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6.2. SOMEEXAMPLESOFINCOMPRESSIBLEVISCOUSFLOW
101
6.2.3 Poiseuilleflow
Weconsidernowaflowinacylindricalgeometry.ANewtonianviscousfluidof
constantdensityisinsteadymotiondownacylindricaltubeofradiusRandof
infiniteextentinbothdirections.Becauseofviscousstressesatthewallsofthe
tube,weexpect theretobeapressuregradientdownthetube. Lettheaxisof
thetubebethez-axis,rtheradialvariable,andu=(u
z
,u
r
,u
θ
)=(u
z
(r),0,0)
the velocity y fieldin cylindricalpolar coordinates. . We e note here, , for r future
reference,theformoftheNavier-Stokesequationsinthesecoordinates:
∂u
z
∂t
+u·∇u
z
+
1
ρ
∂p
∂z
=ν∇
2
u
z
,
(6.13)
∂u
r
∂t
+u·∇u
r
u
2
θ
r
+
1
ρ
∂p
∂r
Lu
r
2
r2
∂u
θ
∂θ
,
(6.14)
∂u
θ
∂t
+u·∇u
θ
+
u
r
u
θ
r
+
1
∂p
∂θ
Lu
θ
+
2
r2
∂u
r
∂θ
,
(6.15)
∂u
z
∂z
+
1
r
∂(ru
r
)
∂r
+
1
r
∂u
θ
∂θ
=0.
(6.16)
Here
u·∇=u
z
∂z
+u
r
∂r
+
u
θ
r
∂θ
,
(6.17)
2
=
2
(·)
∂z2
+
1
r
∂r
r
∂(·)
∂r
+
1
r2
2
(·)
∂θ2
, L=∇
2
1
r2
.
(6.18)
Fortheproblemathand,wesetp=−Gz+constanttoobtainthefollowing
equationforu
z
(r):
µ∇
2
u
z
=−G=µ
2
u
z
∂r2
+
1
r
∂u
z
∂r
.
(6.19)
Theno-slipconditionappliesatr=R,sotherelevantsolutionof(6.19)is
u
z
=
G
(R
2
−r
2
).
(6.20)
Thusthevelocityprofileisparabolic.Thetotalfluxdownthetubeis
Q≡2π
R
0
ru
z
dr=
πGR
4
.
(6.21)
IfatubeoflengthLissubjectedtoapressuredifference∆patthetwoends,
thenwecanexpecttodriveatotalvolumefloworfluxQ=
π∆pR
4
8µL
downthe
tube. TherateWatwhichworkisdonetoforcethefluiddownatubeoflength
Listhepressuredifferencebetweentheendsofthetubetimesthevolumeflow
rateQ,i.e.
W=
πG
2
LR
4
(6.22)
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102CHAPTER6. VISCOSITYANDTHENAVIER-STOKESEQUATIONS
Poiseuilleflowcanbeeasilyobservedinthelaboratory,particularlyintubesof
smallradius,andmeasurementsofflowratesthroughsmalltubesprovidesone
wayofdeterminingafluid’sviscosity.Ofcoursealltubesarefinite,thevelocity
profile (6.20) isnot establishedatonce whenfluidis introducedintoa tube.
Thisentryeffectcanpersistforsubstantialdistancesdownthetube,depending
onthe viscosity andthe tube radius, , andalso o onthevelocity profileat t the
entrance. Anotherinterestingquestionconcerns s thestabilityofPoiseuilleflow
inadoublyinfinitepipe;thiswasstudiedbytheengineerOsborneReynoldsin
the1870’s. Heobservedinstabilityandtransitiontoturbulenceinlongtubes.
AnapplicationofPoiseuilleflowofsomeimportanceistobloodflow;andinthe
arterialsystemtherearemanybrancheswhicharetooshorttoescapesignificant
entryeffects.
AgeneralizationofPoiseuilleflowtoanarbitrarycylinder,boundedbygen-
erators parallelto o the z-axisandhavingacross s sectionS is easily obtained.
Theequationforu
z
isnow
2
u
z
=
2
u
z
∂x2
+
2
u
z
∂y2
=−G/µ, u
z
=0on∂S.
(6.23)
Thesolutionisnecessarily≥0forG>0andcanbefoundbystandardmethods
fortheinhomogeneousLaplaceequation.
6.2.4 Flowdownanincline
Weconsidernowtheflowofaviscousfluiddownanincline,seefigure6.4. The
velocityhastheform(u,v,w)=(u(z),0,0)andthepressureisafunctionofz
alone. Thefluidisforceddowntheinclinebythegravitationalbodyforce. The
equationstobesatisfiedare
ρgsinα+µ
d
2
u
dz2
=0,
dp
dz
+ρgcosα=0.
(6.24)
On the e free surface z z = = H H the e stress s must equal the e normalstress s due
to the constant t pressure, p
0
say, abovethefluid. . Thus s σ
xz
= ν
du
dz
= 0and
σ
zz
= −p p = = −p
0
whenz = = H. . Sincetheno-slipconditionapplies, , wehave
u(0)=0. Therefore
u=
ρgsinα
z(2H−z), p=p
0
+ρg(H−z)cosα.
(6.25)
Thevolumeflowperunitlengthinthey-directionis
U
0
udz=
gH
3
sinα
.
(6.26)
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6.2. SOMEEXAMPLESOFINCOMPRESSIBLEVISCOUSFLOW
103
H
z
α
g
x
Figure6.4:Flowofaviscousfluiddownanincline.
6.2.5 Flowwithcircularstreamlines
Weconsideravelocityfieldincylindricalpolarcoordinatesoftheform(u
z
,u
r
,u
θ
)=
(0,0,u
θ
(r,t)),withp=p(r,t).From(6.13)-(6.18)theequationforu
θ
is
∂u
θ
∂t
2
u
θ
∂r2
+
1
r
∂u
θ
∂r
u
θ
r2
,
(6.27)
withtheequation
∂r
∂r
=
ρ
r
u
2
θ
(6.28)
determiningthepressure. Thevorticityis
ω=
1
r
∂ru
θ
∂r
.
(6.29)
From(6.27)wethenfindanequationforthevorticity
∂ω
∂t
2
ω
∂r2
+
1
r
∂ω
∂r
=ν∇
2
ω.
(6.30)
Thisequation,whichisthesymmetricformoftheheat equationintwospace
dimensions,maybeusedtostudythedecayofapointvortexintwodimensions,
seeproblem6.2.
6.2.6 TheBurgersvortex
Thimplicationof(6.30)isthatvorticityconfinedtocircularstreamlinesintwo
dimensions willdiffuse likeheat, , never r reachinga a non-trivial steady state in
R
2
. WenowconsiderasolutionoftheNavier-Stokesequationswhichinvolves
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104CHAPTER6. VISCOSITYANDTHENAVIER-STOKESEQUATIONS
atwo-dimensionalvorticityfieldω=(ω
z
r
θ
))=(ω(r),0,0). Theideaisto
prevent thevorticity fromdiffusingbyplacingitinasteadyirrotationalflow
fieldof theform(u
z
,u
r
,u
θ
)=(αz,−αr/2,0). Thus s thefullvelocityfieldhas
theform
(u
z
,u
r
,u
θ
)=(αz,−αr/2,u
θ
(r,t)).
(6.31)
Nowthez-componentofthevortiityequationis,with(6.31),
∂ω
∂t
αr
2
∂ω
∂r
−αω=ν
1
r
∂r
r
∂ω
∂r
, ω=
1
r
∂ru
θ
∂r
.
(6.32)
First note that if ν ν = = 0,so o that t there e is s nodiffusionofω, , we mysolvethe
equationtoobtain
ω=e
αt
F(r
2
e
αt
),
(6.33)
where F(r
2
) is the initialvalue of ω. . Thissolutionexhibitsthe e exponential
growthofvorticitycomingfromthestretchingofvortextubes inthestraining
flow(αz,−αr/2,0).
Ifnowwerestoretheviscosity,welookforasteadysolutionof(6.32),repre-
sentingavortexinforwhichdiffusionisbalancedbytheadvectionofvorticity
towardthez-axis. Wehave
1
r
∂r
α
2
r
2
ω+νr
∂ω
∂r
=0.
(6.34)
Integratingandenforcingthecondition2thatr
2
ωandr
∂ω
∂r
vanishwhenr=∞,
wehave
α
2
rω+ν
dr
=0.
(6.35)
Thus
ω(r)=Ce
−αr
2
,
(6.36)
sothat
u
θ
=
Γ
1−e
−αr
2
r
,
(6.37)
where we have redefined d the constant t to o exhibit t the total circulationof the
vortex. Notethatasν ν decreases s thesizeofthevortextubesshrinks. . WithΓ
fixedthiswouldmeanthatthevorticityofthetubeisincreased.
6.2.7 Stagnation-pointflow
Inthis example we attempt to o modifythe two-dimensional stagnationpoint
flowwithstreamfunctionUL
−1
xytoasolutioniny>0of theNavier-Stokes
equationswithconstantdensity,satisfyingtheno-slipconditionony=0. The
vorticitywillsatisfy
u
∂ω
∂x
+v
∂ω
∂y
−ν
2
ω
∂x2
+
2
ω
∂y2
=0.
(6.38)
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6.3. DYNAMICALSIMILARITY
105
0
0.5
1
1.5
2
2.5
3
0
0.2
0.4
0.6
0.8
1
1.2
eta
fprime
Figure6.5:f
versusηfortheviscousstagnatioinpointflow.
Ifwesetψ=UL
−1
xF(y),thenω=−UL
−1
yF

. Insertionin(6.38)gives
F
F

−FF

−Re
−1
F

,
(6.39)
where Re = UL/ν. . The e boundary conditionsare that F(0) = F
(0) = 0to
makeψ,u,vvanishonthewally=0,andF ∼yasy→∞,sothatweobtain
theirrotationalstagnationpointflowaty=∞.
Oneintegrationof(6.39)canbecarriedouttoobtain
F
2
−FF

−Re
−1
F

=1.
(6.40)
WithF=Re
−1/2
f(η),η=Re
1/2
y,(6.40)becomes
f
2
−ff

−f

=1,
(6.41)
withconditionsf
(∞)=1,f(0)=f
(0)=0.Weshowinfigure6.5thesolution
f
(η)ofthisODEproblem. Thisrepresentsagradualtransitionthroughalayer
ofthicknessoforder
UL/nubetweenthenullvelocityontheboundary and
thevelocityU(x/L)whichuhasatthewallintheirrotationalstagnationpoint
flow. Weshallbereturningtoadiscussionofsuchtransitionlayersinchapter
7,wherewetakeupthestudyofboundarylayers.
6.3 Dynamicalsimilarity
Inthestagnationpointexamplejustconsidered, thedimensionalcombination
Re=UL/νhasoccurredasaparameter. Thisparameter,calledtheReynolds
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106CHAPTER6. VISCOSITYANDTHENAVIER-STOKESEQUATIONS
numberinhonorofOsborneReynolds, arose because wechosetoexhibitthe
probleminadimensionlessnotation.ConsidernowtheNavier-Stokesequations
withconstantdensityittheirdimensionalform:
∂u
∂t
+u·∇u+
1
ρ
∇p−ν∇
2
u=0, ∇·u=0.
(6.42)
Wemaydefinedimensionless(starred)variablesasfollows:
u
=u/U,x
=x/L,p
=p/ρU
2
.
(6.43)
HereU,Lareassumedtobeavelocityandlengthcharacteristicoftheproblem
beingstudied.Inthecaseofflowpastabody,Lmightbeabodydiameterand
U theflowspeedatinfinity. Inthesestarredvariablesitiseasilycheckedthat
theequationsbecome
∂u
∂t
+u
·∇
u
+∇
p
1
Re
∗2
u
=0, ∇
·u
=0.
(6.44)
Thus Re survives asthe onlydimensionless parameter inthe equations. . For
a givenvalue of Re e a a givenproblem m willhave e asolutionor solutions which
arefullydeterminedbythevalueofRe.Nevertheless theset t ofsolutions is
fullydeterminedbyReandRe alone. . Thusweareabletomakeacorrespon-
dencebetweenvariousproblemshavingdifferentUandLbutthesamevalueof
Re. Wecallthiscorrespondencedynamicalself-similarity.Twoflowswhichare
self-similarinthisrespectbecomeidenticalwhichexpressedinthestarred,di-
mensionlessvariables(6.43).Inasensethestatement“theviscosityνissmall”
conveysnodynamicalinformation,althoughtheintendedimplicationmightbe
thatRe1. IfLisalso“small”,thenitcouldwellbethatRe=1ore1.
Theonlymeaningfulwaytostate thatafluidis “almostinviscid”is through
theReynoldsnumber,Re1. Ifwewanttoconsiderfluidswhoseviscosityis
dominantcomparedtoinertialforces,weshouldrequireRe1.Theseremarks
underline theoft-repeateddefinitionofRe as “the ratioofinertialtoviscous
forces”. Thisisbecause
ρu·∇u
µ∇2u
=Re
u·∇u
2
u
∼Re
(6.45)
sinceweregardallstarredvariablesasoforderunity.
example6.1: ThedragDperunitlengthofacircularcylinderofradiusL
inatwo-dimensionaluniformflowofspeedUmustsatisfyD=ρU
2
LF(Re)for
somefunctionF. Notethat t weareassumingherethat cylindersarefullyde-
terminedbytheirradius.Inexperimentsotherfactors,suchassurfacematerial
orroughness,slightellipticity,etc. mustbeconsidered.
Problemset6
3It is not alwaysthe casethatwell-formulatedboundary-valueproblemsfor theNavier-
Stokesequationshaveuniquesolutions.Seetheexampleofviscousflowinadivergingchannel,
page79ofLandauanLifshitz.
6.3. DYNAMICALSIMILARITY
107
H
2L
θ
Q
θ
Figure6.6:BifurcatingPoiseuillflow.Assumeaparabolicprofileineachsection.
1. Considerthefollowingoptimzationproblem: : ANewtonianviscousfluid
ofconstant density flows throughacylindricaltubeofradius R
1
,whichthen
bifurcatesintotwostraighttubesofradiusR
2
,seethefigure.AvolumeflowQ
isintroducedintotheuppertube,whichdividesintoflowsofequalfluxQ/2at
thebifurcation.Becauseofthematerialcompositionofthetubes,itisdesirable
thatthewallstress µdu/dr,evaluatedatthewall,bethesameinbothtubes.
IfLandHaregivenandfixed,whatistheangleθwhichminimizestherateof
workingrequiredtosustaintheflowQ?. Besuretoverifythatyouhaveatrue
minimum.
2. Look k for a solutionof f (6.30)of f the form ω ω = = t
−1
F(r/
t), satisfying
ω(∞,t)=0,2π
0
rω(r,t)dr=1,t>).Show,bycomputingu
θ
withu
θ
(∞,t)=
0,thatthisrepresentsthedecayofapointvortexofunitstrengthinavbiscous
fluid,i.e.
lim
t→0+
u
θ
(r,t)=
1
2πr
,r>0.
(6.46)
3.ANavier-Stokesfluidhasconstantρ,µ,nobodyforces.Consideramotion
inafixedboundeddomainV withno-slipconditiononitsrigidboundary.Show
that
dE/dt=−Φ,E=
V
ρ|u|
2
/2dV,Φ=µ
V
(∇×u)
2
dV.
Thisshowsthatforsuchafluidkineticenergyisconvertedintoheatatarate
Φ(t). This s lastfunctionoftime givesthe netviscousdissipationforthefluid
containedinV. (Hint: : ∇×(∇×u)=∇(∇·u)−∇
2
u. Also∇·(A×B)=
∇×A·B−∇×B·A.)
108CHAPTER6. VISCOSITYANDTHENAVIER-STOKESEQUATIONS
4. In n twodimensions, withstreamfunction ψ, where e (u,v) = = (ψ
y
,−ψ
x
),
showthattheincompressibleNavier-Stokesequationswithoutbody forces for
afluidofconstantρ,µreduceto
∂t
2
ψ−
(∂(ψ,∇2ψ)
∂(x,y)
−ν∇
4
ψ=0.
Interms of ψ, , what aretheboundary conditions s ona a rigidboundary ifthe
no-slipconditionissatisfiedthere?
5. Findthe e time-periodic2D flowinachannel−H < y< H,filledwith
viscousincompressiblefluid,giventhatthepressuregradient is dp/dx= A+
Bcos(ωt),whereA,B,ωareconstants.Thisisanoscillating2DPoiseuilleflow.
Youmayassumethatu(y,t)iseveninyandperiodicintwithperiod2π/ω.
6.verify(6.33).
7. Theplanez z = 0isrotating g about t the z-axis withanangularvelocity
Ω. ANewtonianviscousfluidofconstantdensityandviscosityoccupies s z>0
andthefluidsatisfiestheno-slipconditionontheplane,i.e. atz=0thefluid
rotateswiththeplane. Bycentrifugaleffectweexpectthefluidneartheplane
tobethrownout radiallyandacompensatingflowof fluiddownwardtoward
theplane.
Usingcylindricalpolarcoordinates,lookforasteadysolutionoftheNavier-
Stokesequationsoftheform
(u
z
,u
r
,u
θ
)=(f(z),rg(z),rh(z)).
(6.47)
Weassumethatthevelocitycomponentu
θ
vanishesasz→∞.Showthatthen
p
ρ
df
dz
1
2
f
2
+F,
(6.48)
whereF isafunctionofralone. Nowarguethat,ifh(∞)=0,i.e. norotation
atinfinity,thenF mustinfactbeaconstant. Fromtherandθcomponentof
themomentumequationtogetherwith∇·u=0,findequationsforf,g,hand
justifythefollowingconditions:
f=
df
dz
=0,h=Ω, z=0; ; f
,h→0, z→∞.
(6.49)
(Thesolutionoftheseequationsisdiscussedonpp. 75-76ofL&L L and290-92
ofBatchelor.)
Chapter7
Stokesflow
Wehaveseen insection6.3thatthedimensionlessform m of theNavier-Stokes
equations foraNewtonianviscousfluidofconstantdensity andconstantvis-
cosityis,nowdroppingthestars,
∂u
∂t
+u·∇u+∇p−
1
Re
2
u=0, ∇·u=0.
(7.1)
TheReynoldsnumberReistheonlydimensionlessparameter intheequa-
tionsofmotion.Inthepresent chapter weshallinvestigatethefluiddynamics
resultingfromtheaprioriassumptionthattheReynoldsnumberisverysmall
comparedto unity,Re e  1. . Since e Re = UL/ν,thesmallness of Re canbe
achievedbyconsideringextremelysmalllengthscales,orbydealingwithavery
viscous liquid, or by y treating flows of very small velocity, so-called d creeping
flows.
ThechoiceRe1isanveryinterestingandimportantassumption,foritis
relevanttomanypracticalproblems,especiallyinaworldwheremanyproducts
of technology, , including those e manipulating g fluids, , are shrinking g insize. . A
particularlyinterestingapplicationistotheswimmingofmicro-organisms. In
all of these areas we shall,withthis assumption, unveil a specialdynamical
regimewhichisusuallyreferredtoasStokesflow,inhonorofGeorgeStokes,
whoinitiatedinvestigationsintothisclassoffluidproblems. Weshallalsorefer
tothisgeneralareaoffluiddynamicsastheStokesianrealm,incontrasttothe
theoriesofinviscidflow,whichmightbetermedtheEulerianrealm.
WhataretheprinciplecharacteristicsoftheStokesianrealm? Since e Reis
indicativeoftheratioofinertialtoviscousforces,theassumptionofsmallRe
willmeanthat viscous forces dominatethedynamics. . Thatsuggests s that we
maybeabletodropentirelythetermDu/DtfromtheNavier-Stokesequations,
rendering the system linear. . Thiswillindeedbethe e case,withsomecaveats
discussedbelow.Thelinearityoftheproblemwillbeamajorsimplification.
Lookingat(7.1)intheform
Re
∂u
∂t
+u·∇u+∇p
=∇
2
u, ∇·u=0,
(7.2)
109
110
CHAPTER7. STOKESFLOW
Itis temptingtosaythat thesmallnessofRemeansthat wecanneglectthe
left-handsideofthefirstequation,leadingtothereduced(linear)system
2
u=0, ∇·u=0.
(7.3)
Indeedsolutionsof(7.3)belongtotheStokesianrealmandarelegitimate.
Example 7.1: : Consider r the e velocity fieldu=
A×R
R3
inthree dimensions
withAaconstant vector andR R = = (x,y,z). . Note e that u= = ∇×
A
R
, andso
∇·u=0andalso∇
2
u=0,R>0since
1
R
isaharmonicfunctionthere. This
infactaninterestingexampleofaStokes flow. . Considerasphereofradius s a
rotatinginaviscousfluidwithangularvelocityΩ. Theonthesurfaceofthe
spherethevelocityisΩ×Riftheno-slipconditionholds.Comparingthiswith
ourexampleweseethatifA=Ωa
3
wesatisfythisconditionwithaStokesflow.
ThuswehavesolvedtheStokesflowproblemofaspherespinninginaninfinite
expanseofviscousfluid.
Itisnot difficulttosee,however,that(7.3) doesnot encompassallofthe
Stokesflowsofinterest. Thereasonisthatthepressurehasbeenexpelledfrom
thesystem, whereas s thereisnophysicalreasonforthis. . If,inthe e process of
writingthedimensionlessequations,wehaddefinedthedimensionlesspressure
aspL/(µU)insteadofp/(ρU
2
,(7.2)wouldbechangedto
Re
∂u
∂t
+u·∇u
+∇p=∇
2
u, ∇·u=0,
(7.4)
leadinginthelimit→0to
∇p−∇
2
u=0, ∇·u=0.
(7.5)
Weseethatanysolutionof(7.5)willhavetheformu=∇φ+vwhere∇
2
φ=p
and∇
2
v = 0,∇·v= −p. . Thislargerclassofflows, , validfor r Re small,are
calledStokesflows.Thespecialfamilyofflowswithzeropressureformasmall
subsetofallStokesflows.
7.0.1 Somecaveats
WenotedabovethatthedroppingoftheinertialtermsinStokesflowmighthave
tobequestionedinsome cases, andweconsidertheseexceptionsnow. . First,
it canhappen n that t there e is more e thanone possible Reynolds number which
canbe formed,involvingoneor moredistinctlengths, and/ora frequency of
oscillation,etc. It t canthenhappenthatthe timederivativeofuneeds tobe
kepteventhoughtheu·∇unonlineartermmaybe dropped. . Anexampleis
awalladjacenttooaviscousfluid,executingastandingwavewithamplitude
A, frequency y ω ω and wavelengthL. . If f ωL
2
/ν is s of order unity, and we e take
U =ωL,thentheReynoldsnumberUL/ν ν isoforderunityandnotermsmay
bedropped. HowevertheactualvelocityisoforderωA,andifALthenthe
nonlineartermsarenegligible.
Anotherunusualsituationisassociatedwiththenon-uniformityoftheStokes
equations inthreedimensions near infinity,insteady flowpast afinite body.
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