1.2. THEMATERIALDERIVATIVE
11
1.2.2 Theconvectiontheorem
SupposethatS
t
isaregionoffluidparticlesandletf(x,t)beascalarfunction.
Formingthevolumeintegralover S
t
, F =
S
t
fdV
x
,we seek tocompute
dF
dt
.
NowdV
x
=dx
1
···dx
N
=Det(J)da
1
···a
N
=Det(J)dV
a
. Thus
dF
dt
=
d
dt
S
0
f(x(a,t),t)Det(J)dV
a
=
S
0
Det(J)
d
dt
f(x(a,t),t)dV
a
+
S
0
f(x(a,t),t)
d
dt
Det(J)dV
a
=
S
0
Df
Dt
+fdiv(u)
Det(J)dV
a
,
andso
dF
dt
=
S
t
Df
Dt
+fdiv(u)
dV
x
.
(1.17)
Theresult (1.17) is calledthe convection n theorem. . We e cancontrast this
calculationwithoneoverafixedfiniteregionRofspacewithboundary∂R.In
thatcasetherateofchangeoffcontainedinRisjust
d
dt
R
fdV
x
=
R
∂f
∂t
dV
x
.
(1.18)
Thedifferencebetweenthetwocalculationsinvolvesthefluxoff throughthe
boundaryofthedomain. Indeedwecanwritetheconvectiontheorem m inthe
form
dF
dt
=
S
t
∂f
∂t
+div(fu)
dV
x
.
(1.19)
Usingthe divergence (or Gauss’) theorem, andconsidering theinstant when
S
t
=R,wehave
dF
dt
=
R
∂f
∂t
dV
x
+
∂R
fu·ndS
x
,
(1.20)
wherenistheouternormaltotheregionanddS
x
istheareaelementof∂R.The
secondtermontherightisfluxoffoutoftheregionR. Thustheconvection
theoremincorporatesintothechangeinf withinaregion,thefluxoffintoor
out oftheregion,duetothemotionof theboundaryof theregion. . Once e we
identifyf withausefulphysicalpropertyofthefluid,theconvectiontheorem
willbeusefulforexpressingtheconservationofthisproperty,seechapter2.
1.2.3 Materialvectorfields: : TheLiederivative
Certainvectorfieldsinfluidmechanics,andnotablythevorticityfield,ω(x,t)=
∇×u,seechapter3,canincertaincasesbehaveasamaterialvectorfield. To
understandtheconceptofamaterialvectoronemustimaginethedirectionof
thevectortobedeterminedbynearbymaterialpoints. Itiswrongtothinkof
amaterialvectorasattachedtoafluidparticleandconstantthere. Thiswould
amounttoasimpletranslationofthevectoralongtheparticlepath.
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12
CHAPTER1. THEFLUIDCONTINUUM
Instead, we e want t the direction of the e vector to be that t of f a a differential
segmentconnectingtwonearby fluidparticles,dl
i
=J
ij
da
j
. Furthermore,the
lengthof the materialvector r is s to o be proportionalto this s differentiallength
as timeevolvesandthe particles move. . Consequently, , once theparticles s are
selected,thefutureorientationandlengthofamaterialvectorwillbecompletely
determinedbytheJacobianmatrixoftheflow.
Thus we define amaterialvector fieldas oneof the form (inLagrangian
variables)
v
i
(a,t)=J
ij
(a,t)V
j
(a)
(1.21)
Of course, giventhe e inversea(x,t) we can n express s v v as s afunction n of f x,t to
obtainitsEulerianstructure.
Wenowdeterminethetimerateofchangeofamaterialvectorfieldfollowing
thefluidparcel. Toobtainthiswedifferentiatev(a,t)withrespect t totimefor
fixeda,anddeveloptheresultusingthechainrule:
∂v
i
∂t
a
=
∂J
ij
∂t
a
V
j
(a)=
∂u
i
∂a
j
V
j
=
∂u
i
∂x
k
∂x
k
∂a
j
V
j
=v
k
∂u
i
∂x
k
.
(1.22)
Introducingthematerialderivative,weseethatamaterialvectorfieldsatisfies
thefollowingequationinEulerianvariables:
Dv
Dt
=
∂v
∂t
x
+u·∇v−v·∇u≡v
t
+L
u
v=0
(1.23)
IndifferentialgeometryL
u
iscalledtheLiederivativeofthevectorfieldvwith
respect tothevectorfieldu.
Thewaythisworkscanbeunderstoodbymovingneighboringpointalong
particlepaths.
Figure1.5:Computingthetimederivativeofamaterialvector.
Let v=
AB=∆xbeasmallmaterialvectorattimet,seefigure1.5. At
time ∆tlater, the vector r has s become
CD. The e curved d lines s arethe particle
paths throughA,B ofthevector fieldu(x,t). . SelectingA A asx, we see that
afterasmalltimeinterval∆tthepointCisA+u(x,t)∆tandDisthepoint
B+u(x+∆x,t)∆t. Consequently
CD−
AB
∆t
=u(x+∆x,t)−u(x,t).
(1.24)
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1.2. THEMATERIALDERIVATIVE
13
The left-handside of (1.24) is approximatelyDv/Dt, andright-handside is
approximatelyv·∇u, sointhe line∆x,∆t→ → 0 0 we get t (1.23). . A A material
vector fieldhastheproperty thatits magnitudecanchangebythestretching
propertiesoftheunderlyingflow,anditsdirectioncanchangebytherotation
ofthefluidparcel.
ProblemSet1
1. Considertheflowinthe e (x,y) planegivenbyu= −y,v = x+t. . (a)
Whatis theinstantaneous streamlinethroughtheoriginatt=1?(b)whatis
thepathofthefluidparticleinitiallyattheorigin,0<t<6π? (c)Whatisthe
streaklineemanatingformtheorigin,0<t<6π?
2. Considerthe“pointvortex”flowintwodimensions,
(u,v)=UL(
−y
x2+y2
,
x
x2+y2
),x
2
+y
2
=0,
where U,L are reference valuesof speedandlength. . (a) ) Show w that t theLa-
grangiancoordinatesforthisflowmaybewritten
x(a,b,t)=R
0
cos(ωt+θ
0
), y(a,b,t)=R
0
sin(ωt+θ
0
)
whereR
2
0
=a
2
+b
2
0
=arctan(b/a),andω=UL/R
2
0
.(b)Consider,att=0a
smallrectangleofmarkedfluidparticlesdeterminedbythepointsA(L,0),B(L+
∆x,0),C(L+∆x,∆y),D(L,∆y).Ifthepointsmovewiththefluid,oncepoint
Areturnstoitsinitialpositionwhatistheshapeofthemarkedregion? Since
(∆x,∆y) are small,youmayassumetheregionremains aparallelogram. . Do
this,first,bycomputingtheentry∂y/∂aintheJacobian,evaluatedatA(L,0).
Thenverifyyourresultbyconsideringthe“lag”ofparticleBasitmovesona
slightlylargercircleataslightlyslowerspeed,relativetoparticleA,foratime
takenbyAtocompleteonerevolution.
3.Aswasnotedinclass,Lagrangiancoordinatescanuseanyuniquelabeling
offluidparticles. Toillustratethis,considertheLagrangiancoordinatesintwo
dimensions
x(a,b,t)=a+
1
k
e
kb
sink(a+ct), y=b−
1
k
e
kb
cosk(a+ct),
where k,careconstants. . Note e here a,barenotequalto(x,y) foranyt
0
By
examiningthedeterminantoftheJacobian,verifythatthisgivesauniquelabel-
ingoffluidparticlesprovidedthatb=0.Whatisthesituationifb=0?(These
waves,whichwere discoveredby Gerstner in1802, represent gravitywaves if
c
2
=g/kwhereg is theaccelerationofgravity. . They donothaveanysimple
Eulerianrepresentation. ThesewavesarediscussedinLamb’sbook.)
4. Inonedimension,theEulerianvelocityisgiventobeu(x,t)=2x/(1+t).
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14
CHAPTER1. THEFLUIDCONTINUUM
(a)FindtheLagrangiancoordinatex(a,t).(b)FindtheLagrangianvelocityas
afunctionofa,t. (c)FindtheJacobean∂x/∂a=Jasafunctionofa,t.
5.Forthestagnation-pointflowu=(u,v)=U/L(x,−y),showthatafluid
particleinthefirstquadrantwhichcrossestheliney=Lattimet=0,crosses
thelinex=L at timet=
L
U
log(UL/ψ)onthe streamlineUxy/L =ψ. . Do
this intwoways. . First, , consider r theline integralofu·
ds/(u
2
+v
2
)alonga
streamline. Second,useLagrangianvariables.
6. Let t S bethesurfaceofadeformablebodyinthree dimension,andlet
I =
S
fndS forsome scalar functionf,nbeingthe outwardnormal. . Show
that
d
dt
fndS=
S
∂f
∂t
ndS+
S
(u
b
·n)∇fdS.
(1.25)
(Hint: First t convert toavolumeintegral betweenS S andanouter r surface S
whichis fixed. . Thendifferentiate e andapplytheconvectiontheorem. . Finally
convertbacktoasurfaceintegral.)
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Chapter2
Conservationofmassand
momentum
2.1 Conservationofmass
Everyfluidweconsiderisendowedwithanon-negativedensity,usuallydenoted
byρ,whichisintheEuleriansettingisascalarfunctionofx,t.Itsunitaremass
per unit volume. . Water r hasadensityofabout1gramper cubiccentimeter.
For airthe density is about 10
−3
grams per r cubic c centimeter, , but t ofcourse
pressure and temperature e affect air density significantly. . The e air ina room
of athousandcubic meters= 10
9
cubic centimeters weighs abouta thousand
kilograms,ormorethanaton!
2.1.1 Eulerianform
Letussupposethatmassisbeingaddedorsubtractedfromspaceasafunction
q(x,t),ofdimensionsmassperunitvolumeperunittime.Theconservationof
massinafixedregionRcanbeexpressedusing(1.20)withf=ρ:
d
dt
R
ρdV
x
=
R
∂ρ
∂t
dV
x
+
∂R
ρu·ndS
x
.
(2.1)
Now
d
dt
R
ρdV
x
=
R
qdV
x
(2.2)
andifwebringthesurfaceintegralin(2.1)backintothevolumeintegralusing
thedivergencetheoremwearriveat
R
∂ρ
∂t
+div(uρ)−q
dV
x
=0.
(2.3)
SinceourfunctionsarecontinuousandR is anarbitraryopensetinR
N
,the
integrandin(2.3)mustvanish,yieldingtheconservationof massequationin
15
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16
CHAPTER2. CONSERVATIONOFMASSANDMOMENTUM
theEulerianform:
∂ρ
∂t
+div(uρ)=q.
(2.4)
Notethatthislastequationcanalsobewritten
Dt
+ρdivu=q.
(2.5)
Theconservationofmassequationineitheroftheseformsissometimescalled
(forobscurereasons)theequationofcontinuity.
Theform(2.5)showsthatthematerialderivativeofthedensitychangesin
twoways,eitherbysourcesandsinksofmassq>0orq<0respectively,orelse
bythenon-vanishingofthedivergenceofthevelocityfield. Apositivevalueof
thedivergence,asforu=(x,y,z),isassociatedwithanexpansiveflow,thereby
reducinglocaldensity.Thiscanbeexaminedmorecloselyasfollows.LetV be
asmallvolumeoffluidwhere the densityisessentiallyconstant. . ThenρV V is
themasswithinthisfluidparcel,whichisamaterialinvariantD(ρV)/Dt=0.
ThusDρ/Dt+ρV
−1
DV/Dt=0.Comparingthiswith(2.5)wehave
divu=
1
V
DV
Dt
.
(2.6)
Example2.1: AswehaveseeninChapter1,anincompressiblefluidsatis-
fiesdivu=0.Forsuchafluid,freeofsourcesorsinksofmass,wehave
Dt
=0,
(2.7)
that is,nowdensitybecomes amaterialproperty. . Thisdoes s notsay thatthe
density isconstanteverywhere inspace,onlythatisconstant at agivenfluid
parcel,as it movesabout . . (Notethat t weuseparcelheretosuggest thatwe
havetoaverageoverasmallvolumetocomputethedensity.) Howeverafluidof
constantdensitywithoutmassadditionmustbeincompressible. Thisdifference
isimportant. Seawaterisessentiallyincompressiblebutdensitychangesdueto
salinityareanimportantpartofthedynamicsoftheoceans.
2.1.2 Lagrangianform
Ifq=0theLagrangianformoftheconservationofmassisverysimplebecause
ifwemovewiththefluidthedensitychangesthatweseeareduetoexpansion
anddilationofthefluidparcel,whichiscontrolledbyDet(J).Letaparcelhave
volumeV
0
initially,withessentiallyconstantinitialdensityρ
0
. Thenthemass
oftheparcelisρ
0
V
0
,andisamaterialinvariant. Atlatertimesthedensityis
ρandthevolumeisV
0
Det(J),soconservationofmassisexpressedby
DetJ(a,t)=
ρ
0
ρ
.
(2.8)
Ifq=0theLagrangianconservationofmassmustbewritten
∂t
a
ρDet(J)=Det(J)q(x(a,t),t).
(2.9)
2.1. CONSERVATIONOFMASS
17
ItiseasytogetfromEuleriantoLagrangianformusing(1.14).Assumingq=0,
Dt
+ρdivu=0=
Dt
DDet(J)/Dt
Det(J)
=
1
Det(J)
D
Dt
(ρDet(J))
(2.10)
andtheconnectioniscomplete.
Example 2.2: : Consider, , in one dimension, the unsteady y velocity y field
u(x,t)=
2xt
1+t2
. Weassumenosources s ofsinks of mass, , andset t ρ(x,0)= = x.
Whatisthedensityfieldatlatertimes,inbothEulerianandLagrangianforms?
First note thatthis isareasonablequestion,sincewehaveaconservationof
mass equationto evolve e the density intime. . First t deriving the e Lagrangian
coordinates,wehave
dx
dt
=
2xt
1+t2
, x(0)=a.
(2.11)
Thesolutionisx=a(1+t
2
).TheJacobianisthenJ=1+t
2
.Theequationof
conservationofmassinLagrangianform,giventhatρ
0
(a)=a,isρ=a/(1+t
2
).
Sincea=x/(1+t
2
),the Eulerianformofthedensityisρ=x(1+t
2
)
2
. Itis
easytocheckthatthislastexpressionsatisfiestheEulerianconservationofmass
equationinonedimensionρ
t
+(ρu)
x
=0.
Example 2.3Considerthetwo-dimensionalstagnation-pointflow(u,v)=
(x,−y) withinitialdensity ρ
0
(x,y) =x
2
+y
2
andq =0. . Theflowis s incom-
pressible,soρismaterial. InLagrangianform,ρ(a,b,t)=a
2
+b
2
. Tofindρ
asafunctionofx,y,t,wenotethattheLagrangiancoordinatesofthefloware
(x,y)=(ae
t
,be
−t
),andso
ρ(x,y,t)=(xe
−t
)
2
+(ye
t
)
2
=x
2
e
−2t
+y
2
e
2t
.
(2.12)
Thelinesofconstant density,whichareinitiallycircles centeredattheorigin,
areflattenedintoellipsesbytheflow.
2.1.3 Anotherconvectionidentity
Frequently fluidproperties s are e most conveniently thought of asdensities per
unitmassratherthanper unitvolume. . Iftheconservationsuchaquantity,f
say,istobeexamined,wewillneedtoconsiderρf toget“fperunitvolume”
andsobeabletocomputetotalamountbyintegrationoveravolume. Consider
then
d
dt
S
t
ρfdV
x
=
S
t
∂ρf
∂t
+div(ρfu)
dV
x
.
(2.13)
We now assume conservationof mass withq = = 0. . From m theproduct ruleof
differentiationwehavediv(ρfu)=fdiv(ρu)+ρu·∇f,andsotheintegrandsplits
intoapartwhichvanishesbyconservationofmass,andamaterialderivativeof
f timethedensity:
d
dt
S
t
ρfdV
x
=
S
t
ρ
Df
Dt
dV
x
.
(2.14)
Thustheeffectofthemultiplierρistoturnthederivativeoftheintegralinto
anintegralofamaterialderivative.
18
CHAPTER2. CONSERVATIONOFMASSANDMOMENTUM
2.2 Conservationofmomentuminanidealfluid
Themomentumofafluidisdefinedtobeρu,perunitvolume.Newton’ssecond
lawofmotionstates that momentumis conservedby amechanicalsystem of
massesifnoforcesactonthesystem. Wearethusinapositiontouse(2.14),
wherethe“sourcesandsinks”ofmomentumareforces.
If F(x,t)is the forceacting on n the e fluid, per r unit t volume, thenwehave
immediately(assumingconservationofmasswithq=0),
ρ
Du
Dt
=F.
(2.15)
Sincewehaveseenthat
Du
Dt
isthefluidacceleration,(2.15)statesNewton’sLaw
thatmasstimesaccelerationequalsforce,inbothmagnitudeanddirection.
OfcoursetheLagrangianformof(2.15)isobtainedbyreplacingtheaccel-
erationbyitsLagrangiancounterpart:
ρ
2
x
∂t2
a
=F.
(2.16)
Themainissues involvedwithconservationofmomentumarethoseconnected
withtheforceswhichareonaparceloffluid.Therearemanypossibleforcesto
consider:pressure,gravity,viscous,surfacetension,electromotive,etc. Eachhas
aphysicaloriginandamathematicalmodelwithasupportingsetofobservation
andanalysis. Inthepresent t chapter weconsider onlyanidealfluid. . Theonly
new fluidvariablewewillneedtointroduceisthepressure,ascalar r function
p(x,t).
IngeneraltheforceFappearingin(2.15)isassumedtotaketheform
F
i
=f
i
+
∂σ
ij
∂x
j
.
(2.17)
Here f f is s abody force (exertedfrom the “outside”),andσis asecond-order
tensor calledthe e stresstensor. . Integrated d over r aregionR, , the force onthe
regionis
R
FdV
x
=
R
fdV
x
+
∂R
σ·ndS
x
,
(2.18)
usingthedivergencetheorem. Wecanthusseethattheeffectofthestresstensor
istoproduceaforceontheboundaryofanyfluidparcel,thecontributionfrom
anareaelement to o this s force beingσ
ij
n
j
dS
x
for anoutwardnormaln. . The
remainingbodyforcef willsometimesbetakentobe auniformgravitational
fieldf = = ρg, , where g = = constant. . Onthesurface e of the earth h gravity y acts
towardtheEarth’scenterwithastrengthg≈980cm/sec
2
. Wealsointroduce
ageneralforcepotentialΦ,suchthatf=−ρ∇Φ.
2.2. CONSERVATIONOFMOMENTUMINANIDEALFLUID
19
2.2.1 Thepressure
Anidealfluidisdefinedbyastresstensoroftheform
σ
ij
=−pδ
ij
=
−p
0
0
0
−p
0
0
0
−p
,
(2.19)
where δ
ij
= 1,i= j,= 0otherwise. . Thuswhenpressure e ispositivetheforce
onthesurfaceofaparcelisoppositetotheouternormal,asintuitionsuggests.
Notethatnow
divσ=−∇p.
(2.20)
Foracompressiblefluidthepressureaccountsphysicallyforthe resistance
tocompression.Butpressurepersistsasafundamentalsourceofsurfaceforces
foranincompressiblefluid,anditsphysicalmeaningintheincompressiblecase
issubtle.
1
Anidealfluidwithnomassadditionandnobodyforcethussatisfies
ρ
Du
Dt
+∇p=0,
(2.21)
togetherwith
Dt
+ρdivu=0.
(2.22)
ThissystemofequationforanidealfluidarealsooftenreferredtoasEuler’s
equations. ThetermEulerflowisalsoinwideuse.
With Euler’s s system m we e have N+1 equations s for the N+2 unknowns
u
1
,...,u
N
,ρ,p.Anotherequationwillbeneededtocompletethesystem. One
possibilityistheincompressibleassumptiondivu=0. Acommonoptionisto
assumeconstantdensity.Thenρiseliminatedasanunknownandtheconserva-
tionofmassequationisreplacedbytheincompressibilitycondition. Forgases
themissingrelationisanequationofstate,whichbringsinthethermodynamic
propertiesofthefluid.
The pressure force as s we e have defined d it t above is isotropic, inthe sense
thepressure isthesameindependently oftheorientationof theareaelement
on whichit t acts. . A A simple two-dimensionaldiagram willillustrate why this
isso, see figure2.1. . Supposethatthepressure e isp
i
onthe faceoflengthL
i
.
Equatingforces,wehavep
1
L
1
cosθ=p
2
L
2
,p
1
L
1
sinθ=p
3
L
3
. ButL
1
cosθ=
L
2
,L
1
sinθ=L
3
,soweseethat p
1
=p
2
=p
3
. Soindeedthepressure e sensed
byafacedoesnotdependupontheorientationoftheface.
2.2.2 Lagrangianformofconservationofmomentum
TheLagrangianformoftheaccelerationhasbeennotedabove.Themomentum
equationofanidealfluidrequiresthatweexpress∇pasaLagrangianvariable.
1One aspect of the incompressiblecaseshouldbe noted here, namely that the pressure
is arbitraryup toanadditiveconstant. . Consequentlyit t is s onlypressure differences s which
matter. Thisisnotthecaseforacompressiblegas.
20
CHAPTER2. CONSERVATIONOFMASSANDMOMENTUM
Figure2.1:Isotropicityofpressure.
Thatis,ifpistobeafunctionofa,tthensince∇hereisactuallythexgradient
x
,wehave∇
x
p=J−1
a
p. ThisappearanceoftheJacobianisanawkward
feature ofLagrangianfluiddynamics, andis oneofthereasons thatweshall
emphasizeEulerianvariablesindiscussingthedynamicsofafluid.
2.2.3 Hydrostatics: : theArchimedeanprinciple
Hydrostaticsisconcernedwithfluidsatrest(u=0),usuallyinthepresenceof
gravity. Weconsider r hereonlythe caseofafluidstratifiedinone dimension.
Tofixthecoordinatesletthez-axisbeverticalup,andg=−gi
z
,wheregisa
positiveconstant. Wesupposethatthedensity y is afunctionofz alone. . This
allows,forexample,abodyofwaterbeneathastratifiedatmosphere.Letasolid
three-dimensionalbody(anydeformationofasphereforexample)besubmerged
inthefluid. Archimedesprinciplesays s thattheforceexertedbythepressure
onthesurfaceofthebodyisequaltothetotalweightofthefluiddisplacedby
thebody. Wewanttoestablishthisprincipleinthecaseconsidered.
Nowthepressure satisfies ∇p=−gρ(z)i
z
. The e pressureforce is givenby
F
pressure
= −
pndS taken over r the e surface e of the body. . But t this s surface
pressure is s just thesame as wouldbe actingonavirtualsurface withinthe
fluid,nobodypresent. Usingthedivergence e theorem,wemayconvertthisto
anintegralovertheinteriorofthissurface. Ofcourse,thereisnofluidwithin
thebody. Wearejust t usingthemathto o evaluatethe surfaceintegral. . The
result isF
pressure
=gi
z
ρdV. Thisisaforceupwardequaltotheweightof
thedisplacedfluid,asstated.
2.3
Steadyflowofafluidofconstantdensity
Thisspecialcasegivesusanopportunitytoobtainsomeusefulresults rather
easilyinaclassofproblemsofsomeimportance. Weshallallowabodyforceof
theformf=−ρ∇Φ,sothemomentumequationmaybewritten,afterdivision
bytheconstantdensity,
u·∇u+ρ
−1
∇p+∇Φ=0.
(2.23)
Wenotenowavectoridentitywhichwillbeuseful:
A×(∇×B)+B×(∇×A)+A·∇B+B·∇A=∇(A·B).
(2.24)
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