904

120588119891

] (11989110158402minus 11989111989110158401015840)

+119872(119875119903)13

(119896119891)13

(119862119901119891)minus13

(120588119891)minus23

1198911015840

12057910158401015840=[(1 minus 120601) + 120601 (120588

119904119862119901119904120588119891119862119901119891)]

119896nf119896119891(21198911015840120579 minus 119891120579

1015840) 119875119903

(10)


119891 (0) = 0 120579 (0) = 11

(1 minus 120601)2511989110158401015840(0) = minus2

1198911015840(infin) = 0 120579 (infin) = 0

(11)


0(120574119886)23


119906119908(119909) = 3radic

(1205900120574119886)2

120588119891120583119891

1199091198911015840(0) (12)

Nu119909=

119909119902119908(119909)

119896119891 [119879 (119909 0) minus 119879 (119909infin)]

(13)


is given by

119902119908(119909) = minus119896nf(

120597119879

120597119910)119910=0

(14)


Nu119909= minus

119896nf119896119891

11986221205791015840(0) (15)


Nu119871= minus

119896nf119896119891

(Ma119871

Pr)13

1205791015840(0) (16)


Ma119871=(120597120590120597119879) (Δ119879) 119871

120572119891120583119891

(17)



0 05 1 15 2 25 3 35 40

02

04

06

08

1 Pr = 62

119872 = 0 100 200 500

120578

120601 = 01

Cu-water

119891(120578

)


Pr = 62120601 = 01

119872 = 0 100 200 500

0 05 1 15 2 25 3 35 4120578

0

02

12

04

06

08

1

119891(120578

)

Al2O3-water




2O3-water andTiO




Pr = 62120601 = 01

119872 = 0 100 200 500

0

02

12

04

06

08

1

119891(120578

)

0 05 1 15 2 25 3 35 4120578

TiO2-water


0 02 04 06 08 1 12 14 16 18 20

01

02

03

04

05

06

07

08

09

1Cu-water

Pr = 62120601 = 01

119872 = 0 100 200 500

120578

120579(120578)


Pr = 62120601 = 01

119872 = 0 100 200 500

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

Al2O3-water


119872

Pr = 62120601 = 01

119872 = 0 100 200 500

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

TiO2-water


0 05 1 15 2 25 3 35 4 45 50

05

1

15

Cu

119872 = 10

Pr = 62

120601 = 0 01 02 03119891(120578

)

120578

Al2O3

TiO2






119872 = 10

Pr = 62

120601 = 0 01 02 03

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

CuAl2O3

TiO2


Pr = 2 4 6

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

0 05 1 15 2 25 3120578

119872 = 10

120601 = 01

CuAl2O3

TiO2




4 Conclusion


2 3 4 5 616

2

24

28

32

36

Pr

119872 = 10

120601 = 01

minus1205791(0)

Cu Al2O3 TiO2


0 005 01 015 02

25

3

35

4

45

5

Cu Al2O3 TiO2

119872 = 10

Pr = 62

minus1205791(0)

120601



120601 11989110158401015840(0) minus1205791015840(0)

0 2 46842958601 153686694 34036131502 11448668 258644151


2O3 and TiO

2were taken into





2-H2O

References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891101584010158401015840= (1 minus 120601)

25

[(1 minus 120601) + 120601120588119904

120588119891

] (11989110158402minus 11989111989110158401015840)

+119872(119875119903)13

(119896119891)13

(119862119901119891)minus13

(120588119891)minus23

1198911015840

12057910158401015840=[(1 minus 120601) + 120601 (120588

119904119862119901119904120588119891119862119901119891)]

119896nf119896119891(21198911015840120579 minus 119891120579

1015840) 119875119903

(10)


119891 (0) = 0 120579 (0) = 11

(1 minus 120601)2511989110158401015840(0) = minus2

1198911015840(infin) = 0 120579 (infin) = 0

(11)


0(120574119886)23


119906119908(119909) = 3radic

(1205900120574119886)2

120588119891120583119891

1199091198911015840(0) (12)

Nu119909=

119909119902119908(119909)

119896119891 [119879 (119909 0) minus 119879 (119909infin)]

(13)


is given by

119902119908(119909) = minus119896nf(

120597119879

120597119910)119910=0

(14)


Nu119909= minus

119896nf119896119891

11986221205791015840(0) (15)


Nu119871= minus

119896nf119896119891

(Ma119871

Pr)13

1205791015840(0) (16)


Ma119871=(120597120590120597119879) (Δ119879) 119871

120572119891120583119891

(17)



0 05 1 15 2 25 3 35 40

02

04

06

08

1 Pr = 62

119872 = 0 100 200 500

120578

120601 = 01

Cu-water

119891(120578

)


Pr = 62120601 = 01

119872 = 0 100 200 500

0 05 1 15 2 25 3 35 4120578

0

02

12

04

06

08

1

119891(120578

)

Al2O3-water




2O3-water andTiO




Pr = 62120601 = 01

119872 = 0 100 200 500

0

02

12

04

06

08

1

119891(120578

)

0 05 1 15 2 25 3 35 4120578

TiO2-water


0 02 04 06 08 1 12 14 16 18 20

01

02

03

04

05

06

07

08

09

1Cu-water

Pr = 62120601 = 01

119872 = 0 100 200 500

120578

120579(120578)


Pr = 62120601 = 01

119872 = 0 100 200 500

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

Al2O3-water


119872

Pr = 62120601 = 01

119872 = 0 100 200 500

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

TiO2-water


0 05 1 15 2 25 3 35 4 45 50

05

1

15

Cu

119872 = 10

Pr = 62

120601 = 0 01 02 03119891(120578

)

120578

Al2O3

TiO2






119872 = 10

Pr = 62

120601 = 0 01 02 03

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

CuAl2O3

TiO2


Pr = 2 4 6

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

0 05 1 15 2 25 3120578

119872 = 10

120601 = 01

CuAl2O3

TiO2




4 Conclusion


2 3 4 5 616

2

24

28

32

36

Pr

119872 = 10

120601 = 01

minus1205791(0)

Cu Al2O3 TiO2


0 005 01 015 02

25

3

35

4

45

5

Cu Al2O3 TiO2

119872 = 10

Pr = 62

minus1205791(0)

120601



120601 11989110158401015840(0) minus1205791015840(0)

0 2 46842958601 153686694 34036131502 11448668 258644151


2O3 and TiO

2were taken into





2-H2O

References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Pr = 62120601 = 01

119872 = 0 100 200 500

0

02

12

04

06

08

1

119891(120578

)

0 05 1 15 2 25 3 35 4120578

TiO2-water


0 02 04 06 08 1 12 14 16 18 20

01

02

03

04

05

06

07

08

09

1Cu-water

Pr = 62120601 = 01

119872 = 0 100 200 500

120578

120579(120578)


Pr = 62120601 = 01

119872 = 0 100 200 500

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

Al2O3-water


119872

Pr = 62120601 = 01

119872 = 0 100 200 500

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

TiO2-water


0 05 1 15 2 25 3 35 4 45 50

05

1

15

Cu

119872 = 10

Pr = 62

120601 = 0 01 02 03119891(120578

)

120578

Al2O3

TiO2






119872 = 10

Pr = 62

120601 = 0 01 02 03

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

CuAl2O3

TiO2


Pr = 2 4 6

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

0 05 1 15 2 25 3120578

119872 = 10

120601 = 01

CuAl2O3

TiO2




4 Conclusion


2 3 4 5 616

2

24

28

32

36

Pr

119872 = 10

120601 = 01

minus1205791(0)

Cu Al2O3 TiO2


0 005 01 015 02

25

3

35

4

45

5

Cu Al2O3 TiO2

119872 = 10

Pr = 62

minus1205791(0)

120601



120601 11989110158401015840(0) minus1205791015840(0)

0 2 46842958601 153686694 34036131502 11448668 258644151


2O3 and TiO

2were taken into





2-H2O

References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119872 = 10

Pr = 62

120601 = 0 01 02 03

0 02 04 06 08 1 12 14 16 18 2120578

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

CuAl2O3

TiO2


Pr = 2 4 6

0

01

02

03

04

05

06

07

08

09

1

120579(120578)

0 05 1 15 2 25 3120578

119872 = 10

120601 = 01

CuAl2O3

TiO2




4 Conclusion


2 3 4 5 616

2

24

28

32

36

Pr

119872 = 10

120601 = 01

minus1205791(0)

Cu Al2O3 TiO2


0 005 01 015 02

25

3

35

4

45

5

Cu Al2O3 TiO2

119872 = 10

Pr = 62

minus1205791(0)

120601



120601 11989110158401015840(0) minus1205791015840(0)

0 2 46842958601 153686694 34036131502 11448668 258644151


2O3 and TiO

2were taken into





2-H2O

References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











2-H2O

References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article the effect of heat transfer on ... - hindawi

Documents