ACourseinFluidMechanics

withVectorFieldTheory

by

DennisC.Prieve

DepartmentofChemicalEngineering

CarnegieMellonUniversity

Pittsburgh,PA15213

AnelectronicversionofthisbookinAdobePDF?formatwasmadeavailableto

studentsof()6-703,DepartmentofChemicalEngineering,

CarnegieMellonUniversity,Fall,2000.

Copyright?2(XX)byDennisC.Prieve

06-7031Fall,2000

TableofContents

ALGEBRAOFVECTORSANDTENSORS1

DefinitionofDyadicProduct2

DECOMPOSITIONINTOSCALARCOMPONENTS3

GeometricMeaningoftheGradient6

ApplicationsofGradient7

CURVILINEARCOORDINATES7

CylindricalCoordinates7

SphericalCoordinates8

DIFFERENTIATIONOFVECTORSW.R.T.SCALARS9

VFCTORFlFLDSI；

FluidVelocityasaVectorFieldII

PARTIAL&MATERIALDERIVATIVES12

CALCULUSOFVECTORFIELDS14

DIVERGENCE,CURL,ANDGRADIENT16

PhysicalInterpretationofDivergence16

Calculationo/V-vinR.C.C.S16

Evaluationo廣.v,VxvandVvinCurvilinearCoordinates19

PhysicalInterpretationofCurl20

VECTORFIELDTHEORY22

CorollariesoftheDivergenceTheorem24

TheContinuityEquation24

ReynoldsTransportTheorem26

VelocityCirculation:PhysicalMeaning28

DERIVABLEFROMASCALARPOTENTIAL29

TRANSPOSEOFATENSOR,IDENTITYTENSOR31

DIVERGENCEOFATENSOR32

INTRODUCTIONTOCONTINUUMMECHANICS*34

CONTINUUMHYPOTHESIS34

HYDROSTATICEQUILIBRIUM37

FLOWOFIDEALFLUIDS37

IRROTATIONALFLOWOFANINCOMPRESSIBLEFLUID42

PotentialFlowAroundaSphere45

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Two-DFLOWS54

AXISYMMETRICFLOW(CYLINDRICAL)55

STREAMLINES,PATHLINESANDSTREAKLINES57

PHYSICALMEANINGOFSTREAMFUNCTION58

VISCOUSFLUIDS62

TENSORIALNATUREOFSURFACEFORCES62

GENERALIZATIONOFEULER'SEQUATION66

RESPONSEOFELASTICSOLIDSTOUNIAXIALSTRESS70

RESPONSEOFELASTICSOLIDSTOPURESHEAR72

RESPONSEOFAVISCOUSFLUIDTOPURESHEAR75

GENERALIZEDNEWTON'SLAWOFVISCOSITY76

EXACTSOLUTIONSOFN-SEQUATIONS80

FlowinLongStraightConduitofUniformCrossSection81

FlowofThinFilmDownInclinedPlane84

PROBLEMSWITHNON-ZEROINERTIA89

CREEPINGFLOWAPPROXIMATION91

CREEPINGFLOWAROUNDASPHERE(Re->0)96

Scaling97

VelocityProfile99

DisplacementofDistantStreamlines101

CORRECTINGFORINERTIALTERMS106

FLOWAROUNDCYLINDERASRE—>0109

BOUNDARY-LAYERAPPROXIMATION110

MATHEMATICALNATUREOFBOUNDARYLAYERS111

MATCHED-ASYMPTOTICEXPANSIONS115

MAE'sAPPLIEDTO2-DFLOWAROUNDCYLINDER120

InnerExpansion120

BoundaryLayerThickness120

PRANDTUSB.L.EQUATIONSFOR2-DFLOWS120

ALTERNATEMETHOD：PRANDTL*SSCALINGTHEORY120

TimeOut:FlowNexttoSuddenlyAcceleratedPlate120

TimeIn:BoundaryLayeronFlatPlate120

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SOLUTIONFORASYMMETRICCYLINDER120

DragCoefficientandBehaviorintheWakeoftheCylinder120

THELUBRICATIONAPPROXIMATION157

TRANSLATIONOFACYLINDERALONGAPLATE163

TURBULENCE176

TURBULENTFLOWINPIPES177

TIME-SMOOTHINGOFCONTINUITYEQUATION180

TIME-SMOOTHINGOFTHENAVIER-STOKESEQUATION180

ANALYSISOFTURBULENTFLOWINPIPES182

PRANDTL'SMIXINGLENGTHTHEORY184

PRANDTL'S“UNIVERSAL“VELOCITYPROFILE187

PRANDTfSUNIVERSALLAWOFFRICTION189

ELECTROHYDRODYNANUCS120

GOUY-CHAPMANMODELOFDOUBLELAYER120

ELECTROSTATICBODYFORCES120

ELECTROKINETICPHENOMENA120

SMOLUCHOWSKI'SANALYSIS(CA.1918)120

ELECTRO-OSMOSISINCYLINDRICALPORES120

SURFACETENSION120

BOUNDARYCONDITIONSFORFLUIDFLOW120

INDEX211

Copyright?2000byDennisC.Prieve

06-7031Fall,2000

AlgebraofVectorsandTensors

Whereasheatandmassarescalars,fluidmechanicsconcernstransportofmomentum,whichisa

vector.Heatandmassfluxesarevectors,momentumfluxisatensor.Consequently,themathematical

descriptionoffluidflowtendstobemoreabstractandsubtlethanforheatandmasstransfer.Inan

efforttomakethestudentmorecomfortablewiththemathematics,wewillstartwithareviewofthe

algebraofvectorsandanintroductiontotensorsanddyads.Abriefreviewofvectoradditionand

multiplicationcanbefoundinGreenberg,*pages132-139.

Scalar-aquantityhavingmagnitudebutnodirection(e.g.temperature,density)

Vector-(a.k.a.1stranktensor)aquantityhavingmagnitudeanddirection(e.g.velocity,force,

momentum)

(2ndrank)Tensor-aquantityhavingmagnitudeandtwodirections(e.g.momentumflux,

stress)

VECTORMULTIPLICATION

Giventwoarbitraryvectorsaandb,therearethreetypesofvectorproducts

aredefined:

NotationResultDefinition

DotProducta-bscalarabcos0

CrossProductaxbvectorab|sinGIn

where0isaninteriorangle(0<0<TT)andnisaunitvectorwhichisnormaltobothaandb.The

senseofnisdeterminedfromthe"right-hand-rule"?

DyadicProductabtensor

*Greenberg,M.D.,FoundationsOfAppliedMathematics,Prentice-Hall,1978.

?The^right-handrule”:withthefingersoftherighthandinitiallypointinginthedirectionofthefirst

vector,rotatethefingerstopointinthedirectionofthesecondvector;thethumbthenpointsinthe

directionwiththecorrectsense.Ofcourse,thethumbshouldhavebeennormaltotheplanecontaining

bothvectorsduringtherotation.Inthefigureaboveshowingaandb,axbisavectorpointingintothe

page,whilebxapointsoutofthepage.

Copyright?2000byDennisC.Prieve

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Intheabovedefinitions,wedenotethemagnitude(orlength)ofvectorabythescalara.Boldfacewill

beusedtodenotevectorsanditalicswillbeusedtodenotescalars.Second-ranktensorswillbe

denotedwithdouble-underlinedboldface;e.g.tensorJ.

DefinitionofDyadicProduct

Reference:AppendixBfromHappel&Brenner.*Thewordudyad^^comesfromGreek:“dy”

meanstwowhile“ad”meansadjacent.Thusthenamedyadreferstothewayinwhichthisproductis

denoted:thetwovectorsarewrittenadjacenttooneanotherwithnospaceorotheroperatorin

between.

ThereisnogeometricalpicturethatIcandrawwhichwillexplainwhatadyadicproductis.It*sbest

tothinkofthedyadicproductasapurelymathematicalabstractionhavingsomeveryusefulproperties:

DyadicProductab-thatmathematicalentitywhichsatisfiesthefollowingproperties(wherea,

b,v,andwareanyfourvectors):

1.ab-v=a(b?v)[whichhasthedirectionofa;notethatba-v=b(a?v)whichhasthedirectionof

b.Thusab工basincetheydon'tproducethesameresultonpost-dottingwithv.]

2.v-ab=(v-a)b[thusv?abab-v]

3.abxv=a(bxv)whichisanotherdyad

4.vxab=(vxa)b

5.ab:vw=(a?w)(b-v)whichissometimesknownastheinner-outerproductorthedouble-dot

product：

6.a(v+w)=av+aw(distributiveforaddition)

7.(v+w)a=va+wa

8.(s+/)ab=sab+rab(distributiveforscalarmultiplication-alsodistributivefordotandcross

product)

9.sab=(sa)b=a(sb)

“Happel,J.,&H.Brenner,LowReynoldsNumberHydrodynamics.Noordhoff,1973.

“Brennerdefinesthisas(a?v)(b?w).Althoughthetwodefinitionsarenotequivalent,eithercanbe

used-aslongasyouareconsistent.Inthesenotes,wewilladoptthedefinitionaboveandignore

Brennefsdefinition.

Copyright?2000byDennisC.Prieve

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DECOMPOSITIONINTOSCALARCOMPONENTS

Threevectors(saye］,e?，and03)aresaidtobelinearlyindependentifnonecanbeexpressed

asalinearcombinationoftheothertwo(e.g.i,j,andk).GivensuchasetofthreeLIvectors,any

vector(belongingtoE3)canbeexpressedasalinearcombinationofthisbasis:

v=vlel+v2e2+v3e3

wherethev,arecalledthescalarcomponentsofv.Usually,forconvenience,wechoose

orthonormalvectorsasthebasis:

lifi=j

3號=而=

OifiWj

althoughthisisnotnecessary.即iscalledtheKroneckerdelta.Justasthefamiliardotandcross

productscanwrittenintermsofthescalarcomponents,socanthedyadicproduct:

VW=(v1e222+^3。3)(卬遇1+卬2。2+叩3￡3)

=(vje1)(w1eI)+(V)ej)(w2e2)+...

=?［卬遇匹］+1/］卬20遇2+…(nineterms)

wherethee’e,areninedistinctunitdyads.Wehaveappliedthedefinitionofdyadicproductto

performthesetwosteps:inparticularitems6,7and9inthelistabove.

Moregenerallyanynthranktensor(inE^)canbeexpressedasalinearcombinationofthe券unitn-

ads.Forexample,ifn=2,3〃=9andann-adisadyad.Thusageneralsecond-ranktensorcanbe

decomposedasalinearcombinationofthe9unitdyads:

1=7，llelel+7，12ele2+-='=1,3?廣1,37瀘

tensors

Althoughadyad(e.g.vw)isanexampleofasecond-ranktensor,notall'\

2ndranktensorsIcanbeexpressedasadyadicproductoftwovectors,1/山-^\I

Toseewhy,notethatageneralsecond-ranktensorhasninescalar\\//

componentswhichneednotberelatedtooneanotherinanyway.By\—

contrast,the9scalarcomponentsofdyadicproductaboveinvolveonlysix

distinctscalars(the3componentsofvplusthe3componentsofw).

Afterawhileyougettiredofwritingthesummationsignsandlimits.Soan

abbreviationwasadoptedwherebyrepeatedappearanceofanindeximpliessummationoverthethree

allowablevaluesofthatindex:

工=丁恒

Copyright?2000byDennisC.Prieve

06-7034Fall,2000

ThisissometimescalledtheCartesian(implied)summationconvention.

SCALARFIELDS

SupposeIhavesomescalarfunctionofposition(x,y,z)whichiscontinuouslydifferentiable,that

is

anddf/dx,df/dy,anddf/dzexistandarecontinuousthroughoutsome3-Dregioninspace.This

functioniscalledascalarfield.Nowconsider/atasecondpointwhichisdifferentiallyclosetothe

first.Thedifferenceinfbetweenthesetwopointsis

calledthetotaldifferentialof/:

f(x+dxfy+dyfz+dz)-/(x,y,z)=df

Foranycontinuousfunction/(x,y,z),dfislinearlyrelated

tothepositiondisplacements,dx,dyandThat

linearrelationisgivenbytheChainRuleof

differentiation:

Insteadofdefiningpositionusingaparticularcoordinate

system,wecouldalsodefinepositionusingapositionvectorr:

r=xi++zk

Thescalarfieldcanbeexpressedasafunctionofavectorargument,representingposition,insteadofa

setofthreescalars:

/=/(r)

Consideranarbitrarydisplacementawayfromthepointr,whichwedenoteasdrtoemphasizethatthe

magnitude|dr\ofthisdisplacementissufficiendysmallthatf(r)canbelinearizedasafunctionof

positionaroundr.Thenthetotaldifferentialcanbewrittenas

Copyright?2000byDennisC.Prieve

06-7035Fall,2000

df=/(r+Jr)-/(r)

GRADIENTOFASCALAR

Wearenowisapositiontodefineanimportantvectorassociated

withthisscalarfield.Thegradient(denotedasV/)isdefined

suchthatthedotproductofitandadifferentialdisplacement

vectorgivesthetotaldifferential:

dfmdrZf

EXAMPLE:ObtainanexplicitformulaforcalculatingthegradientinCartesian*coordinates.

Solution'.r=xi++zk

r+c/r=(x+dx)i+(y+dy)j+(z+dz)k

subtracting:dr=(dx)i+(dy)j+?z)k

v=(w+(w+(w

dr?Vf=[(dx)i+..J?KW)/+???]

df=(Nf)xdx+(yf)ydy+⑴

UsingtheChainrule:df=(df/dx)dx+(df/dy)dy+(df/dz)dzQ)

Accordingtothedefinitionofthegradient,(1)and(2)areidentical.Equatingthemandcollectingterms:

+KV/y(型力)"+=o

Thinkofdx,dy,anddzasthreeindependentvariableswhichcanassumeaninfinitenumberofvalues,

eventhoughtheymustremainsmall.Theequalityabovemustholdforallvaluesofdx,dy,anddz.The

onlywaythiscanbetrueisifeachindividualtermseparatelyvanishes:**

*NamedafterFrenchphilosopherandmathematicianReneDescartes(1596-1650),pronounced"day-

cart",whofirstsuggestedplotting/(x)onrectangularcoordinates

**Foranyparticularchoiceofdx,dy,anddz,wemightobtainzerobycancellationofpositiveand

negativeterms.Howeverasmallchangeinoneofthethreewithoutchangingtheothertwowouldcause

thesumtobenonzero.Toensureazero-sumforallchoices,wemustmakeeachtermvanish

independently.

Copyright?2000byDennisC.Prieve

06-7036Fall,2000

So(Y/)x=df/dx,(yf)y=^f/dy,and(V/)2=df/dz,

leaving如i+蛆j+/k

dx3ydz

Otherwaystodenotethegradientinclude:

V/'=gradf=df/dr

GeometricMeaningoftheGradient

1)direction:V/(r)isnormaltothe尸constsurfacepassingthroughthepointrinthedirectionof

increasing/.V/,alsopointsinthedirectionofsteepestascentoff.

2)magnitude:|V/]istherateofchangeoffwith

distancealongthisdirection

Whatdowemeanbyan7=constsurface1*?Consideran

example.

Example'.Supposethesteadystatetemperatureprofile

insomeheatconductionproblemisgivenby:

T(x,y,Z)=H+,2+

PerhapsweareinterestedinNTatthepoint(3,3,3)

where7=27.NTisnormaltotheT=constsurface:

+y2+/=27

whichisasphereofradiusV27.*

Proofof1),Let'susethedefinitiontoshowthatthesegeometricmeaningsarecorrect.

df=dr-Vf

*Averticalbarintheleftmargindenotesmaterialwhich(intheinterestoftime)willbeomittedfromthe

lecture.

Copyright?2000byDennisC.Prieve

06-7037Fall,2000

Consideranarbitraryf.Aportionofthe六constsurface

containingthepointrisshowninthefigureatright.Choosea

drwhichliesentirelyon/^=const.Inotherwords,thesurface

containsbothrandr+dr,so

{r)=/(r+dr)

anddf=f(r+dr)-f(r)=0

Substitutingthisintothedefinitionofgradient:

df=0=dr-Vf=|dr||V7|

SinceIdrIandIV/|areingeneralnotzero,weareforced

totheconclusionthatcos0=Oor0=90°.ThismeansthatV/isnormaltodrwhichliesinthesurface.

2)canbeprovedinasimilarmanner:choosedrtobeparalleltoV/.DoesNfpointtowardhigheror

lowervaluesof/?

ApplicationsofGradient

?findavectorpointinginthedirectionofsteepestascentofsomescalarfield

?determineanormaltosomesurface(neededtoapplyb.c/s1汰en-v=0foraboundarywhichis

impermeable)

?determinetherateofchangealongsomearbitrarydirection:ifnisaunitvectorpointingalongsome

path,then

nF嗎

OS

istherateofchangeoffwithdistance(s)alongthispathgivenbyn.df/dsiscalledthedirected

derivativeoff.

CURVILINEARCOORDINATES

Inprinciple,allproblemsinfluidmechanicsandtransportcouldbesolvedusingCartesian

coordinates.Often,however,wecantakeadvantageofsymmetryinaproblembyusinganother

coordinatesystem.Thisadvantagetakestheformofareductioninthenumberofindependentvariables

(e.g.PDEbecomesODE).Afamiliarexampleofanon-Cartesiancoordinatesystemis:

Copyright?2000byDennisC.Prieve

06-7038Fall,2000

CylindricalCoordinates

r=(]2+y2)l/2x=rcOs0

0=tan-1(y/x)y-rsin0

z=zz=z

Vectorsaredecomposeddifferently.Insteadof

inR.C.C.S.:v=vxi+

incylindricalcoordinates,wewrite

incyl.coords.:v=vrer+v0e0+

whereer,eg,ande.arenewunitvectorspointingther,0andzdirections.Wealsohaveadifferent

setofnineunitdyadsfordecomposingtensors:

erer,ereg,erez,ege,.,etc.

LiketheCartesianunitvectors,theunitvectorsincylindricalcoordinatesformanorthonormalsetof

basisvectorsforE?.UnlikeCartesianunitvectors,theorientationoferande°dependonposition.In

otherwords:

er=er(0)

%=。(。)

Copyright?2000byDennisC.Prieve

06-7039Fall,2000

SphericalCoordinates

SphericalcoordinatesaredefinedrelativetoCartesiancoordinatesassuggestedinthe

figuresabove(twoviewsofthesamething).Thegreensurfaceisthexy-plane,theredsurfaceisthe

xz-plane,whilethebluesurface(atleastintheleftimage)istheyz-plane.Thesethreeplanesintersectat

theorigin(0,0,0),whichliesdeeperintothepagethan(1,1,0).Thestraightredline,drawnfromthe

origintothepoint(/\0,(|))*haslengthr,Theangle0istheangletheredlinemakeswiththez-axis(the

redcirculararclabelled0hasradiusrandissubtendedbytheangle0).Theangle。(measuredinthe

xy-plane)istheanglethesecondblueplane(actuallyifsonequadrantofadisk)makeswiththexy-

plane(red).Thisplanewhichisaquadrantofadiskisa(j)=constsurface:allpointsonthisplanehave

thesame(|)coordinate.Thesecondred(circular)arclabelled。isalsosubtendedbytheangle([>.

*Thisparticularfigurewasdrawnusingr-1,0=TC/4and([)=兀/3.

Copyright?2000byDennisC.Prieve

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Anumberofother(|)=constplanesare

showninthefigureatright,alongwitha

sphereofradiusr=l.Alltheseplanes

intersectalongthez-axis,whichalsopasses

throughthecenterofthesphere.1-

x=rsin0cos(|)0.5-

y=rsin0sin(|)0=tan-1+產°-

-0.5-

z=rcosG0=tan-1(y/x)

-1-

Thepositionvectorinsphericalcoordinates

isgivenby0-0.5

yX

r=xi+)j+zk=rer(0?<l))

Inthiscaseallthreeunitvectorsdependon

position:

er=e.。,#,ee=ee(0,(|)),ande。=e/Q)

whereeristheunitvectorpointingthedirectionofincreasingr,holding0and(|)fixed;e0istheunit

vectorpointingthedirectionofincreasing0,holdingrand0fixed;ande,istheunitvectorpointingthe

directionofincreasing%holdingrand0fixed.

Theseunitvectorsareshowninthefigureatright.

Noticethatthesurface0=constisaplanecontainingthe

pointritself,theprojectionofthepointontothexy-plane

andtheorigin.Theunitvectorserandeelieinthisplanez

aswellastheCartesianunitvectork(sometimes*

denotedez).

Ifwetiltthis(|)=constplane

intotheplaneofthepage(asinthesketchatleft),wecanmoreeasilysee

therelationshipbetweenthesethreeunitvectors:

unitcircleone.=(cos0)er-(sin0)ee

(j>=const

surface

Thisisobtainedbydeterminedfromthegeometiyoftherighttrianglein

thefigureatleft.Whenanyoftheunitvectorsispositiondependent,we

saythecoordinatesare:

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curvilinear-atleastoneofthebasisvectorsispositiondependent

Thiswillhavesomeprofoundconsequenceswhichwewillgettoshortly.Butfirst,weneedtotake

“time-out“todefine:

DIFFERENTIATIONOFVECTORSW.R.T.SCALARS

Supposewehaveavectorvwhichdependsonthescalarparametert:

v=v(r)

Forexample,thevelocityofasatellitedependsontime.Whatdowemeanbythe“derivative”ofa

vectorwithrespecttoascalar.AsintheFundamentalTheoremofCalculus,wedefinethederivative

as:

dvlimv(r+Ar)-v(r)

—=

dtNt

Notethatdy/dtisalsoavector.

EXAMPLE:ComputedeJdBincylindricalcoordinates.

Solution:Fromthedefinitionofthederivative:

de「_lim|巧.一+一與(田]_lim{Aer

c/SAe^0[A6JAB-^0[A0

Sincethelocationofthetailofavectorisnotpart

ofthedefinitionofavector,let'smoveboth

vectorstotheorigin(keepingtheorientation

fixed).Usingtheparallelogramlaw,weobtainthe

differencevector.Itsmagnitudeis:

|er(e+A0)-er(G)|=2sin^

Itsdirectionisparalleltoe0(6+A0/2),so:

er(e+A0)-er(G)=2si噂e?(6+豹

Recallingthatsinxtendstoxasx70,wehave

Copyright?2000byDennisC.Prieve

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lim{er(0+A0)-er(e)}=A0ee(0)

DividingthisbyAO,weobtainthederivative:

deJdB=e。

Similarly,=-er

Oneimportantapplicationof''differentiationwithrespecttoa

scalar“isthecalculationofvelocity,givenpositionasafunctionof

time.Ingeneral,ifthepositionvectorisknown,thenthevelocity

canbecalculatedastherateofchangeinposition:

r=r(r)

v=dr/dt

Similarly,theaccelerationvectoracanbecalculatedasthe

derivativeofthevelocityvectorv:

a=dx/dt

EXAMPLE:Giventhetrajectoryofanobjectin

cylindricalcoordinates

r=r(0,0=0(0,andz=z(Z)

Findthevelocityoftheobject.

Solution:First,weneedtoexpressrinintermsofthe

unitvectorsincylindricalcoordinates.Usingthefigureat

right,wenotebyinspectionthat*

r(八e,z)=rer(0)+zez

NowwecanapplytheChainRule:

Recallingthatr=xi+yj+zkinCartesiancoordinates,youmightbetemptedtowriter=rer+0ee+

zezincylindricalcoordinates.Ofcourse,thistemptationgivesthewrongresult(inparticular,theunits

oflengthinthesecondtermaremissing).

Copyright?2000byDennisC.Prieve

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「an(加、

Jr=—Jr+—<70+—dz=erdr+i-e^dQ+e-dz

1氏人,0

er”e「(。)與

r

d。

ee

Dividingbydi,weobtainthevelocity:

drdr(t')t/0(z)以(。

dtdtdtdtz

v

ri'evz

VECTORFIELDS

Avectorfieldisdefinedjustlikeascalarfield,exceptthatit'savector.Namely,avectorfieldisa

position-dependentvector:

v=v(r)