A revised model.docx
《A revised model.docx》由会员分享，可在线阅读，更多相关《A revised model.docx（22页珍藏版）》请在冰豆网上搜索。
Arevisedmodel
Naturalconvectiveboundary-layerflowofananofluidpastavertical
plate:
A.V.Kuznetsov
a,
*,D.A.Nield
b
a
DepartmentofMechanicalandAerospaceEngineering,NorthCarolinaStateUniversity,CampusBox7910,Raleigh,NC27695-7910,USA
b
DepartmentofEngineeringScience,UniversityofAuckland,PrivateBag92019,Auckland1142,NewZealand
articleinfo
Articlehistory:
Received5May2013
Receivedinrevisedform
23July2013
Accepted17October2013
Availableonline4December2013
Keywords:
Nanofluid
Brownianmotion
Thermophoresis
Naturalconvection
Boundarylayer
Verticalplate
abstract
Theproblemofnaturalconvectiveboundary-layerflowofananofluidpastaverticalplateisrevisited.
Themodel,whichincludestheeffectsofBrownianmotionandthermophoresis,isrevisedsothatthe
nanofluidparticlefractionontheboundaryispassivelyratherthanactivelycontrolled.Inthisrespectthe
modelismorerealisticphysicallythanthatemployedbypreviousauthors.
_2013ElsevierMassonSAS.Allrightsreserved.
1.Introduction
ThemodelforananaofluidincludingtheeffectsofBrownian
motionandthermophoresis,introducedbyBuongiorno[1],was
appliedbyKuznetsovandNield[2]totheclassicalproblemstudied
byPohlhausen,KuikenandBejan[3e6],namelyconvective
boundarylayerflowpastaverticalplate.Intheirpioneeringpaper
KuznetsovandNield[2]employedboundaryconditionsonthe
nanoparticlefractionanalogoustothoseonthetemperature.Inthis
notetheproblemisrevisitedandaboundaryconditionthatismore
realisticphysicallyisapplied.Itisnolongerassumedthatonecan
controlthevalueofthenanoparticlefractionatthewall,butrather
thatthenanoparticlefluxatthewalliszero.Thischangenecessitatesarescalingoftheparametersthatareinvolved.
2.Analysis
ThefollowinganalysiscloselyfollowsthatinRef.[2]andsois
describedbrieflyhere.Atwo-dimensionalproblemisconsidered.A
coordinateframeinwhichthex-axisisalignedverticallyupwards
isutilized.Averticalplateisaty¼0.Weassumethataty¼0the
temperatureTtakestheconstantvalueTw.Thefluxofthenanoparticlefractionaty¼0istakentobezero.Theambientvalueof
temperatureisTNandtheambientvalueofthenanoparticlefractionisfN;theambientvaluesareattainedataninfinitedistance
fromthewall.
WeusedtheOberbeck-Boussinesqapproximation.Thegoverningequationsexpressingtheconservationoftotalmass,momentum,thermalenergy,andnanoparticlesare,respectively:
V$v¼0;
（1）
rf
_
vv
vt
þv$Vv
_
¼_VpþmV
2
vþ
h
frp
þð1_fÞ
_
n
rf
ð1_bðT_TNÞÞ
oi
g;
（2）
ðrcÞ
f
_
vT
vt
þv$VT
_
¼kV
2
TþðrcÞ
p
½DBVf$VTþðDT=TNÞVT$VT_;
（3）
vf
vt
þv$V4¼DBV
2
fþðDT=TNÞV
2
T:
（4）
InEqs.
（1）e（4）thefieldvariablesarethevelocityv[wewrite
v¼（u,v）],thetemperatureTandthenanoparticlevolumefraction
*Correspondingauthor.
E-mailaddresses:
avkuznet@ncsu.edu（A.V.Kuznetsov）,d.nield@auckland.ac.nz
（D.A.Nield）.
ContentslistsavailableatScienceDirect
InternationalJournalofThermalSciences
journalhomepage:
1290-0729/$eseefrontmatter_2013ElsevierMassonSAS.Allrightsreserved.
http:
//dx.doi.org/10.1016/j.ijthermalsci.2013.10.007
InternationalJournalofThermalSciences77（2014）126e129
f.Also,rfisthedensityofthebasefluidandm,kandbarethe
viscosity,thermalconductivityandvolumetricexpansioncoefficientofthenanofluid,andrPisthedensityoftheparticles.We
denotedthegravitationalaccelerationbyg.InEqs.（3）and（4）the
coefficientsDBandDTaretheBrowniandiffusioncoefficientandthe
thermophoreticdiffusioncoefficient,respectively,eachnondimensionalizedintermsoftheambientvalueofthetemperature.Itisbeingassumedthetemperaturedoesnotvarymuchfrom
theambienttemperature,andsoDBandDTmayeachbetreatedasa
constant.
Eqs.
（1）e（4）mustbesolvedsubjecttothefollowingboundary
conditions:
u¼v¼0;T¼Tw;DB
vf
vy
þ
DB
TN
vT
vy
¼0aty¼0;（5a,b,c）
u¼v¼0;T/TN;f/fNasy/N:
（6a,b,c）
Asteadystateflowisconsidered.Eq.（5c）isastatementthat,
withthermophoresistakenintoaccount,thenormalfluxofnanoparticlesiszeroattheboundary[7,8].
WeusedtheOberbeck-Boussinesqapproximation.Wealso
madeanassumptionthatthenanoparticleconcentrationisdilute.
Usingasuitablechoiceforthereferencepressure,welinearizedthe
momentumequationandrecastEq.
（2）asfollows:
rf
_
vv
vt
þv$Vv
_
¼_VpþmV
2
vþ
h_
rp_rfN
ðf_fNÞ
þð1_fNÞrfNbðT_TNÞ
i
g:
（7）
Basedonascaleanalysis,wenowemploythestandard
boundary-layerapproximation,andexpressthegoverningequationsas
vu
vx
þ
vv
vy
¼0;（8）
vp
vx
¼m
v
2
u
vy
2
_rf
_
u
vu
vx
þv
vu
vy
_
þ
h
ð1_iNÞrfNbgðT_TNÞ
_
_
rp_rfN
gðf_fNÞ
i
;（9）
vp
vy
¼0;（10）
u
vT
vx
þv
vT
vy
¼aV
2
Tþs
"
DB
vf
vy
vT
vy
þ
_
DT
TN
__
vT
vy
_2
#
;（11）
u
vf
vx
þv
vf
vy
¼DB
v
2
f
vy
2
þ
_
DT
TN
_
v
2
T
vy
2
;（12）
where
a¼
k
ðrcÞ
f
;s¼
ðrcÞ
p
ðrcÞ
f
:
（13）
Weusedcross-differentiationtoeliminatepfromEqs.（8）and
（9）.Wealsointroducedastreamfunctionjdefinedby
u¼
vj
vy
;v¼_
vj
vx
:
（14）
Eq.（8）isnowsatisfiedidentically.
Thisleavesuswiththefollowingthreeequations:
vj
vy
v
2
j
vxvy
_
vj
vx
v
2
j
vy
2
_n
v
3
j
vy
3
¼ð1_fNÞrfNbgðT_TNÞ
_
_
rp_rfN
gf（15）
vj
vy
vT
vx
_
vj
vx
vT
vy
¼aV
2
Tþs
"
DB
vf
vy
vT
vy
þ
_
DT
TN
__
vT
vy
_2
#
;（16）
Nomenclature
DBBrowniandiffusioncoefficient
DTthermophoreticdiffusioncoefficient
frescalednanoparticlevolumefraction,definedbyEq.
（20）
ggravitationalaccelerationvector
kthermalconductivity
LeLewisnumber,definedbyEq.（28）
Nrbuoyancyeratioparameter,definedbyEq.（25）
NbBrownianmotionparameter,definedbyEq.（26）
Ntthermophoresisparameter,definedbyEq.（27）
NuNusseltnumber,definedbyEq.（31）
NurreducedNusseltnumber,Nu/Rax
1/4
PrPrandtlnumber,definedbyEq.（24）
ppressure
q
00
wallheatflux
RaxlocalRayleighnumber,definedbyEq.（18）
sdimensionlessstreamfunction,definedbyEq.（20）
Ttemperature
Twtemperatureattheverticalplate
TNambienttemperatureattainedasytendstoinfinity
vvelocity,（u,v）
（x,y）Cartesiancoordinates（x-axisisalignedvertically
upwards,plateisaty¼0）
Greeksymbols
athermaldiffusivity
bvolumetricexpansioncoefficientofthefluid
hsimilarityvariable,definedbyEq.（19）
qdimensionlesstemperature,definedbyEq.（20）
mdynamicviscosityofthefluid
nkinematicviscosity,m/rfN
rffluiddensity
rpnanoparticlemassdensity
（rc）fheatcapacityofthefluid
（rc）peffectiveheatcapacityofthenanoparticlematerial
sparameterdefinedbyEq.（13）,（rc）p/（rc）f
fnanoparticlevolumefraction
fNambientnanoparticlevolumefractionattainedasy
tendstoinfinity
jstreamfunction,definedbyEq.（14）
A.V.Kuznetsov,D.A.Nield/InternationalJournalofThermalSciences77（2014）126e129127
vj
vy
vf
vx
_
vj
vx
vf
vy
¼DB
v
2
f
vy
2
þ
_
DT
TN
_
v
2
T
vy
2
:
（17）
Eq.（15）wasderivedbyintegrationwithrespecttoy;theuse
wasalsomadeoftheboundaryconditionsatinfinity.InEq.（15）
n¼m/rfN.
ThelocalRayleighnumberRaxwasdefinedby
Rax¼
ð1_fNÞbgðTw_TNÞx
3
na
;（18）
andthesimilarityvariablewasdefinedby
h¼
y
x
Ra
1=4
x
:
（19）
Thesimilarityvariablewaschosenonthebasisofscaleanalysis.
Ourreviewshowsthatmostnanofluidshavelargevaluesforthe
LewisnumberLe.ForthisreasonweconcentrateonthecaseLe>1.
Wealsoassumethatheattransfer,ratherthanmasstransfer,drives
theflow.Intermsofourmodel,thisimpliesthatthebuoyancye
ratioparameterNrdefinedbyEq.（25）belowissmallincomparison
withunity.ThisalsoimpliesthattheLewisnumberLedefinedby
Eq.（28）belowisgreaterthanunity.
Thedimensionlessvariabless,q,andfwereintroducedbythe
followingequations
sðhÞ¼
j
aRa
1=4
x
;qðhÞ¼
T_TN
Tw_TN
;fðhÞ¼
f
fN
:
（20）
WesubstitutedthequantitiesdefinedbyEq.（20）intoEqs.（15）e
（17）andobtainedthefollowingordinarydifferentialequations
s
000
þ
1
4Pr
_
3ss
00
_2s
02
þq_Nrf¼0;（21）
q
00
þ
3
4
sq
0
þNbf
0
q
0
þNtq
02
¼0;（22）
f
00
þ
3
4
Lesf
0
þ
Nt
Nb
q
00
¼0:
（23）
ThefivedimensionlessparametersinEqs.（21）e（23）are
Pr¼
n
a
;（24）
Nr¼
_
rp_rfN
fN
rfNbðTw_TNÞ
;（25）
Nb¼
ðrcÞ
pDBfN
ðrcÞ
fa
;（26）
Nt¼
ðrcÞ
pDT
ðTw_TNÞ
ðrcÞ
faTN
;（27）
Le¼
a
DB
:
（28）
HereNristhebuoyancyratio,NbistheBrownianmotion
parameter,Ntisthethermophoresisparameter,andLetheLewis
number.
WesolvedEqs.（21）e（23）subjecttothefollowingboundary
conditions:
Ath¼0:
s¼0;s
0
¼0;q¼1;Nbf
0
þNtq
0
¼0;（29a,b,c）
Ash/N:
s
0
¼0;q¼0;f¼0:
（30a,b,c）
ForthecasewhenNr,NbandNtareallequaltozero,thereare
justtwoindependentvaluablesinvolvedinEqs.（21）and（22）
（namelysandq）.Theboundary-valueproblemforthesetwovariablesreducestotheclassicalPohlhausen-Kuiken-Bejanproblem,
whiletheboundaryvalueproblemforfthenbecomesill-posedand
hasnophysicalsignificance.
WedefinedtheNusseltnumberNuas
Nu¼
q
00
x
kðTw_TNÞ
:
（31）
Hereq
00
isthewallheatflux.ThereducedNusseltnumber
Nu/Rax
1/4
whichwedenotedbyNur,isfoundas_q
0
（0）.The
readerwillnotethatthedimensionlessmassfluxrepresented
byaSherwoodnumberShisnowidenticallyzero.
3.Resultsanddiscussion
Asabaselineforcomparison,wefirstpresentinTable1some
resultsforthelimitingcaseofaregularfluid.Wethenpresentthe
resultsofourinvestigationoftheeffectoftheparametersNr,Nb
andNtonNur,forvariousvaluesofPrandLe.
In
升级会员 