Class Notes Topic 31

8/2/2019 Class Notes Topic 31

1/31

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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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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2.5FLUJODEDOSLQUIDOSIMMISIBLESADYACENTES

Situacindelainterfasededoslquidos,estosfluyenendireccindelejedezconunalongitud

LyunanchoW,esteflujobajoungradientedepresinhorizontalexpresadocomo(p0p)/L.el

flujode

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de

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dividan

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sus

densidades.

El

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deba

ser

lo

suficientementelentoparaquenopresenteninestabilidadenlainterfasedeestos,estopara

encontrarelflujodemomentumylavelocidaddedistribucin.

Ecuacindiferencialparaflujodemomentun

Alintegrallaecuacinanteriorseobtiene

Dosflujos

immisibles

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aplicacin

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de

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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

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velocidaden

la

interfase,

el

plano

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del

estrs

cortante,

yla

friccin

en

las

paredes.

Anteriormentesehansolucionadoproblemasdeflujosviscosos.Sehantratadosolo

componentesrectilneosconuncomponentedevelocidad.Elflujoalrededordeunaesfera

aplicadoscomponentesnonvanishingdelavelocidad,vryvnosepuedeexplicar

convenientementeporlastcnicasexplicadasalprincipiodeestecaptulo.Unabrevediscusin

delflujoalrededordeunaesferasedeterminaaqudebidoalaimportanciadelflujoalrededor

deobjetos.Enelcaptulo4sedemuestracmoobtenerlasdistribucionesdelavelocidadyde

presin.Aqusemuestralosresultadosycomopuedenserutilizadosparaciertasderivaciones

posteriormente.Aqu

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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.

Class Notes Topic 31

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