class notes topic 31
TRANSCRIPT
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Topic3:
TransportPhenomena2nd
EditionR.ByronBird,WarrenE.Stewart,EdwinN.
Lightfoot;Chapter2pg.4074
Chapter2:
ShellMomentumBalancesandVelocityDistributionsinLaminarFlow
Introduction:
In this chapter showhow toobtain the viscosityprofiles for laminar flows in simple
systems.Weusethedefinitionofviscosity,theexpressions for themolecularandconvective
momentum fluxes,andtheconceptofamomentumbalance.Toobtain interestasquantities
suchasthemaximumvelocity,theaveragevelocity,ortheshearstressatasurface.Themethods
andproblems in thischapterapplyonly tosteady flowwithLaminar flow.Bysteadywemean
thatthepressure,density,andvelocitycomponentsateachpointinthestreamdonotchangewith
time. Laminar flow is the orderly flow that is observed, for example, in tube flow at velocities
sufficiently lowthattinyparticles injected intothetubemovealong inathin line.This is insharp
contrastwiththewildlychaotic"turbulentflow"atsufficientlyhighvelocitiesthattheparticlesare
flungapartanddispersedthroughouttheentirecrosssectionofthetube.
A) Laminar flow, the fluid layers movesmoothlyoveroneanotherinthedirectionof
flow.
B) Turbulent Flow, the flow pattern iscomplex and timedependent, with
considerable motion perpendicular to the
principalflowdirection.
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2.1 SHELLMOMENTUMBALANCESANDBOUNDARYCONDITIONS
Momentumbalanceforsteadyflow:
0
Thisstatementhasarelationwiththelawofconservationofmomentum. InthemomentumbalanceweneedtheexpressionsfortheconvectivemomentumfluxesgiveninTable1.71and
themolecularmomentumfluxesgiveninTable1.21. Isimportantthatthemolecular
momentumfluxincludesboththepressureandtheviscouscontributions.
Themomentumbalanceisappliedonlytosystemsinwhichthereisjustonevelocitycomponentinthischapter,butitcanbeappliedtosysteminwhichhasmorethanonevelocity
component,whichdependsononlyonespatialvariable,alsotheflowmustberectilinear.
Thestepsforsettingupandsolvingviscousproblemsare:
1. Identifythenonvanishingvelocitycomponentandthespatialvariableonwhichitdepends.2. Applythemomentumbalanceoverathinshellperpendiculartotherelevantspatial
variable.
3. Findthelimitwhenthethicknessoftheshellapproachzeroandmakeuseofthedefinitionofthefirstderivativetoobtainthecorrespondingdifferentialequationforthemomentum
flux.
4. Thenintegratethisequationtogetthemomentumfluxdistribution.5. InsertNewton'slawofviscosityandobtainadifferentialequationforthevelocity.6. Integratethisequationtogetthevelocitydistribution.7. Usethevelocitydistributiontogetotherquantities,suchasthemaximumvelocity,average
velocity,orforceonsolidsurfaces.
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Thesestepsmentionedsomeintegration,severalconstantsofintegrationappear,andtheseareevaluatedbyusingboundaryconditionsthatisstatementsaboutthevelocityorstress
attheboundariesofthesystem.Themostcommonlyusedboundaryconditionsareas
follows:
A. Atsolidfluidinterfacesthefluidvelocityequalsthevelocitywithwhichthesolidsurfaceismoving. Thisstatementisappliedtoboththetangentialandthenormal
componentofthevelocityvector.Theequalityofthetangentialcomponentsisreferred
toasthe"noslipcondition.
B. Atliquidliquidinterfacialplaneofconstantx,thetangentialvelocitycomponentsVyandVzarecontinuousthroughtheinterface(the"noslipcondition")asarealsothe
molecularstresstensorcomponentsp+ xx, xyand xz.
C. Ataliquidgasinterfacialplaneofconstantx,thestresstensorcomponents xyand xzaretakentobezero,providedthatthegassidevelocitygradientisnottoolarge.Thisis
logical,sincetheviscositiesofgasesaremuchlessthanthoseofliquids.
Inalloftheseboundaryconditionsitissupposedthatthereisnomaterialpassingthroughtheinterfacethatis,thereisnoadsorption,absorption,dissolution,evaporation,melting,or
chemicalreactionatthesurfacebetweenthetwophases.
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2.2FLOWOFAFALLINGFILM
ThisexampleshowaflowofaliquidaninclinedflatplateoflengthLandwidthW,asshownintheFigure.Weconsidertheviscosityanddensityofthefluidtobeconstant.Acomplete
descriptionoftheliquidflowisdifficultbecauseofthedisturbancesattheedgesofthe
system(z=0,z=L,y=0,y=W).
Adescriptioncanoftenbeobtainedbyneglectingsuchdisturbances,particularlyifWandLarelargecomparedtothefilmthickness .
Forsmallflowratesweexpectthattheviscousforceswillpreventcontinuedaccelerationoftheliquiddownthewall,sothatV,willbecomeindependentofzinashortdistancedown
theplate.
AsaresultitseemsreasonabletopostulatethatVz=Vz(x),Vx,=0andVy=0andfurtherthatp=p(x).Thenonvanishingcomponentsof arethen xz= zx,= (dVz/dx).
Selectas the "system"a thin shellperpendicular to thexdirection.Thenwe setup a zmomentumbalanceoverthisshell,whichisaregionofthicknessx,boundedbytheplanes
z=0andz=L,andextendingadistanceWintheydirection.
Usingthecomponentsofthe"combinedmomentumfluxtensor"definedintables1.71to3,wecanincorporateallthepotentialmechanismsformomentumtransportatonce:
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Usingthequantitiesxzandzzweaccountforthezmomentumtransportbyallmechanisms,convectiveandmolecular.
The"in"and"out"directionsinthedirectionofthepositivex andzaxes(inthisproblemthesehappentocoincidewiththedirectionsofzmomentumtransport).
Whenthesetermsaresubstitutedintothezmomentumbalance,weget:
LW(xz x xz x+x)+Wx(zz z=0 zz z=L)+(LWX)(gcos)=0
Inthisfigure x isthethicknessoverwhichazmomentumbalance ismade.Arrowsshowthemomentumfluxesrelatedwiththesurfacesoftheshell.SinceVxandVyarebothzero,
VyVzand VyVzarezero.Vydoesnotdependonyandz, yz=0and zz=0.Alsothedashed
underlinedfluxesdonotneedtobeconsidered.BothpandVzVzarethesameatz=0andz
=L,andasaresultdonotappearinthebalanceofzmomentum.
Rateofzmomentuminacrosssurfaceatz=0 (Wx)zz/z=0
Rateofzmomentumoutacrosssurfaceatz=L (Wx)zz/z=L
Rateofzmomentuminacrosssurfaceatx (LW)(xz)/x
Rateofzmomentumoutacrosssurfacex+ x (LW)(xz)/x+x
Gravityforceactingonfluidinthezdirection (LWx)(gcos)
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Shellmomentumbalanceofafluidinafallingfilm:
I. Assumption1. L>>>W2. L>>>3. Length=z
Width=y
Thickness=x
4. Flowindirectionz
II. Momentumfluxtensor,
ij= ij+ vivj
zz= zz+ vzvz= zz+p+ vzvz =1 ,i=j
xz= xz+ vxvz =0 ,i j
yz= yz+ vyvz =0 ,i j
III. Velocityandcomponents(Note:Vzdoesnotcancel)
Vz=directionofflux Vz(z)=0Vx=0 Vz(x) 0dependenceofVzinx
Vy=0 Vz(y)=0
p=p(x)
IV. MomentumBalance
ij=ij + vivj
i = coordinate
j = flux direction
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In Out
Z Z=0
Wxzz|z=0
Z=L
Wxzz|z=L
X X=x
LWxz|x=x
X=x+ x
LWxz|x=x+ x
Y Y=0
Lx yz|y=0
Y=W
Lx yz|y=W
Forceofgravity:(LWx) gcos
V. BalanceSubstitutionWx[zz|z=0 zz|z=L]+LW[xz|x=x xz|x=x+ x]+Lx[yz|y=0 yz|y=W]+(LWx) gcos =
0
in= out
Velocitydoesnotdependofy
zz= zz+p+ vzvz
xz= xz+ vxvz
yz= yz+ vyvz
W
[zz|z=0 zz|z=L]+
LW
[xz|x=x xz|x=x+ x]+
L
[yz|y=0 yz|y=W]+
LW
gcos
=0
| |L
+
| |
+ gcos =0
zdoesnotvary
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lim | |
= gcos
gcos
DifferentialEquationofMomentum
xz=
xz= xz+ vxvz = xz
gcos
SeparableIntegration:
xz=( gcos )x+C1
Boundaryconditions:xz(x=0)=0
xz=0=(gcos)*0+C1
xz=( gcos )x
xz=
(gcos)x
Vz=
BoundaryConditions:Vz(x=)=0
Vz=0=
(
2)+C2
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C2=
(
2)
Vz=
(x
2
2)
=
(
2 x
2)(
Vz=
(1
)
VI. VelocityandStressProfile
WithVzcanbecalculated:
Velocityaverage:
Force:
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Thickness:
Massrate:
MaximumVelocity:
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2.3FLOWTHROUGHACIRCULARTUBEWhenanalyzinglaminarflowthroughacircularpipe,cylindricalcoordinatesareused.Lets
considerthisexample.Aliquidflowingdownwardundertheinfluenceofapressuredifference
andgravitythroughaverticaltubeoflengthLandradiusR.So,youmusttakeinto
considerationthefollowingassumptions:
SteadyState
LaminarFlow
constantdensity,
constantviscosity,
NoEndEffects(tubelengthisverylargewithrespecttothetuberadius,sothattheseend
effectswillbeunimportantL>>R)
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Postulates:(Lookatthecoordinatesysteminthediagram) Vz=Vz(r) Vr=0 V=0 Vz(z)=0 Vz()=0 Vz(r)0 p=p(z)
FromthesetermswhenyougotoTableB.1/AppendixBonyourBSLbook(pg. 844)the
nonvanishingcomponentsofare rzand zr,becauseofthepostulatesshownabove.Whenmakingamomentumbalance,youfirstneedtolookatwherethemomentumis
generatedwhenthefluidisflowingdownward.Momentumisgeneratedinzandrdirections
asseeninthecoordinatesystembelowandwecanputthiscoordinatesysteminhalfofour
cylindertoanalyzeit.
Thequantitiesof
and
accountforthe
momentumtransportbyallpossiblemechanisms,convectiveandmolecular.Asforthevalues
ofthose momentums,andyourenotsurehowtoevaluatethem,Table1.21(pg.17),Table1.71(pg.35)andequation1.72(pg.36)canhelp.Rememberyouwilleventuallyneedthese
valueswhenmakingshellbalances.Thereforeifwehave,
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Then,
Ok,sobacktoourmomentumbalance.Weselectoursystemasacylindrialshellofthickness
andlengthL.Weevaluatethemomentuminandoutofthisshellandwecanthenlistthecontributions:
Directions In Out
R r=r 2| 2|
Z z=02| z=L2|
Norateofmomentuminthis
direction
Norateofmomentuminthis
direction
Gravityforceactinginzdirectiononcylindricalshell2* Notethatthoseinandoutareinthepositivedirectionoftherandzaxes.
Wenowmakeourmomentumbalancebasedonequation2.11(pg.41)fromyourBSLbook:
2| | 2| | 2 0Wethendividethisequationby2toget:
| | | | 0
Thesameas,
| | | |
Bytakingthelimitoftheequationontheleftsidewhenr0,weget:
lim| |
| |
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Andbydefinition,
lim| |
Therefore,
| |
NowweevaluatethecomponentsandwiththevaluesinAppendixB.1: 2 (*Remember:Vr=0) Bysubstitutingthesevaluesinand: 2
Wenowhavethefollowingsimplifications:
1) BecausewehaveVz=Vz(r),theterm
willbethesameatbothendsofthetube.
| | 2) BecausewehaveVz=Vz(r),theterm 2 willbethesameatbothendsofthetube.
2 | 2 |
Sonowourequationsturnsinto:
0
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Withthesepressuredifferences,wecannowusemodifiedpressures.Letstakealookatthe
diagramfirst:
0 | wherePisthemodifiedpressure
|
0
Byusingseparableequationsandintegrating:
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TolookforthevalueofconstantC1,weusecertainboundaryconditionstosimplifyour
problem.Letslookatthefollowingdiagram,wherewecanwatchthevelocityprofile:
BoundaryCondition1:
Whenr=0,=0Therefore,C1=0andbysubstitutingwithNewtonsLawofViscosity(obtainedfromApendixB.2) weobtain:
2
Integratingthisfirstorderdifferentialequationweobtain:
4 ThisnewconstantC2isevaluatedfromtheboundarycondition
B.C.2: atr=R, vz=0
Then,fromthisC2isfoundtobe: 4 .Hence,thevelocitydistributionis:
4 1 Weseethatthevelocitydistributionforlaminar,incompressibleflowofaNewtonianfluidina
longtubeisparabolic.
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Oncethevelocityprofilehasbeenestablished,variousderivedquantitiescanbeobtained:
(i) Themaximumvelocity,occursatr=0andis:
,
4
(ii) Theaveragevelocityisisobtainedbydividingthetotalvolumetricflowratebythecrosssectionalarea
8 12 ,
(iii)Themassrateflowwistheproductofthecrosssectionalarea
,thedensity ,andtheaveragevelocity
8
ThisratherfamousresultiscalledtheHagenPoiseuille equation.Itisused,alongwith
experimentaldatafortherateofflowandthemodifiedpressuredifference,to
determinetheviscosityoffluids(seeExample2.31)inacapillaryviscometer.
(iv)Thezcomponentoftheforce,oftheFluidonthewettedsurfaceofthepipeisjust
theshearstressintegratedoverthewettedarea
2 | Theresultstatesthattheviscousforceiscounterbalancedbythenetpressureforceandthegravitationalforce.
Theresultsofthissectionareonlyasgoodasthepostulatesintroducedatthebeginningofthesection,namelythat and .
ExperimentshaveshownthatthesepostulatesareinfactrealizedforReynoldsnumbersupto2100;abovethatvalue,theflowwillbeturbulentifthereareanyappreciable
disturbancesinthesystem,thatis,wallroughnessorvibrations.
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ForcirculartubestheReynoldsnumberisdefinedby ,whereD=2Risthetubediameter.
WenowsummarizealltheassumptionsthatweremadeinobtainingtheHagenPoiseuille
equation.
(a)Theflowislaminar(Re Le.(f) Thefluidbehavesasacontinuum,thisassumptionisvalid,exceptforverydilutegasesorverynarrowcapillarytubes,inwhichthemolecularmeanfreepathis
comparabletothetubediameter(theslipflowregion)ormuchgreaterthanthe
tubediameter(theKundsenfloworfreemoleculeflowregime).
(g) Thereisnoslipatthewall,sothatB.C.2isvalid;thisisanexcellentassumptionfotpurefluidsundertheconditionsassumedin(f).
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2.4FCaso
fluid
entre
Com
envo
ante
Tng
ygra
obte
La c
LUJOATRAparticulare
incompre
doscilindr
nzamos ef
ltura cilnd
iormentep
aseencuen
vedadact
ner
onstante C
VSDEUNncoordena
sible fluye
scirculares
ectuando
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araelflujoe
taquepara
anendirec
1 no pue
NULOdascilndric
enestado
coaxialesd
n balance
ga a la m
nuntubo
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e determi
asdeunflu
stacionario
eradioskR
de cantida
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stas.Esta
narse de f
idoviscoso
a travsd
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gz,puestoq
cuacindif
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atravsde
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comprendid
re una fin
ha obtenid
asdepresi
integra,pa
to que n
n
a
a
o
n
a
,
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disp
enni
existi
r=R,
Teniquel
Nte
habe
laec
ecua
Integ
Ahor
sigui
Subs
ecua
Sere
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sequees
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espectoar:
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tascondici
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ncuentra
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ransformae
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de cantid
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tesdeinteg
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ara r=R
enlaecuaci
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v
nanterior
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obtieneest
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o
e
n
a
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Subs
la diveloc
conc
ituyendoe
tribucin didad, para
ntricosson
tosvalores
e densidadel flujo inc
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es
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2.5FLUJODEDOSLQUIDOSIMMISIBLESADYACENTES
Situacindelainterfasededoslquidos,estosfluyenendireccindelejedezconunalongitud
LyunanchoW,esteflujobajoungradientedepresinhorizontalexpresadocomo(p0p)/L.el
flujode
estos
es
ajustado
de
manera
que
se
dividan
por
sus
densidades.
El
flujo
deba
ser
lo
suficientementelentoparaquenopresenteninestabilidadenlainterfasedeestos,estopara
encontrarelflujodemomentumylavelocidaddedistribucin.
Ecuacindiferencialparaflujodemomentun
Alintegrallaecuacinanteriorseobtiene
Dosflujos
immisibles
entre
dos
places
paralelas
don
aplicacin
de
un
gradiente
de
presin
HaciendousoinmediatodeBoundaryconditions,dondeelfluidode momentun escontinuode
lainterfaselquido liquido
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B.C.1: atx=0, =
Las y sernlaconstantesdelaintegra,estasiguala
Lasustituir
la
leu
de
viscosidad
de
newtons,
en
Fig.
2.5
2y2.5
3obtenemos
Estassepuedenentegrarparaobtener
LastresconstantesdeintegracionsepuedendeterminarsiguiendoNoslipB.C.
B.C.2: atx=0, vIz=V
IIz
B.C.3: atx=b, v=0
B.C.4: atx=+b, v=0
Cuandoestastrescondicionessonaplicadas,conseguimostresecuacionessimultneas
paralasconstantesdelaintegracin:
Deestastresecuacionesconseguimos
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Losresultadosdelflujodemomentunyperfildevelocidadson
Siambasviscosidadessoniguales,despusdeladistribucindelavelocidadesparablica
Lavelocidadmediaencadacapapuedeserobtenidayresultara
Lasdistribucionesdelavelocidaddadasarriba,sepodraobtenerlavelocidadmxima,la
velocidaden
la
interfase,
el
plano
cero
del
estrs
cortante,
yla
friccin
en
las
paredes.
Anteriormentesehansolucionadoproblemasdeflujosviscosos.Sehantratadosolo
componentesrectilneosconuncomponentedevelocidad.Elflujoalrededordeunaesfera
aplicadoscomponentesnonvanishingdelavelocidad,vryvnosepuedeexplicar
convenientementeporlastcnicasexplicadasalprincipiodeestecaptulo.Unabrevediscusin
delflujoalrededordeunaesferasedeterminaaqudebidoalaimportanciadelflujoalrededor
deobjetos.Enelcaptulo4sedemuestracmoobtenerlasdistribucionesdelavelocidadyde
presin.Aqusemuestralosresultadosycomopuedenserutilizadosparaciertasderivaciones
posteriormente.Aqu
como
en
el
captulo
4,
se
trabaja
con
el
arrastre
del
flujo.
(este
en
un
flujo
lento)
ConsideramosaquelflujodeunlquidoincompresiblesobreunaesferaslidadelradioRydel
dimetroDsegnlasindicacionesdefig.2.61.Ellquido,conladensidadpylaviscosidad
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2.6 CREEPINGFLOWAROUNDASPHERE
Theproblemtreatedhereisconcernedwith"creepingflow"thatis,veryslowflow.Thistypeof
flowisalsoreferredtoas"Stokesflow." Weconsiderheretheflowofanincompressiblefluid
aboutasolidsphereofradiusRanddiameterDasshowninFig.2.61.Thefluid,withdensity
andviscosity ,approachesthefixedsphereverticallyupwardinthezdirectionwithauniform
velocity . Forthisproblem,"creepingflow"meansthattheReynoldsnumberRe=D/,is
lessthanabout0.1.Thisflowregimeischaracterizedbytheabsenceofeddyformation
downstreamfromthesphere.
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Thevelocityandpressuredistributionsforthiscreepingfloware:
InthelastequationthequantityP0isthepressureintheplanez=0farawayfromthesphere.
Theterm pgzisthehydrostaticpressureresultingfromtheweightofthefluid,andthetermcontainingvisthecontributionofthefluidmotion.
Equations2.61,2,and3showthatthefluidvelocityiszeroatthesurfaceofthesphere. Furthermore,inthelimitasr ,thefluidvelocityisinthezdirectionwithuniform
magnitudev;thisfollowsfromthefactthatvz=vrcos Vsin ,andvx=vy=0.
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Thecomponentsofthestresstensor rinsphericalcoordinatesmaybeobtainedfrom the
velocitydistributionabovebyusingTableB.1.Theyare
andallothercomponentsarezero.Notethatthenormalstressesforthisflowarenonzero,
exceptatr=R.
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IntegrationoftheNormalForce
Ateachpointonthesurfaceofthespherethefluidexertsaforceperunitarea (p+
rr)/r=Ronthesolid,actingnormaltothesurface.Sincethefluidisintheregionofgreaterrand
thesphereintheregionoflesserr,wehavetoaffixaminussigninaccordancewiththesign
conventionestablishedin1.2.Thezcomponentoftheforceis (p+ rr)/r=R(cos).Wenow
multiplythisbyadifferentialelementofsurfaceR2sindd togettheforceonthesurface
element(seeFig.A.82).Thenweintegrateoverthesurfaceofthespheretogettheresultant
normalforceinthezdirection:
AccordingtoEq.2.65,thenormalstress rriszero5atr=Randcanbeomittedintheintegralin
Eq.2.67.Thepressuredistributionatthesurfaceofthesphereis,accordingtoEq.2.64,
WhenthisissubstitutedintoEq.2.67andtheintegrationperformed,thetermcontainingp0
giveszero,thetermcontainingthegravitationalaccelerationggivesthebuoyantforce,andthe
termcontainingtheapproachvelocityv givesthe"formdrag"asshownbelow:
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Thebuoyantforceisthemassofdisplacedfluid(4/3 R3)timesthegravitationalacceleration
(g).
IntegrationoftheTangentialForce
Ateachpointonthesolidsurfacethereisalsoashearstressactingtangentially.The
forceperunitareaexertedinthe directionbythefluid(regionofgreaterr)onthesolid
(regionoflesserr)is+r/r=R .Thezcomponentofthisforceperunitareais(r/r=R)sin.We
nowmultiplythisbythesurfaceelementR2sindd andintegrateovertheentirespherical
surface.Thisgivestheresultantforceinthezdirection:
Theshearstressdistributiononthespheresurface,fromEq.2.66,is
SubstitutionofthisexpressionintotheintegralinEq.2.610givesthe"frictiondrag"
HencethetotalforceFofthefluidonthesphereisgivenbythesumofEqs.2.69and2.612:
or
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Thefirsttermisthebuoyantforce,whichwouldbepresentinafluidatrest;itisthemassofthedisplacedfluidmultipliedbythegravitationalacceleration.
Thesecondterm,thekineticforce,resultsfromthemotionofthefluid. TherelationFk=6R(2.615)isknownasStokeslaw. Itisusedindescribingthemotionofcolloidalparticlesunderanelectricfield,inthe
theoryofsedimentation,andinthestudyofthemotionofaerosolparticles.
Stokes'lawisusefulonlyuptoaReynoldsnumberRe=Dv/ ofabout0.1. AtRe=1,Stokes'lawpredictsaforcethatisabout10%.toolow.
Example
Derivearelationthatenablesonetogettheviscosityofafluidbymeasuringthe
terminalvelocity tofasmallsphereofradiusRinthefluid.
Ifasmallsphereisallowedtofallfromrestinaviscousfluid,itwillaccelerateuntilitreachesaconstantvelocitytheterminalvelocity.
Whenthissteadystateconditionhasbeenreachedthesumofalltheforcesactingonthespheremustbezero.
Theforceofgravityonthesolidactsinthedirectionoffall,andthebuoyantandkineticforcesactintheoppositedirection:
Herepsandparethedensitiesofthesolidsphereandthefluid.Solvingthisequationfortheterminalvelocitygives
ThisresultmaybeusedonlyiftheReynoldsnumberislessthanabout0.1.