Heat Transfer and Mass Diffusion in Nanofluids over a Moving Permeable Convective Surface

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 254973 7 pageshttpdxdoiorg1011552013254973

Research ArticleHeat Transfer and Mass Diffusion in Nanofluids over a MovingPermeable Convective Surface

Muhammad Qasim1 Ilyas Khan23 and Sharidan Shafie2

1 Department of Mathematics COMSATS Institute of Information Technology Park Road Chak Shahzad Islamabad 44000 Pakistan2Department ofMathematical Sciences Faculty of Science Universiti TeknologiMalaysia (UTM) Johor Barhu 81310 SkudaiMalaysia3 College of Engineering Majmaah University PO Box 66 Majmaah 11952 Saudi Arabia

Correspondence should be addressed to Ilyas Khan ilyaskhanqauyahoocom

Received 14 July 2013 Revised 13 September 2013 Accepted 13 September 2013

Academic Editor Waqar Khan

Copyright copy 2013 Muhammad Qasim 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

Heat transfer and mass diffusion in nanofluid over a permeable moving surface are investigated The surface exhibits convectiveboundary conditions and constant mass diffusion Effects of Brownian motion and thermophoresis are considered The resultingpartial differential equations are reduced into coupled nonlinear ordinary differential equations using suitable transformationsShooting technique is implemented for the numerical solution Velocity temperature and concentration profiles are analyzed fordifferent key parameters entering into the problem Performed comparative study shows an excellent agreement with the previousanalysis

1 Introduction

Usually the conventional heat transfer fluids such as oilwater and ethylene glycol mixtures are poor heat transferfluids because of their poor thermal conductivity Thereforeseveral attempts have been made by many researchers toenhance the thermal conductivity of these fluids by suspend-ing nanomicroparticles in liquids Amongst them Choi [1]was the first who introduced a fluid with enhanced thermalconductivity known as nanofluid At present the flow prob-lem involving nanofluids has attracted the investigators tothe field These fluids are engineered colloidal suspensionsof nanoparticles (nanometer-sized particles ofmetals oxidesnitrides carbides or nanotubes) in the ordinary base fluidThermal conductivity of nanofluids is higher than the basefluids Such fluids over a moving surface with heat transferseem to be very important in microelectronics fuel cellshybrid-powered engines and pharmaceutical processes Itshould be pointed out that several metallurgical processesinvolve the cooling of continuous strips or filaments bydrawing them through a nanofluid [2] Having that in mindthe boundary layer flows of nanofluids have been studied

through different approaches in the recent attempts [3ndash15]there has been an increasing interest of the recent researchersin the studies of boundary layer flows over a moving surfacewith heat transfer This interest in fact stems from severalapplications of such flows in aerodynamic extrusion processpaper production food process glass fiber production met-allurgical process and so forth Sakiadis [16] made an initialattempt for boundary layer flow induced by a continuoussolid surface moving with constant speed Tsou et al [17]extended the work of Sakiadis [16] to heat transfer conceptin the boundary layer flow on a continuous moving surfaceAbdelhafez [18] studied the boundary layer flow over acontinuous moving flat surface in a parallel free stream Hestudied the case when surface and free stream move in thesame direction Afzal et al [19] revisited similar problem andreported the case when the surface and free stream movein the opposite directions Ishak et al [20] extended thework of Afzal et al [19] by considering viscous dissipationand mass transfer effects They investigated both the caseswhen surface and free streammove in the same direction andopposite directions They obtained the numerical solution ofthe problem by a finite difference scheme known as Keller

2 Mathematical Problems in Engineering

uinfin Tinfin Cinfin

y

x u

Boundary layerflow region

uw Tw Cw

Tf

Wind-up roll

Figure 1 Physical flow model

box method Very recently Aziz [21] obtained a similaritysolution for Blasius flow of a viscous fluid employing convec-tive boundary conditions Hayat et al [22] investigated theflow of Maxwell fluid over a stretching sheet with convectiveboundary conditions Some more interesting problems withconvective boundary conditions have been reported [23ndash29]

The present paper concentrates on the numerical studyof the boundary layer flow of a nanofluid over a permeablemoving surface Mathematical model is constructed in thepresence of Brownian and thermophoresis effects Governingnonlinear analysis is computed by shooting method Resultsare compared and analyzed in detail

2 Problem Statement

We study flow of nanofluid over a moving permeable surfacewith constant velocity 119906

119908in the parallel direction of the

uniform free stream velocity 119906infin The constant temperature

and concentration of wall are 119879119908

and 119862119908 respectively

The ambient values of temperature and concentration arerespectively 119879

infinand 119862

infin In Cartesian coordinate system

119909- and 119910-axes are chosen parallel and perpendicular to themoving surface (see Figure 1)

The boundary layer equations (in absence of viscousdissipation) for the physical problems under examination are

120597119906

120597119909

+

120597V

120597119910

= 0 (1)

119906

120597119906

120597119909

+ V120597119906

120597119910

= minus

1

120588119891

120597119901

120597119909

+ ]1205972

119906

1205971199102 (2)

119906

120597119879

120597119909

+ V120597119879

120597119911

= 120572

1205972

119879

1205971199102+ 120591 [119863

119861(

120597119862

120597119910

120597119879

120597119910

) +

119863119879

119879infin

(

120597119879

120597119910

)

2

]

(3)

119906

120597119862

120597119909

+ V120597119862

120597119911

= 119863119861

1205972

119862

1205971199102+

119863119879

119879infin

1205972

119879

1205971199102 (4)

where 119906 and V are the components of the velocity along the119909- and 119910-directions respectively 120588

119891is the density of the base

fluid ] (= 120583120588119891) is the kinematic viscosity 120572 is the thermal

diffusivity 119863119861is the Brownian motion coefficient 119863

119879is the

thermophoretic diffusion coefficient and 120591 = ((120588119888)119901(120588119888)119891) is

the ratio of effective heat capacity of the nanoparticlematerialto the heat capacity of the fluid

The boundary conditions in view of physics of the presentproblem are

119906 = 119906119908 V = V

119908

minus119896

120597119879

120597119910

= ℎ119891(119879119891minus 119879) 119862 = 119862

119908at 119910 = 0

119906 997888rarr 119906infin 119879 997888rarr 119879

infin

119862 997888rarr 119862infin

as 119910 997888rarr infin

(5)

It seems worth mentioning to point out that 119906119908

= 0

corresponds to the Blasius problem and for 119906infin

= 0 we havethe Sakiadis problem Here ℎ is the heat transfer coefficientand 119879

119891is the convective fluid temperature below the moving

surface A stream of cold fluid at temperature 119879infin

movingover the right surface of the plate with uniform velocity119906infin

while the left surface of the plate is heated below bythe convection from the hot fluid of temperature 119879

119891which

provides a heat transfer coefficient ℎ119891 As a result convective

boundary conditions ariseWe define the dimensionless quantities given by

120595 = radic119909]119880119891 (120578) 120579 (120578) =

119879 minus 119879infin

119879119891minus 119879infin

120601 (120578) =


119862119908minus 119862infin

120578 = radic119880

119909]119910

(6)

where 119880 = 119906119908+ 119906infin and the free stream function 120595 satisfies

119906 =

120597120595

120597119910

V = minus

120597120595

120597119909

(7)

The above expression also satisfies the continuity equation (1)and (2)ndash(5) are reduced to the following forms

119891101584010158401015840

+

1

2

11989111989110158401015840

= 0

12057910158401015840

+

Pr2

1198911205791015840

+ Pr1198731198871205791015840

1206011015840

+ Pr11987311990512057910158402

= 0

12060110158401015840

+

Le2

1198911206011015840

+

119873119905

119873119887

12057910158401015840

= 0

119891 (0) = 119878 1198911015840

(0) = 120598 1198911015840

(infin) = 1 minus 120598

1205791015840

(0) = minus120574 (1 minus 120579 (0)) 120579 (infin) 997888rarr 0

120601 (0) = 1 120601 (infin) = 0

(8)

Here primes denote differentiationwith respect to 120578 119891(0) = 119878

with 119878 gt 0 corresponding to suction case and 119878 lt 0 implyinginjection Pr (= ]120572) is the Prandtl number Le (= ]119863

119861)

is the Lewis number 119873119887(= (120588119888)

119901119863119861(119862119908minus 119862infin)(120588119888)

119891]) is

the Brownian motion parameter 119873119905(= (120588119888)

119901119863119879(119879119891

minus

119879infin)(120588119888)

119891119879infin]) is the thermophoresis parameter 120574 = (ℎ

119891

119896radic]119909119880) is the Biot number and 120598 (= 119906119908119880) is the velocity

ratio parameter Further we noticed that 120598 = 0 corresponds

Mathematical Problems in Engineering 3f998400 (120578)

10

08

06

04

02

2 4 6 8120578

120598 = 00120598 = 02

120598 = 04

120598 = 06120598 = 08

120598 = 10

Figure 2 Velocity profile 1198911015840(120578) for various values of 120598when 119878 = 10

to the flow over a stationary surface caused by the free streamvelocity while 120598 = 1 is subjected to a moving plate in anambient fluid respectivelyThe case 0 lt 120598 lt 1 holds when theplate and fluid are moving in the same direction If 120598 lt 0 thefree stream is directed towards the positive 119909-direction whilethe plate moves towards negative 119909-direction On the otherhand if 120598 gt 1 the free stream is directed towards negative119909-direction while the plate moves towards the positive 119909-direction Here we only discussed the case when 0 le 120598 le 1

Expressions for the local Nusselt number Nu119909and the

local Sherwood number Sh119909are

Nu119909=

119909119902119908

119896 (119879119908minus 119879infin)

Sh119909=

119909119895119908

119863119861(119862119908minus 119862infin)

(9)

where the wall heat flux 119902119908and the mass flux 119895

119908are given by

119902119908= minus119896(

120597119879

120597119910

)

119910=0

119895119908= minus119863119890(

120597119862

120597119910

)

119910=0

(10)

Dimensionless form of (10) is given by

(Re119908minus Reinfin)minus12Nu

119909= minus1205791015840

(0)

(Re119908minus Reinfin)minus12Sh

119909= minus1206011015840

(0)

(11)

where the Reynolds numbers are defined as

Re119908=

119906119908119909

] Re

infin=

119906infin119909

] (12)

3 Results and Discussion

Here the velocity temperature and concentration profiles areanalyzed for the velocity ratio 120598 suction parameter 119878 Brown-ianmotion parameter119873

119887 thermophoresis parameter119873

119905 and

S = minus10S = minus05

S = 00

S = 05S = 10

S = 20

f998400 (120578)

10

08

06

04

02

2 4 6 12108120578


Lewis number Le Such theme is achieved through the plots ofFigures 2ndash12 which are sketched Figure 2 describes the effectof 120598 on 119891

1015840 It is found that initially 1198911015840 decreases but after

120578 = 10 it increases when 120598 decreases Figures 3 and 4 studythe influence of 119878 on1198911015840 when 120598 = 00 and 120598 = 03 respectivelyThe boundary layer thickness is found to decrease withthe increasing values of 119878 Sucking fluid particles throughporous wall reduces the growth of the boundary layer Thisis quite reliable as the fact that suction causes reductionin the boundary layer thickness Hence a porous characterof wall provides a powerful mechanism for controlling themomentumboundary layer thickness Influence of parameter120574 on dimensionless temperature 120579 is seen in Figure 5 Agradual increase in 120574 increases the thermal boundary layerthickness This is expected because the thermal resistanceon the hot fluid side is proportional to ℎ

119891 Hence when 120574

increases the hot fluid side convection decreases and conse-quently the surface temperature increases Also for 120572 rarr infinthe result approaches the classical solution for the constantsurface temperature For fixed values of cold fluid propertiesand free stream velocity 120572 at any location 119909 is directlyproportional to heat transfer coefficient associated with thehot fluid namely ℎ

119891 The thermal resistance on the hot fluid

side is inversely proportional to ℎ119891 Thus when 120572 increases

then hot fluid side convection resistance decreases andconsequently the surface temperature increases [21] Figure 6elucidates the effects of 119878 on 120579 Temperature field 120579 decreaseswhen 119878 increases The thermal boundary layer thicknessalso decreases by increasing 119878 Effects of thermophoresisparameter 119873

119905and Brownian motion parameter 119873

119887on the

temperature 120579 are shown in Figures 7 and 8 An appreciableincrease in the temperature and thermal boundary layerthickness is noticed with an increase in 119873

119905and 119873

119887 The

Brownian motion of nanoparticles contributes to thermalconduction enhancement and hence both the temperatureand thermal boundary layer thickness increase It is alsonoticed that such increase is larger in the case of 119873

119887when

compared with 119873119905 Figure 9 illustrates the effect of Lewis

number Le onmass fraction field 120601 An increase in Le leads to


Table 1 Values of 120579(0) and minus1205791015840

(0) for various values of 120574 when119873119887= 0 = 119873

119905= 119878 = 120598 = 00

120574

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

Pr = 01 Pr = 10

[21] Present [21] Present [21] Present [21] Present005 02536 025362 00373 003733 00643 006427 00468 004683010 04046 040463 00594 005941 01208 012075 00879 008778020 05761 057613 0848 084821 02155 021552 01569 015685040 07310 073104 01076 010762 03546 035458 02582 025823060 08030 080306 01182 011824 04518 045177 03289 032877080 08446 084463 01243 012434 05235 052354 03812 0381341 08717 087172 01283 012833 05787 057868 04213 0421255 09714 097144 01430 014395 08729 087289 06356 06355810 09855 098554 01450 014597 09321 093207 06787 06785620 09927 099275 01461 014607 09649 096491 07026 070246

05


S = 00

S = 05S = 10

S = 20

f998400 (120578)

07

06

04

2 4 6 8120578


a decrease inmolecular diffusivityThus increasing the valuesof Le gradually decreases the concentration of boundarylayer Figures 10 and 11 are plotted to show the effects of ther-mophoresis parameter 119873

119905and Brownian motion parameter

119873119887 respectively on concentration of field 120601 It is observed

that an increase in119873119887increases the concentration boundary

layer whereas an increase in 119873119905causes a decrease in 120601

Figure 12 shows the effect of 120574 on themass fraction field120601 It isalso observed that119873

119905= 0 = 119873

119887corresponds to the case when

there is no transport driven by the moment of nanoparticlesfrom the surface to the fluid Further for 120598 = 00 our resultsare in excellent agreement with those presented in [21] (seeTable 1)

31 Conclusions An incompressible two-dimensionalboundary layer flow of nanofluids past a permeable movingsurface with convective boundary conditions is studiednumerically The governing boundary layer equations areconverted into highly nonlinear coupled ordinary differentialequations using some suitable transformationsThe resulting

120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

10

08

06

04

02

2 4 6 8120578

120579(120578)

Figure 5 Temperature profile 120579(120578) for various values of 120574when Pr =07 119878 = 10119873

119905= 119873119887= 01 and 120598 = 03


S = 00

S = 05S = 10

S = 20

08

06

04

02

2 4 6 108120578

120579(120578)

Figure 6 Temperature profile 120579(120578) for various values of 119878 whenPr = 07 120574 = 10119873

119887= 02 Le = 50 and 120598 = 03

Mathematical Problems in Engineering 5

03

06

05

04

01

02

2 4 6 8120578

120579(120578)

Nt = 01 Pr = 07

Nt = 01 Pr = 07

Nt = 03 Pr = 07

Nt = 01 Pr = 30Nt = 02 Pr = 30

Nt = 03 Pr = 30

Figure 7 Temperature profile 120579(120578) for various values of 119873119905when

Pr = 07 120574 = 10119873119887= 02 Le = 50 and 120598 = 03

Nb = 01 Pr = 07

Nb = 03 Pr = 07

Nb = 05 Pr = 07

Nb = 01 Pr = 30Nb = 03 Pr = 30

Nb = 05 Pr = 30

03

06

05

04

01

02

2 4 6 8120578

120579(120578)


Pr = 07 120574 = 10 119873119905= 03 Le = 50 and 120598 = 03

Le = 10Le = 20

Le = 30

Le = 40Le = 50

Le = 100

120601(120578)

10

08

06

04

02

2 4 6 8120578

Figure 9 Concentration profile 120601(120578) for various values of Le whenPr = 07 120574 = 10119873

119905= 03119873

119905= 01 and 120598 = 03

120601(120578)

10

08

06

04

02

2 4 6 8120578

Nt = 01 Le = 10

Nt = 02 Le = 10

Nt = 03 Le = 10

Nt = 01 Le = 50Nt = 02 Le = 50

Nt = 03 Le = 50

Figure 10 Concentration profile 120601(120578) for various values of119873119905when

Pr = 07 120574 = 10119873119887= 03 and 120598 = 03

Nb = 01 Le = 30

Nb = 02 Le = 30

Nb = 03 Le = 30

Nb = 01 Le = 50Nb = 02 Le = 50

Nb = 03 Le = 50

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


Pr = 07 120574 = 10 119873119905= 03 and 120598 = 03

120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

120601(120578)

10

08

06

04

02

1 32 4 5 6120578

Figure 12 Concentration profile 120601(120578) for various values of 120574 whenPr = 07119873

119887= 01119873

119905= 03 and 120598 = 03


equations are solved numerically using shooting techniqueBased on the results the following conclusions are drawn

(i) 1198911015840 decreases initially but after 120578 = 10 it increaseswhen 120598 decreases

(ii) Boundary layer thickness decreases with the increas-ing values of 119878

(iii) Porous character of wall provides a powerful mech-anism for controlling the momentum of boundarylayer thickness

(iv) As 120574 increases it increases the thermal boundary layerthickness

(v) Temperature decreases when 119878 increases(vi) Temperature and thermal boundary layer thickness

increase with increasing119873119905and119873

119887

(vii) An increase in 119873119887increases the concentration of

boundary layer whereas an increase in 119873119905causes a

decrease in 120601(viii) Results in [21] are found to be special cases of the

present work

References

[1] S Choi ldquoEnhancing thermal conductivity of fluids with nano-particlerdquo in Developments and Applications of Non-NewtonianFlows D A Siginer and H P Wang Eds vol 66 pp 99ndash105ASME FED 1995

[2] A B Rosmila R Kandasamy and I Muhaimin ldquoLie symmetrygroup transformation for MHD natural convection flow ofnanofluid over linearly porous stretching sheet in presence ofthermal stratificationrdquoAppliedMathematics andMechanics vol33 no 5 pp 593ndash604 2012

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

[4] 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

[5] R Kandasamy P Loganathan and P P Arasu ldquoScaling grouptransformation for MHD boundary-layer flow of a nanofluidpast a vertical stretching surface in the presence of suctioninjectionrdquo Nuclear Engineering and Design vol 241 no 6 pp2053ndash2059 2011

[6] M A A Hamad and M Ferdows ldquoSimilarity solution ofboundary layer stagnation-point flow towards a heated porousstretching sheet saturated with a nanofluid with heat absorp-tiongeneration and suctionblowing a Lie group analysisrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 132ndash140 2012

[7] A Mahdy ldquoUnsteady mixed convection boundary layer flowand heat transfer of nanofluids due to stretching sheetrdquoNuclearEngineering and Design vol 249 pp 248ndash255 2012

[8] M I Anwar I Khan S Sharidan and M Z Salleh ldquoConjugateeffects of heat and mass transfer of nanofluids over a nonlinearstretching sheetrdquo International Journal of Physical Sciences vol7 pp 4081ndash4092 2012

[9] M I Anwar I Khan M Z Salleh A Hasnain and S SharidanldquoMagnetohydrodynamic effects on stagnation-point flow of

Nanofluids towards a non-linear stretching sheetrdquo WulfeniaJournal vol 19 pp 367ndash383 2012

[10] W A Khan and I Pop ldquoBoundary layer flow past a stretchingsurface in a porous medium saturated by a nanofluid Brink-man-Forchheimer modelrdquo PLoS ONE vol 7 no 10 Article IDe47031 2012

[11] O DMakaindeW A Khan and Z H Khan ldquoBuoyancy effectson MHD stagnation point flow and heat transfer of a nanofluidpast a convectively heated stretchingshrinking sheetrdquo Interna-tional Journal of Heat and Mass Transfer vol 62 pp 526ndash5332013

[12] S Nadeem R U Haq and Z H Khan ldquoNumerical solution ofnon-Newtonian nanofluid flow over a stretching sheetrdquo AppliedNanoscience 07s13204-013-0235-8

[13] A Aziz and W A Khan ldquoNatural convective boundary layerflow of a nanofluid past a convectively heated vertical platerdquoInternational Journal of Thermal Sciences vol 52 no 1 pp 83ndash90 2012

[14] S Nadeem R Mehmood and N S Akbar ldquoNon-orthogonalstagnation point flow of a nano non-Newtonian fluid towardsa stretching surface with heat transferrdquo International Journal ofHeat and Mass Transfer vol 57 pp 679ndash689 2013

[15] W A Khan A Aziz and N Uddin ldquoBuongiorno model fornanofluid blasius flow with surface heat and mass fluxesrdquoJournal of Thermophysics and Heat Transfer vol 27 no 1 pp134ndash141 2013

[16] B C Sakiadis ldquoBoundary-layer behavior on continuous solidsurfaces I Boundary-layer equations for two-dimensional andaxisymmetric flowrdquo AIChE Journal vol 7 pp 26ndash28 1961

[17] F K Tsou E M Sparrow and R J Goldstein ldquoFlow and heattransfer in the boundary layer on a continuousmoving surfacerdquoInternational Journal ofHeat andMass Transfer vol 10 no 2 pp219ndash235 1967

[18] T A Abdelhafez ldquoSkin friction and heat transfer on a continu-ous flat surface moving in a parallel free streamrdquo InternationalJournal of Heat and Mass Transfer vol 28 no 6 pp 1234ndash12371985

[19] N Afzal A Badaruddin and A A Elgarvi ldquoMomentum andheat transport on a continuous flat surface moving in a parallelstreamrdquo International Journal of Heat andMass Transfer vol 36no 13 pp 3399ndash3403 1993

[20] A Ishak R Nazar and I Pop ldquoThe effects of transpiration onthe flow and heat transfer over a moving permeable surface ina parallel streamrdquo Chemical Engineering Journal vol 148 no 1pp 63ndash67 2009

[21] 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

[22] T Hayat S A Shehzad M Qasim and S Obaidat ldquoSteady flowof Maxwell fluid with convective boundary conditionsrdquo Zeits-chrift fur Naturforschung A vol 66 no 6-7 pp 417ndash422 2011

[23] O D Makinde ldquoThermal stability of a reactive viscous flowthrough a porous-saturated channel with convective boundaryconditionsrdquo Applied Thermal Engineering vol 29 no 8-9 pp1773ndash1777 2009

[24] 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


[25] W A Khan and R S R Gorla ldquoMixed convection of water at4∘C along a wedge with a convective boundary condition in aporous mediumrdquo Special Topics and Reviews in Porous Mediavol 2 no 3 pp 227ndash236 2011

[26] M A A HamadM J Uddin andA IM Ismail ldquoInvestigationof combined heat and mass transfer by Lie group analysis withvariable diffusivity taking into account hydrodynamic slip andthermal convective boundary conditionsrdquo International Journalof Heat and Mass Transfer vol 55 no 4 pp 1355ndash1362 2012

[27] N A Yacob A Ishak I Pop and K Vajravelu ldquoBoundary layerflow past a stretchingshrinking surface beneath an externaluniform shear flow with a convective surface boundary condi-tion in a nanofluidrdquo Nanoscale Research Letters vol 6 article314 2011

[28] O D Makinde T Chinyoka and L Rundora ldquoUnsteady flowof a reactive variable viscosity non-Newtonian fluid through aporous saturated medium with asymmetric convective bound-ary conditionsrdquo Computers amp Mathematics with Applicationsvol 62 no 9 pp 3343ndash3352 2011

[29] R C Bataller ldquoRadiation effects for the Blasius and Sakiadisflows with a convective surface boundary conditionrdquo AppliedMathematics and Computation vol 206 no 2 pp 832ndash8402008

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


uinfin Tinfin Cinfin

y

x u

Boundary layerflow region

uw Tw Cw

Tf

Wind-up roll

Figure 1 Physical flow model

box method Very recently Aziz [21] obtained a similaritysolution for Blasius flow of a viscous fluid employing convec-tive boundary conditions Hayat et al [22] investigated theflow of Maxwell fluid over a stretching sheet with convectiveboundary conditions Some more interesting problems withconvective boundary conditions have been reported [23ndash29]

The present paper concentrates on the numerical studyof the boundary layer flow of a nanofluid over a permeablemoving surface Mathematical model is constructed in thepresence of Brownian and thermophoresis effects Governingnonlinear analysis is computed by shooting method Resultsare compared and analyzed in detail

2 Problem Statement

We study flow of nanofluid over a moving permeable surfacewith constant velocity 119906

119908in the parallel direction of the

uniform free stream velocity 119906infin The constant temperature

and concentration of wall are 119879119908

and 119862119908 respectively

The ambient values of temperature and concentration arerespectively 119879

infinand 119862

infin In Cartesian coordinate system

119909- and 119910-axes are chosen parallel and perpendicular to themoving surface (see Figure 1)

The boundary layer equations (in absence of viscousdissipation) for the physical problems under examination are

120597119906

120597119909

+

120597V

120597119910

= 0 (1)

119906

120597119906

120597119909

+ V120597119906

120597119910

= minus

1

120588119891

120597119901

120597119909

+ ]1205972

119906

1205971199102 (2)

119906

120597119879

120597119909

+ V120597119879

120597119911

= 120572

1205972

119879

1205971199102+ 120591 [119863

119861(

120597119862

120597119910

120597119879

120597119910

) +

119863119879

119879infin

(

120597119879

120597119910

)

2

]

(3)

119906

120597119862

120597119909

+ V120597119862

120597119911

= 119863119861

1205972

119862

1205971199102+

119863119879

119879infin

1205972

119879

1205971199102 (4)

where 119906 and V are the components of the velocity along the119909- and 119910-directions respectively 120588

119891is the density of the base

fluid ] (= 120583120588119891) is the kinematic viscosity 120572 is the thermal

diffusivity 119863119861is the Brownian motion coefficient 119863

119879is the

thermophoretic diffusion coefficient and 120591 = ((120588119888)119901(120588119888)119891) is

the ratio of effective heat capacity of the nanoparticlematerialto the heat capacity of the fluid

The boundary conditions in view of physics of the presentproblem are

119906 = 119906119908 V = V

119908

minus119896

120597119879

120597119910

= ℎ119891(119879119891minus 119879) 119862 = 119862

119908at 119910 = 0

119906 997888rarr 119906infin 119879 997888rarr 119879

infin

119862 997888rarr 119862infin

as 119910 997888rarr infin

(5)

It seems worth mentioning to point out that 119906119908

= 0

corresponds to the Blasius problem and for 119906infin

= 0 we havethe Sakiadis problem Here ℎ is the heat transfer coefficientand 119879

119891is the convective fluid temperature below the moving

surface A stream of cold fluid at temperature 119879infin

movingover the right surface of the plate with uniform velocity119906infin

while the left surface of the plate is heated below bythe convection from the hot fluid of temperature 119879

119891which

provides a heat transfer coefficient ℎ119891 As a result convective

boundary conditions ariseWe define the dimensionless quantities given by

120595 = radic119909]119880119891 (120578) 120579 (120578) =


119879119891minus 119879infin

120601 (120578) =


119862119908minus 119862infin

120578 = radic119880

119909]119910

(6)

where 119880 = 119906119908+ 119906infin and the free stream function 120595 satisfies

119906 =

120597120595

120597119910

V = minus

120597120595

120597119909

(7)

The above expression also satisfies the continuity equation (1)and (2)ndash(5) are reduced to the following forms

119891101584010158401015840

+

1

2

11989111989110158401015840

= 0

12057910158401015840

+

Pr2

1198911205791015840

+ Pr1198731198871205791015840

1206011015840

+ Pr11987311990512057910158402

= 0

12060110158401015840

+

Le2

1198911206011015840

+

119873119905

119873119887

12057910158401015840

= 0

119891 (0) = 119878 1198911015840

(0) = 120598 1198911015840

(infin) = 1 minus 120598

1205791015840

(0) = minus120574 (1 minus 120579 (0)) 120579 (infin) 997888rarr 0

120601 (0) = 1 120601 (infin) = 0

(8)

Here primes denote differentiationwith respect to 120578 119891(0) = 119878

with 119878 gt 0 corresponding to suction case and 119878 lt 0 implyinginjection Pr (= ]120572) is the Prandtl number Le (= ]119863

119861)

is the Lewis number 119873119887(= (120588119888)

119901119863119861(119862119908minus 119862infin)(120588119888)

119891]) is

the Brownian motion parameter 119873119905(= (120588119888)

119901119863119879(119879119891

minus

119879infin)(120588119888)

119891119879infin]) is the thermophoresis parameter 120574 = (ℎ

119891

119896radic]119909119880) is the Biot number and 120598 (= 119906119908119880) is the velocity

ratio parameter Further we noticed that 120598 = 0 corresponds


10

08

06

04

02

2 4 6 8120578

120598 = 00120598 = 02

120598 = 04

120598 = 06120598 = 08

120598 = 10





Nu119909=

119909119902119908

119896 (119879119908minus 119879infin)

Sh119909=

119909119895119908

119863119861(119862119908minus 119862infin)

(9)


119908are given by

119902119908= minus119896(

120597119879

120597119910

)

119910=0

119895119908= minus119863119890(

120597119862

120597119910

)

119910=0

(10)



119909= minus1205791015840

(0)


119909= minus1206011015840

(0)

(11)


Re119908=

119906119908119909

] Re

infin=

119906infin119909

] (12)




119905 and


S = 00

S = 05S = 10

S = 20

f998400 (120578)

10

08

06

04

02

2 4 6 12108120578





119891 Hence when 120574






119887on the


119905and 119873

119887 The


119887when






119905= 119878 = 120598 = 00

120574

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

Pr = 01 Pr = 10


05


S = 00

S = 05S = 10

S = 20

f998400 (120578)

07

06

04

2 4 6 8120578








119905= 0 = 119873




120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

10

08

06

04

02

2 4 6 8120578

120579(120578)


119905= 119873119887= 01 and 120598 = 03


S = 00

S = 05S = 10

S = 20

08

06

04

02

2 4 6 108120578

120579(120578)


119887= 02 Le = 50 and 120598 = 03


03

06

05

04

01

02

2 4 6 8120578

120579(120578)

Nt = 01 Pr = 07

Nt = 01 Pr = 07

Nt = 03 Pr = 07

Nt = 01 Pr = 30Nt = 02 Pr = 30

Nt = 03 Pr = 30


Pr = 07 120574 = 10119873119887= 02 Le = 50 and 120598 = 03

Nb = 01 Pr = 07

Nb = 03 Pr = 07

Nb = 05 Pr = 07

Nb = 01 Pr = 30Nb = 03 Pr = 30

Nb = 05 Pr = 30

03

06

05

04

01

02

2 4 6 8120578

120579(120578)


Pr = 07 120574 = 10 119873119905= 03 Le = 50 and 120598 = 03

Le = 10Le = 20

Le = 30

Le = 40Le = 50

Le = 100

120601(120578)

10

08

06

04

02

2 4 6 8120578


119905= 03119873

119905= 01 and 120598 = 03

120601(120578)

10

08

06

04

02

2 4 6 8120578

Nt = 01 Le = 10

Nt = 02 Le = 10

Nt = 03 Le = 10

Nt = 01 Le = 50Nt = 02 Le = 50

Nt = 03 Le = 50


Pr = 07 120574 = 10119873119887= 03 and 120598 = 03

Nb = 01 Le = 30

Nb = 02 Le = 30

Nb = 03 Le = 30

Nb = 01 Le = 50Nb = 02 Le = 50

Nb = 03 Le = 50

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


Pr = 07 120574 = 10 119873119905= 03 and 120598 = 03

120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


119887= 01119873

119905= 03 and 120598 = 03









119887




present work

References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










10

08

06

04

02

2 4 6 8120578

120598 = 00120598 = 02

120598 = 04

120598 = 06120598 = 08

120598 = 10





Nu119909=

119909119902119908

119896 (119879119908minus 119879infin)

Sh119909=

119909119895119908

119863119861(119862119908minus 119862infin)

(9)


119908are given by

119902119908= minus119896(

120597119879

120597119910

)

119910=0

119895119908= minus119863119890(

120597119862

120597119910

)

119910=0

(10)



119909= minus1205791015840

(0)


119909= minus1206011015840

(0)

(11)


Re119908=

119906119908119909

] Re

infin=

119906infin119909

] (12)




119905 and


S = 00

S = 05S = 10

S = 20

f998400 (120578)

10

08

06

04

02

2 4 6 12108120578





119891 Hence when 120574






119887on the


119905and 119873

119887 The


119887when






119905= 119878 = 120598 = 00

120574

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

Pr = 01 Pr = 10


05


S = 00

S = 05S = 10

S = 20

f998400 (120578)

07

06

04

2 4 6 8120578








119905= 0 = 119873




120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

10

08

06

04

02

2 4 6 8120578

120579(120578)


119905= 119873119887= 01 and 120598 = 03


S = 00

S = 05S = 10

S = 20

08

06

04

02

2 4 6 108120578

120579(120578)


119887= 02 Le = 50 and 120598 = 03


03

06

05

04

01

02

2 4 6 8120578

120579(120578)

Nt = 01 Pr = 07

Nt = 01 Pr = 07

Nt = 03 Pr = 07

Nt = 01 Pr = 30Nt = 02 Pr = 30

Nt = 03 Pr = 30


Pr = 07 120574 = 10119873119887= 02 Le = 50 and 120598 = 03

Nb = 01 Pr = 07

Nb = 03 Pr = 07

Nb = 05 Pr = 07

Nb = 01 Pr = 30Nb = 03 Pr = 30

Nb = 05 Pr = 30

03

06

05

04

01

02

2 4 6 8120578

120579(120578)


Pr = 07 120574 = 10 119873119905= 03 Le = 50 and 120598 = 03

Le = 10Le = 20

Le = 30

Le = 40Le = 50

Le = 100

120601(120578)

10

08

06

04

02

2 4 6 8120578


119905= 03119873

119905= 01 and 120598 = 03

120601(120578)

10

08

06

04

02

2 4 6 8120578

Nt = 01 Le = 10

Nt = 02 Le = 10

Nt = 03 Le = 10

Nt = 01 Le = 50Nt = 02 Le = 50

Nt = 03 Le = 50


Pr = 07 120574 = 10119873119887= 03 and 120598 = 03

Nb = 01 Le = 30

Nb = 02 Le = 30

Nb = 03 Le = 30

Nb = 01 Le = 50Nb = 02 Le = 50

Nb = 03 Le = 50

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


Pr = 07 120574 = 10 119873119905= 03 and 120598 = 03

120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


119887= 01119873

119905= 03 and 120598 = 03









119887




present work

References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119905= 119878 = 120598 = 00

120574

120579(0) minus1205791015840

(0) 120579(0) minus1205791015840

(0)

Pr = 01 Pr = 10


05


S = 00

S = 05S = 10

S = 20

f998400 (120578)

07

06

04

2 4 6 8120578








119905= 0 = 119873




120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

10

08

06

04

02

2 4 6 8120578

120579(120578)


119905= 119873119887= 01 and 120598 = 03


S = 00

S = 05S = 10

S = 20

08

06

04

02

2 4 6 108120578

120579(120578)


119887= 02 Le = 50 and 120598 = 03


03

06

05

04

01

02

2 4 6 8120578

120579(120578)

Nt = 01 Pr = 07

Nt = 01 Pr = 07

Nt = 03 Pr = 07

Nt = 01 Pr = 30Nt = 02 Pr = 30

Nt = 03 Pr = 30


Pr = 07 120574 = 10119873119887= 02 Le = 50 and 120598 = 03

Nb = 01 Pr = 07

Nb = 03 Pr = 07

Nb = 05 Pr = 07

Nb = 01 Pr = 30Nb = 03 Pr = 30

Nb = 05 Pr = 30

03

06

05

04

01

02

2 4 6 8120578

120579(120578)


Pr = 07 120574 = 10 119873119905= 03 Le = 50 and 120598 = 03

Le = 10Le = 20

Le = 30

Le = 40Le = 50

Le = 100

120601(120578)

10

08

06

04

02

2 4 6 8120578


119905= 03119873

119905= 01 and 120598 = 03

120601(120578)

10

08

06

04

02

2 4 6 8120578

Nt = 01 Le = 10

Nt = 02 Le = 10

Nt = 03 Le = 10

Nt = 01 Le = 50Nt = 02 Le = 50

Nt = 03 Le = 50


Pr = 07 120574 = 10119873119887= 03 and 120598 = 03

Nb = 01 Le = 30

Nb = 02 Le = 30

Nb = 03 Le = 30

Nb = 01 Le = 50Nb = 02 Le = 50

Nb = 03 Le = 50

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


Pr = 07 120574 = 10 119873119905= 03 and 120598 = 03

120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


119887= 01119873

119905= 03 and 120598 = 03









119887




present work

References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










03

06

05

04

01

02

2 4 6 8120578

120579(120578)

Nt = 01 Pr = 07

Nt = 01 Pr = 07

Nt = 03 Pr = 07

Nt = 01 Pr = 30Nt = 02 Pr = 30

Nt = 03 Pr = 30


Pr = 07 120574 = 10119873119887= 02 Le = 50 and 120598 = 03

Nb = 01 Pr = 07

Nb = 03 Pr = 07

Nb = 05 Pr = 07

Nb = 01 Pr = 30Nb = 03 Pr = 30

Nb = 05 Pr = 30

03

06

05

04

01

02

2 4 6 8120578

120579(120578)


Pr = 07 120574 = 10 119873119905= 03 Le = 50 and 120598 = 03

Le = 10Le = 20

Le = 30

Le = 40Le = 50

Le = 100

120601(120578)

10

08

06

04

02

2 4 6 8120578


119905= 03119873

119905= 01 and 120598 = 03

120601(120578)

10

08

06

04

02

2 4 6 8120578

Nt = 01 Le = 10

Nt = 02 Le = 10

Nt = 03 Le = 10

Nt = 01 Le = 50Nt = 02 Le = 50

Nt = 03 Le = 50


Pr = 07 120574 = 10119873119887= 03 and 120598 = 03

Nb = 01 Le = 30

Nb = 02 Le = 30

Nb = 03 Le = 30

Nb = 01 Le = 50Nb = 02 Le = 50

Nb = 03 Le = 50

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


Pr = 07 120574 = 10 119873119905= 03 and 120598 = 03

120574 = 001120574 = 01

120574 = 05

120574 = 10120574 = 100

120574 = 1000

120601(120578)

10

08

06

04

02

1 32 4 5 6120578


119887= 01119873

119905= 03 and 120598 = 03









119887




present work

References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















119887




present work

References







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Heat Transfer and Mass Diffusion in Nanofluids over a Moving Permeable Convective Surface

Documents

Transcript of Heat Transfer and Mass Diffusion in Nanofluids over a Moving Permeable Convective Surface