WhatdotheNavier-Stokesequationsmean?SimonSchneiderbauer1andMichaelKrieger21ChristianDopplerLaboratoryonParticulateFlowModeling,JohannesKeplerUniversity,Altenbergerstrae69,4040Linz,Austria.2InstituteforFluidMechanicsandHeatTransfer,JohannesKeplerUniversity,Altenbergerstrae69,4040Linz,Austria.Abstract.
TheNavier-Stokesequationsarenon-linearpartialdierentialequationsdescribingthemotionof
uids. Duetotheir complicatedmathematical
formtheyarenotpartofsecondaryschooleducation.
AdetaileddiscussionofthefundamentalphysicstheconservationofmassandNewtonssecondlawmay,however,increasethe
understanding of the behavior of uids. Based on these principles
the Navier-Stokesequationscanbederived.
Thisarticleattemptstomaketheseequationsavailabletoa wider
readership,especially teachers and undergraduate students.
Therefore,in thisarticleaderivationrestrictedtosimpledierential
calculusispresented. Finally, wetry to give answers to the
questions what is a uid?and what do the
Navier-Stokesequationsmean?. correspondingauthor:
[email protected] is a preprint of an article
accepted for publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean? 21. IntroductionThe
Navier-Stokes equations describe the motionof uids andare the
fundamentalequationsofuiddynamics. TheyarenamedafterGeorgGabriel
Stokes(18161903)andLouisMarieHenriNavier(17851836),whoderivedtheseequationindependently.TheNavier-StokesequationsarebasedonworkofLeonhardEuler(17071783).
Eulerconsideredtheuidasacontinuumallowinghimtoderivegoverningequationsforthemotionof
invisciduids basedondierential calculus. His equations
weretherstwritten down non-linear partial dierential equations the
Euler equations. Stokes
andNaviercontributedaviscousdiusiontermtoaccountfortheviscosityofauid.TheNavier-Stokesequationsarewidelyusedinscienceandengineering.
However,their complicated mathematical form mostly restricts
engineers to the numerical solutionoftheseequations[1,2].
Themathematicalproofoftheexistenceofaglobalsolutionof the
Navier-Stokes equations is still one of the millennium problems
[3]. Nevertheless,theNavier-Stokes equations
aresuccessfullyappliedtodesignairfoils [4],
reducethedragof(racing)cars,optimizeparticlelters,understandthewindthrowofforests[5],analyzeoceancurrents[6],studyenvironmentalparticletransport[7]andsoforth.DuetothefactthattheNavier-Stokesequationsarepartialdierentialequationsand
their solutions are non-trivial, these are commonly not included in
secondary schoolcurriculums. A detailed discussion of the
fundamental assumptions of the Navier-Stokesequationsandof
theunderlyingphysicsmay, however,
increasetheunderstandingofthebehaviorofuids.
Especially,thephysicsofuidscanbetopicofin-depthphysicscourses.This
article aims tointroduce the Navier-Stokes equations
tosecondaryschoolteachersandundergraduatestudents.
Wealsointendtoprovideteachingmaterial
forextraordinaryinterestedstudents. Therefore, we restrict our
calculations
tosimpledierentialcalculus,forexample,partialderivatives andTaylor
series expansions.
Thisisincontrasttostandardderivationsusingintegraltheorems.This
article is organized as follows. In section 2 we begin with a
practical denitionofuids. Then,
wediscusstheunderlyingassumptionsoftheNavier-Stokesequationsandthe
basic concepts. Insection5 we derive the two-dimensional
Navier-Stokesequationsusingdierential calculus. Attheendof
thearticle,
somesimpleexamplesfortheexactsolutionoftheNavier-Stokesequationsarediscussed.2.
Whatisauid?Each natural or articial material is characterized by
its distinct state of matter. Thesestatesaresolid, gasandliquid.
Thereisalsoafourthstateof matter, referredtoasplasma, which we do
not discuss in more detail. Intuitively, one is tempted to
categorizeuidsbyitsstateof matter. Forexample,
waterisauidsinceitisliquidatroomtemperatureandatmosphericpressure.
However, suchaclassicationof
uidsseemstobetoorestrictivesincebothgaseousandliquidmatterisconsideredasuid[2,
8].This is a preprint of an article accepted for publication in the
European Journal of PhysicsWhatdotheNavier-Stokesequationsmean?
3Inuidmechanicsitiscommontoassignall materials,
whicharenotclearlysolid, touids.
Thus,eachmaterialcanbedistinctlyassignedtoeithersolidsoruids.F C DA
BC' D'FA BC D D'' C''(a)(b) !A !A!D' C'Figure1.
Anexternalforceisappliedtoa)asolidandb)auid,
whichconsistsofrandomlymovingmolecules. Theamountof deformationof
thesolidisdeterminedbythebalanceof theexternal andtheshearforces.
Theuidestablishesavelocitygradient, whichcounteractstheexternal
force. Theuidproceedsmovingaslongastheforceisapplied.
ThisisindicatedbytheparallelogramsABC
D
andABC
D
.The uidcounterpart of the elasticityof the solids is knownas
viscosity. Thephysical dierence between elasticity and viscosity is
outlined by the following
example.Letusconsiderablockofelasticsolid(Figure1a),whichismountedtothegroundatpointsAandB.
Themoleculesofthesolidholdtogetherbyexertingattractiveforcesoneachother.
Whenanelasticsolidisdeformedittriestodeformbacktoitsinitialstate.
Thus,qualitativelywemaymodeltheseattractiveforcesbysprings(Figure1a).Furthermore,a
force F,which is parallel to AB,is applied at point D. In case of
smalldeformationsweobtainfortheshearstress(= force/area = F/A) kA,
(1)wherekdenotesthestinessofthesprings.
Ourillustratingexampleindicatesthatinelasticsolidstheshearstressisproportional
totheangleof deformation.
Ingeneral,thedisplacementoftheatomsinsolidsbyexternal
shearforcesisreversibleforsmalldeformations(elasticdeformation).Incontrast
tothedeformationof solids, inuids avelocitygradient
establisheswhenanexternal forceis applied(Figure1b) [9, 10, 2].
Thevelocitygradient is
aconsequenceofmomentumdiusionatmolecularscale.
AccordingtoNewtonssecondlawmoleculesatpointDareacceleratedbytheappliedforceF.
Incontrasttosolids,themolecules moverandomlyinauid. Thus, someof
thesefastmolecules nearDmovedowntowardsA(Figure1b). Therefore,
themomentumof theuidinthedirection of the force Fnear A increases.
Similarly, some slow molecules near A
moveuptowardsDleadingtoadecreaseofmomentumintheregionofD.
Mathematically,theshearstressonanareaAiswrittenas[11]= uy,
(2)WhatdotheNavier-Stokesequationsmean? 4where u denotes the
velocity of the uid in lateral direction and ythe spatial
coordinateinvertical direction.
Theviscosity(Ns)isacharacteristicpropertyoftheuidlikethestinessofaspring.
Amoredetailedderivationofequation(2)isgiveninsection5.2. Whenthe
appliedshearforceiswithdrawnthevelocitygradientvanishesuntiltheuidisatrest.
Therandomnatureof themovementof
themoleculesindicatesthatthedisplacementofthemoleculesunderanexternalforceisnotreversible,whichisincontrasttoelasticsolids.Thisexampledemonstratesthatauidatrest,
incontrasttosolids, isnotabletosupportexternal shearforces.
Itimmediatelyreactstotheappliedshearforcebyestablishing a velocity
gradient. Viscous shear forces are completely dierent from
elasticshearforces[10].
Strictlyspeakinguidsdonottransmitshearforces; theseforces(time rate
of change of momentum) rather appear due to the random motion of
the uidmolecules. Asaconsequenceunderexternal forces, i.e.
theearthsgravitational
eld,uidsneedcontainingwallstokeeptheirgeometricshape.Althoughsolidsanduidsbehaveverydierentlywhensubjectedtoshearforces,they
behave similarly under the action of pressure, i.e. normal
compressive stresses [10].However, whereas solids are able to
support both normal tensile and compressive
forces,auidusuallysupportsonlycompression(pressure).3.
ContinuumhypothesisFluidsarecomposedofahugenumberofmolecules,whichareinconstantmotionandundergoing
collisions with each other [10]. Typical orders of magnitudes for
air at 1 barand 0C are dm 41010m for the diameter of the molecules,
m 3109m for theaveragedistancebetweentwomolecules,mfp
6108mforthemeanfreepath(thedistancebetweensubsequentcollisions)and1025moleculesperm3.
Inprinciple, itispossible to study the behavior of uids by
following the trajectory of each single moleculeincludingits
collisions withthesurroundingmolecules [10, 9], as
donebymoleculardynamics. However, it is morepractical toaskfor
themacroscopicbehavior of theuid. The idea is to take account of
the behavior of the molecules and their properties
byconsideringahugeensembleofmolecules[2].
Thus,thediscretemolecularstructureof uids is replacedbycontinuous
distributions, calledcontinuum[9]. For
example,wecandenethemacroscopiccontinuouspressureataconstrainingwall
asthetimerateofchangeofmomentumperunitareaofahugeensembleofcollidingmolecules.Similarly,wecandenethetemperatureofauidasthekineticenergyofanensembleof
molecules; thedensityasthenumberof moleculesperunitvolume;
thevelocityofthe uid as the average velocity of the ensemble. Note,
since we consider the propertiesand behavior of uids by considering
an ensemble of molecules, the molecular nature
ofuidsisnotneglected.
Itremainstodiscusshowmanymoleculesweneedforsuchanensembleandinwhichcasesthecontinuumhypothesisisvalid.This
is a preprint of an article accepted for publication in the
European Journal of PhysicsWhatdotheNavier-Stokesequationsmean?
53.1. DenitionofthemacroscopiccontinuousdensityandpressureIt is
commontodenethedensityof auid(x0, y0) at (x0, y0)
(seealsoFigure2)byconsideringtheaggregatedmassof
moleculesmVinvolumeV xedinspaceat(x0, y0)(x0, y0) =mVV. (3)Since
the molecules are not xed in a lattice but move freely [9], the uid
density has noprecisemeaning.
ThenumberofmoleculesinagivenvolumeVcontinuouslychanges.On the one
hand, molecules enter the volume and on the other hand, molecules
leave thevolume. If the volume is very small, that is in the order
of 3m 1026m3, estimating thedensity would, therefore, result in a
huge uncertainty (Figure 3a). Increasing the volumereduces the
inuence of leaving and incoming molecules since their contribution
becomesnegligiblecomparedtothenumberofmoleculesoccupyingV .
InFigure2thedensityascalculatedfromtheaggregatedmassofthemoleculesisplotted.
Thereisalimitingvolume V 1018m3above whichthe contributionof the
leavingandincomingmolecules gets negligible. For smaller volumes
than V considerable uncertainty in
theevaluationofthedensitymaybeobserved.uid yx!Vp!Am!VFigure2.
Sketchofthedenitionoftheuiddensityanduidpressure.According to the
denition of , we dene the pressure p(x0, y0) by considering thetime
rate of change of momentum of the molecules FpAat a plane of area A
at (x0, y0)(Figure2)p(x0, y0) =FpAA . (4)Thus, the order of
magnitude of the minimum size of A should be O(A) O(V2/3) =O(1012)
to ensure an accurate denition of the pressure. This is also
indicated in Figure3b, wherethelimitingareaA,
abovewhichthepressurecanbepreciselydened,
isapproximately1012m2.This is a preprint of an article accepted for
publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean? 6!V (m3)
10!3010!1810!10microscopic uncertainty macroscopic density (a)pA
(m2) 10!1510!1210!5microscopic uncertainty macroscopic pressure
(b)Figure3.
Uncertaintiesintheestimationofa)theuiddensityasafunctionofthevolumeV
occupiedbytheconsideredmoleculesandb)ofthepressureasafunctionoftheareaunderconsideration[9,2].3.2.
KnudsennumberThe Knudsen number is a dimensionless number (Note in
uid mechanics there are lotsofdimensionlessnumbers),thatisKn =mfpL,
(5)which is the ratio between the mean free path of the molecules
mfpand a characteristicphysical lengthscaleLof
theproblemunderconsideration. Forexample, Lmaybethelengthof
anairfoil. TheKnudsennumbergivesameasureforthevalidityof
thecontinuumhypothesis. If theKnudsennumber isverysmall (Kn 1)
thephysicallength scale is much larger than the mean free path of
the molecules. Thus, the physicalobstacle does only feel the
average molecular behavior and we can apply the
continuumhypothesistothephysical descriptionof theow. Incaseof
ournumerical examples(mfp 6108m, A=1012m2)thelimitingKnudsennumber,
belowwhichthecontinuumhypothesiscanbeused, yieldsKn=mfp/A 6102.
Incaseof
aKnudsennumbergreaterthanKnstatisticalmethodsormoleculardynamicsmustbeused.
Anexampleisaspaceshuttleenteringtheearthsexosphere,
wherethemeanfree path of the molecules is several kilometers (high
temperature and low density), andthus,Kn 1.
Similarly,theowaroundnanobersinaextra-neparticleltershowsKnudsennumbersof
Kn 1. Hence, thecontinuumhypothesiscannotbeappliedtothesecases.3.3.
Finitecontrol volumesInnitesimal uidelementsBasedontheideaof
thecontinuumhypothesis weareabletoderivethegoverningequations of
uiddynamics. It is, therefore, commonto introduce the concept
ofnitecontrol volumes V , whichareveryuseful inuiddynamics [1].
Accordingtothe discussion about the denition of density (section
3.1), one has to keep in mind thatthese nite control volume have to
be greater than V . Then, the fundamental
physicalprinciplescanbeappliedtotheuidinsidethecontrolvolumeV .
SuchavolumecanWhatdotheNavier-Stokesequationsmean?
7eithermovewiththeowoccupyingaxedensembleof uidmolecules,
asshowninFigure4a, orcanbexedinspacewiththeuidpassingvolumeV ,
aspresentedinFigure4b. Therefore, onlyasmall nitecontrol
volumeisconsidered, insteadof thewholeoweldat once[1].
Wecandirectlyderivethegoverningequations of
uiddynamicsbylookingatthephysicalpropertiesoftheuidwithinsuchanitecontrolvolumeV
.
Asnotedabovewehavetodistinguishbetweentwodierentapproaches[2,9,10,12]:Lagrangiandescriptionof
thegoverningequations, whichareobtainedfromthenite uid element
moving along with the ow (Figure 4a). Since a xed
ensembleofuidmoleculesisfollowedthemassofaLagrangiancontrolvolumeisconstant.Euleriandescriptionof
the governing equations, which are obtained from the
niteuidelementxedinspace(Figure4b).WhatdotheNavier-Stokesequationsmean?
7eithermovewiththeowoccupyingaxedensembleof uidmolecules,
asshowninFigure4a, orcanbexedinspacewiththeuidpassingvolumeV ,
aspresentedinFigure4b. Therefore, onlyasmall nitecontrol
volumeisconsidered, insteadof thewholeoweldat once[1].
Wecandirectlyderivethegoverningequations of
uiddynamicsbylookingatthephysicalpropertiesoftheuidwithinsuchanitecontrolvolumeV
.
Asnotedabovewehavetodistinguishbetweentwodierentapproaches[2,9,10,12]:Lagrangiandescriptionof
thegoverningequations, whichareobtainedfromthenite uid element
moving along with the ow (Figure 4a). Since a xed
ensembleofuidmoleculesisfollowedthemassofaLagrangiancontrolvolumeisconstant.Euleriandescriptionof
the governing equations, which are obtained from the
niteuidelementxedinspace(Figure4b).------ControlVolumeVLControlSurfaceAL(a)----ControlVolumeVEControlSurfaceAE(b)Figure4.
Modelsofaow:
a)Finitecontrolvolumemovingwiththeuidsuchthatthesameindividualuidparticlesarealwaysinthiscontrolvolume;b)Finitecontrolvolumewhichisxedinspace.
Theuidismovingthroughit.Thesameideascanalsoberepresentedbyinnitesimal
uidelements. Theuidelementisinnitesimal inthesenseof dierential
calculus[1]. Theinnitesimal uidelement has also to be large enough
to contain a suciently large number of molecules,thatisV V
,sothatthecontinuumhypothesiscancanbeapplied.4.
MaterialderivativeThevelocityuandtheaccelerationaof
aLagrangianuidelementmovingwiththeowaregivenby[2,9,10,12]u(VL, t)
=dxdt
VLand a(VL, t) =dudt
VL,
(6)wherethepositionvectorxandvelocityuintwodimensionalcartesianspacearex
= exx +eyy,u = exu +eyv,(a)WhatdotheNavier-Stokesequationsmean?
7eithermovewiththeowoccupyingaxedensembleof uidmolecules,
asshowninFigure4a, orcanbexedinspacewiththeuidpassingvolumeV ,
aspresentedinFigure4b. Therefore, onlyasmall nitecontrol
volumeisconsidered, insteadof thewholeoweldat once[1].
Wecandirectlyderivethegoverningequations of
uiddynamicsbylookingatthephysicalpropertiesoftheuidwithinsuchanitecontrolvolumeV
.
Asnotedabovewehavetodistinguishbetweentwodierentapproaches[2,9,10,12]:Lagrangiandescriptionof
thegoverningequations, whichareobtainedfromthenite uid element
moving along with the ow (Figure 4a). Since a xed
ensembleofuidmoleculesisfollowedthemassofaLagrangiancontrolvolumeisconstant.Euleriandescriptionof
the governing equations, which are obtained from the
niteuidelementxedinspace(Figure4b).------ControlVolumeVLControlSurfaceAL(a)----ControlVolumeVEControlSurfaceAE(b)Figure4.
Modelsofaow:
a)Finitecontrolvolumemovingwiththeuidsuchthatthesameindividualuidparticlesarealwaysinthiscontrolvolume;b)Finitecontrolvolumewhichisxedinspace.
Theuidismovingthroughit.Thesameideascanalsoberepresentedbyinnitesimal
uidelements. Theuidelementisinnitesimal inthesenseof dierential
calculus[1]. Theinnitesimal uidelement has also to be large enough
to contain a suciently large number of molecules,thatisV V
,sothatthecontinuumhypothesiscancanbeapplied.4.
MaterialderivativeThevelocityuandtheaccelerationaof
aLagrangianuidelementmovingwiththeowaregivenby[2,9,10,12]u(VL, t)
=dxdt
VLand a(VL, t) =dudt
VL,
(6)wherethepositionvectorxandvelocityuintwodimensionalcartesianspacearex
= exx +eyy,u = exu +eyv,(b)Figure4. Modelsofaow:
a)Finitecontrolvolumemovingwiththeuidsuchthatthesameindividualuidparticlesarealwaysinthiscontrolvolume;b)Finitecontrolvolumewhichisxedinspace.
Theuidismovingthroughit.Thesameideascanalsoberepresentedbyinnitesimal
uidelements. Theuidelement is innitesimal in the sense of
dierential calculus [1] but has to be large
enoughtocontainasucientlylarge number of molecules, that is V V ,
sothat thecontinuumhypothesiscancanbeapplied.4.
MaterialderivativeThevelocityuandtheaccelerationaof
aLagrangianuidelementmovingwiththeowaregivenby[2,9,10,12]u(VL, t)
=dxdtVLand a(VL, t) =dudtVL,
(6)wherethepositionvectorxandvelocityuintwodimensionalcartesianspacearex
= exx +eyy,u = exu +eyv,This is a preprint of an article accepted
for publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean? 8with ex and eydenoting
the cartesian unit basis vectors. In equation (6) the symbol
[VLindicatesthatwefollowthetrajectoryoftheuidelementVLofxedmass,whichwehighlighted
at t = 0. However, in uid dynamics it is not practical to use the
governingequations in Lagrangian form. It is common to obtain these
equation based on Euleriancontrol volumes VE(Eulerian form), which
are xed in space and where the velocity anddensity elds are
functions of x and t. It is straightforward to transform the
velocity
ofaLagrangianuidelementintoEulerianformatanarbitrarytimet0u(x, t0)
= u(VL, t0) if VE= VL(t0). (7)Hence,we simply consider a Eulerian
control volume VEequal to VL(t0) at t = t0.
ForuastreamlineubVFigure5. Steadyowthroughaconvergentpipe.
Thetimerateof changeof thevelocity at an arbitrary location x is
zero for all t. A uid element Vmoving throughthe pipe (Figure 5)
experiences acceleration due to the area reduction in the middle
ofthepipe.thederivationof
theaccelerationinEulerianformthesituationismorecomplicated,since
the time rate of change of the velocity at location x at time t0 is
in general not
equaltotheaccelerationoftheLagrangianuidelementpassingxatt0.
Asimpleexampleis showninFigure5. Thegureillustrates
asteadyincompressibleowthroughaconvergent pipe. Thus, thetimerateof
changeof thevelocityat anarbitraryxedlocationxiszeroforall
tsincetheowisstationary. However,
theuidelementVmovingthroughthepipe(Figure5)hastobeacceleratedduetotheareareductioninthemiddleofthepipe.
Theconclusionappliestoastationarywaterfall.
Ifwemonitoraxedlocationatthewaterfall wewill
observethatthevelocityoftheowdoesnotchangewithtime. Incontrast,
thewater is accelerateddownstreamof theobservedlocationbygravity.
Thus,aleaveoatingonthewaterisacceleratedwhenpassingthewaterfall.
Inotherwords, theleaveisequivalenttotheuidelementmovingwiththeow.
Since the uid element is accelerated the time rate of change of the
velocity of theuidelementisnotzero. Therefore,wehavedudtVL,= 0
anddudtVE= 0 (8)inthiscase. Thesymbol
[VEindicatesthatweevaluatethetimerateofchangeoftheuidvelocityforaEuleriancontrolvolumexedinspace.As
anexample, wederiveanexpressionfor theaccelerationinx-directionof
aLagrangianuidelement inEulerianform. Let us consider
aninnitesimallysmallLagrangian uid element moving with the ow in
two-dimensional cartesian space. TheThis is a preprint of an
article accepted for publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean?
9velocitycomponentsuandvarefunctionsofthespacecoordinates(x,
y)andthetimetu = u(x, y, t) and v= v(x, y, t).
(9)Thus,thetotaldierentialofureadsdu =utdt +uxdx +uydy.
(10)Dividingequation(10)bydtyieldsdudt=ut+uxdxdt+uydydt,
(11)where(dx/dt, dy/dt) describes thepathof theuidelement inspace.
Incaseof
aLagrangiannitecontrolelementweobtainfromequation(6),(dx/dt, dy/dt)
= (u, v),whichdenotesthevelocityoftheuidelement.
Therefore,equation(11)readsdudtVL=ut+ uux+ vuy.
(12)InuidmechanicsitiscommontodeneTuTt:=dudtVL. (13)Thesymbol
T/Ttiscalledthematerial orsubstantial derivative,
whichdescribesthetimerateofchangeofascalarquantityofthegivenuidelementasittravelsthroughspace.
Inotherwordsoureyesarelockedontheuidelementandweobservethatthex-component
of the velocity u of the element changes as it moves through a
point (x,
y).IncaseofanEulerianuidelementwetakethederivative(11)ataxedlocationand,therefore,(dx/dt,
dy/dt) = 0. Thus,equation(11)revealsdudtVE=ut.
(14)Notethatu/t,whichindicatesthetimerateofchangeofuatthexedpoint(x,
y),isdierentfromthematerial derivative. Inliteraturethematerial
derivative T/TtiscalledtheLagrangiandescriptionof thedynamicsof
auidwhereasthelocal
partialderivativeu/tistermedastheEuleriandescriptionofuiddynamics[1].Fromequation
(12) we can obtain an expression for the material
derivative[9,10,1,2]TTt=t+ ux+ vy. (15)The expression ux +vyon the
right hand side of equation (15) is called the
convectivederivative, whichdescribesthetimerateof
changeduetothemovementof
theuidelementfromonelocationtoanotherintheoweld,
wheregenerallytheconsideredow property (i.e. u, , p, etc.) is
spatially dierent. The substantial derivative
appliestoanyow-eldvariable[1],forexampleTTt..material
derivative=t..local derivative+ ux+ vy. .convectivederivative.This
is a preprint of an article accepted for publication in the
European Journal of PhysicsWhatdotheNavier-Stokesequationsmean?
105. ConservationlawsInobtainingthebasicequationsof
uiddynamicswehavetochoosetheappropriatefundamentalphysicalprinciplesfromthelawsofphysics[1,9,10,2]:(i)
Massisconserved.
Therearenonuclearreactionsinvolvedinuiddynamics.(ii)
Newtonssecondlaw: F=ddt(mu)5.1.
ConservationofmasscontinuityequationIngeneral,thecontinuityequationdescribesthatthetimerateofdecreaseoftheuiddensityinanarbitrarycontrolvolumeequalsthenetowoutofthisarbitrarycontrolvolumethroughitssurface.
Intuitively,
thecontinuityequationinuiddynamicscanbeeasilyderivedfromtheprincipleofmassconservation.
IfweconsideraLagrangiancontrolvolumemovingwiththeow,theconservationofmassreadsdmVLdt=d(VL)dt=
0, (16)i.e. the mass mVLof aLagrangiancontrol volume VLdoes not
change withtime.However, VLisnophysical eldquantityand, therefore,
wecannotusethematerialderivative to simplify equation (16). It is
more practical to consider an arbitrary Euleriancontrol
volumexedinspace(comparewithFigure6). Sincethevolumeandnottheuid
yx!(x0 !"x / 2, y0, t)!Axu(x0 !!x / 2, y0, t) u(x0 +!x / 2, y0,
t)!(x0 +"x / 2, y0, t)!Ay!VE =!x!y!zp(x0, y0, t)!(x0, y0, t)u(x0,
y0, t)p(x0 !!x / 2, y0, t) p(x0 +!x / 2, y0, t)Figure6. Fluidenters
theEuleriancontrol volumeVEwiththevelocityu(x0 x/2, y0,
t)fromtheleftside. FluidleavesVEwiththevelocityu(x0+ x/2, y0, t)to
the right side, which leads to a change of the density of the uid
occupyingVE= xyz. Thepressurep(x0 x/2, y0,
t)actsontheleftfaceandthepressurep(x0+ x/2, y0,
t)actsontherightfaceof theEuleriancontrol volumeVE, whichleads
toanon-zeronet forceonVE=xyz. Inthetwo-dimensional casez
isequivalenttoz 1.mass of an arbitrary Eulerian control volume is
xed we have to account for the in-
andoutowofmassperunittime,thatis,mVEt= m+..inow m..outow. (17)This
is a preprint of an article accepted for publication in the
European Journal of PhysicsWhatdotheNavier-Stokesequationsmean?
11Note that we have to apply partial (local) derivatives to
Eulerian control volumes.
Thelefthandsideofequation(17)canbewrittenasmVEt=(x, y, t)VEt= VE(x,
y, t)t. (18)For the last step we made use of the fact that the
Eulerian control volume VEdoes notchangewithtime.Expressions for
the in- and outow of mass per unit time can be derived from
Figure6.
Withoutloosinggeneralityweassumethattheowisalignedwiththepositivex-axis.
Hence, thereisnomassowthroughthelowerandupperfaces.
TheinowofmassperunittimeintoVEovertheleftmostarea,therefore,reads
min_kgs_ = density_kgm3_inowvelocity_ms_area_m2.
(19)IntermsofFigure6theinowandoutowofmassperunittimeread m= (x0x/2,
y0, t)u(x0x/2, y0, t)Ax.
(20)Thus,substitutingequation(20)intoequation(17)andusing(18)yieldsVEAx(x0,
y0, t)t= _x0x2, y0, t_u_x0x2, y0, t__x+x2, y, t_u_x+x2, y,
t_.(21)Wecanexpandthetermsontheright handsideof equation(21)
inaTaylor seriesyielding_x0x2_u_x0x2_ = (x0)u(x0)
u(x0)xx0x2(x0)uxx0x2+O. (22)It is reasonable to neglect the higher
order terms as x tends to 0. Thus, by
substitutingthelinearizedin-andoutowofmassperunittimeintoequation(21)yieldsVE(x0,
y0, t)t= Axx_u(x0)xx0(x0)uxx0_. (23)ByusingAx= yz,VE=
Axxequation(23)canbewrittenas(x0, y0, t)t= u(x0)xx0(x0)uxx0.
(24)Inequation(23)thephysicaleldsanduandtheirlocalderivativesareevaluatedat(x0,
y0, t). Sincex0,y0andtarearbitraryweskiptheexplicitdependencies.
Applyingtheproductruletothelasttwotermsontherighthandsideofequation(24)yieldsux
ux= ux. (25)Finally,wehavet= ux(26)forthecontinuityequation.This is
a preprint of an article accepted for publication in the European
Journal of PhysicsWhatdotheNavier-Stokesequationsmean? 12Inthe case
of anon-zerogradient of v iny direction, we canapplythe
sameprinciples we used for the above derivation of the continuity
equation, which then readst+ux+vy= 0.
(27)Forincompressibleuids,i.e. =
const.,thecontinuityequationcanbesimpliedasfollowsux+vy= 0 orux=
vy. (28)5.2.
NewtonssecondlawmomentumequationAsmentionedatthebeginningof
section5, wecanderiveamomentumequationforthe dynamics of the ow by
applying Newtons second law to an arbitrary nite controlvolumeV .
Therearetwosourcesof forcesactingonsuchanitecontrol
volumeV[1,9,10,2]:Bodyforces,whichactdirectlyonthevolumetricmassoftheuidelement.
Theseforces act throughout the nite control volume; examples are
gravitational, electric, andmagnetic forces. In most applications,
only the gravitational force on the uid particlesis taken into
account [1]. At the earths surface the gravitational force on a uid
elementissimplyFg= mVg,where g= (0, g) denotes the standard
acceleration due to gravity at the earths surface.Surfaceforces,
whichactacrossthesurfaceof theuidelement. Theseresultingeneral
fromtherandommotionof theuidmolecules.
Itiscommontodistinguishbetween:Pressure,
whichistheforceactingatrightanglesonthesurface.
Thepressureisimposedbytheoutsideuidsurroundingtheuidelement.
Onamolecularscalethepressureisthetimerateofchangeofcollidinguidmoleculesatthesurfaceofthecontrolvolume.The
shear andnormal stress distributions actingonthe surface; these are
alsoimposedbythe outside uid. These arise
frommomentumdiusiondrivenbytherandommotionof themolecules.
Inliteraturethesearealsointerpretedastugging or pushing of the
surrounding uid on the surface by means of
friction[1].IfweexamineanarbitraryLagrangiannitecontrolvolumemovingwiththeow,theforcebalance,therefore,readsdmVLudt.
.time rate of change of momentum= mVLg. .body forces+Fp+Fm..surface
forces. (29)Notetheright handsideof equation(29) does not
containtimederivatives.
Thus,theappliedforcescanalsobeevaluatedforthetimeindependentandspatiallyxedEuleriancontrolvolumeVE,whichrequiresVL=
VEattimet.This is a preprint of an article accepted for publication
in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean?
13Applyingtheproductruletothelefthandsideofequation(29)yieldsdmVLudt=
mVLdudt+udmVLdt. (30)Inorder toevaluate the rst termonthe right
handside we canuse the materialderivative.
Thesecondtermisequivalenttotheconservationofmass(equation(16))and,therefore,iszero.
Thus,wehavedmVLudt= VLTuTt . (31)Due to these manipulations the
time derivative of the Lagrangian control
volumedisappearsinequation(31)andweareabletosubstituteVLbyVE,whichwechooseequaltoVLattimet.
Thus,inEulerianformequation(29)reads_ut+ uux+ vuy_= fgx+ fpx+ fmx,
(32)_vt+ uvx+ vvy_= fgy+ fpy+ fmy,
(33)wherethefjisdenotetheforcedensities(forcepervolume
Fji/VE)ofthedierentcontributions. The rst term on the left hand
side of equations (32) and (33) is the timerateof changeof
themomentum. Thesecondandthirdtermsof
equations(32)and(33)arethetimerateof changeof
momentumduetothemovementof
aLagrangianuidelementfromonelocationtoanotherintheoweld.Pressure
forces: Before we derive an expression for the force on a uid
element exertedbythe uidpressure, we discuss the pressure inuids
inmore detail. Tosimplifyouranalysiswefocusourattentiononsemi ideal
gases, wherethedimensionsof themolecules are much smaller than the
distances betweenthe molecules. Therefore,thesecanbe consideredas
point particles. Other properties of semi ideal gases are
thatinteraction forces between the molecules are neglected, the
particles move randomly
andthatparticle-particlecollisionsandparticle-wall collisions,
respectively, areconsideredasperfectlyelastic.!VE!AxIdeal gas
Figure7. Semiidealgasconnedbyacontainer:
Randommotionofmolecules.Perdenition(comparewithsection3.1)thepressureistheforceperunitareaappliedinthedirectionperpendiculartoasurface.
Fromthediscussioninsection3.1This is a preprint of an article
accepted for publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean? 14this force can be
calculated from the time rate of change of momentum of the
moleculesataspecicarea.
Therefore,letusconsideranEuleriancontrolvolumeinacontainerlled with
a semi ideal gas at rest. Following Figure 7 the number of
molecules NhittingtheleftsurfaceAxof theEuleriancontrol
volumeVEfromtheleftsidewithinthetimeintervaltcanbeestimatedfrom[2]Nleftin=
nxAxumt,wherenxdenotesthenumberof
moleculesperunitvolumemovinginthepositivex-direction.
Sincethemoleculesmoverandomlynxcanbecalculatedfromnx=16n = nx= ny=
ny= nz= nz,wherenyandnzdenotethenumber of molecules movingin yand
z-directionrespectively. nis the number of molecules per unit
volume andumis the thermalvelocityof themoleculesinanyparticular
direction[13]. Withinadistanceof
mfpfromAxthesemoleculescollidewithmoleculesleavingVEtotheleftsideNleftout=
nxAxumt =16nAxumt.The time rate of change of the momentum of the
incoming colliding particles at surfaceAx,i.e.
theforceperpendiculartoAx,is,therefore,calculatedasFAx=(2mum)Nleftint,where2mumdenotesthechangeofmomentumofasinglecollidingmoleculeandmitsmass.
NotethatFAxtakesthesameformwhetherAxisawallorlocatedwithintheuid.
Recognizingthat =
mnandsimplicationyieldsFAx=13Axum2.DividingbyAxnallygivesthepressureinasemiidealgasp
=13um2.It is important to note that the above analysis shows that
rst, the pressure in a uid isisotropic,i.e.
hasnodirection,sinceonesixthoftheparticleswithinadistanceumtfromeacharbitraryorientedplaneAmovefromonesidetotheotherandviceversa.Second,thepressurearisesfromthemolecularnatureofuids.The
force fpon a uid element can be calculated similarly to the
derivation of thecontinuityequation. Again, weconsider
anEuleriancontrol
volumeVE(Figure6).Notingthatthepressureisisotropicthepressureforceontheleftmostsurfaceof
thecontrolvolumeVEisfp, leftxVE= Axp(x0x/2, y0, t).
(34)Theforceontheuidelementduetothepressureontherightmostsurfacereadsfp,
rightxVE= Axp(x0 + x/2, y0, t). (35)This is a preprint of an
article accepted for publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean?
15Combiningequations(34)and(35)givesfpx(x0, y0, t)VE= Ax_p(x0x/2,
y0, t) p(x + x/2, y,
t)_(36)TaylorseriesexpansionandneglectinghigherordertermsyieldsfpxVE=
Axxpx. (37)SinceVE= Axx,wenallyhavefpx= px, (38)fpy= py.
(39)Fromequations(38)and(39)weobtainthatanon-zeropressuregradientproducesaowintheoppositedirectiontothegradient.
For example, windis
inducedbythepressuregradientbetweenhighandlowpressureareas.
Sincethepressuregradientisdirectedfromlowtohighpressureareas(i.e.
theslopeofthepressureispositivefromalowtoahighpressurearea), theair
ows fromthehighpressureareatothelowpressurearea.Normal
andshearstressesinincompressibleNewtonianuids:
Inthelastsection,wehavediscussedthetime rateof changeof momentumdue
totherandommolecularmotion.
However,ifthelocalvelocitygradients,theseareux,uy,vxandvy,are
non-zero, additional forces on the surface of the control volume
arise from the randommotion of the molecules,which contribute to
fm. The following example illustrates thev1v2! m12!
m21!mfp!mfpyxFigure8. Illustrationof themolecular
momentumdiusion[2]. Railroadworkersshovel coal from their own train
to the other, which induces a change of momentum
ofbothtrains.WhatdotheNavier-Stokesequationsmean?
16underlyingphysics,
whichisreferredtoasmolecularmomentumdiusion[2].
Letusconsidertwoparallel
trainsmovingwithdierentvelocitiesv1andv2(Figure8)withv2> v1.
Ontrain1railroadworkersshovelcoalfromtheirowntraintotrain2witharate
m12(inkg s1).
Ontrain2equallyhardworkingrailroadworkersshovelcoalbacktotrain1witharate
m21(kg s1). Forsimplicityweassumethat m12= m21.
Thus,themassesofthetrainsdonotchangewithtime.
Theshovelingofcoalcausestrain2todeceleratesincethetimerateofchangeofitsmomentumisnegativeF21=
m12v1 m21v2= m12 (v1v2). .0>
0.Inthecaseofaverysmalldistance2mfpbetweenthetrainswemaywritev1=
v[x=0mfpvxx=0,v2= v[x=0 + mfpvxx=0.Inour example, v[x=0is the
meanvelocityof bothtrains and2mfpis theaveragethrowingdistanceof
therailroadworkers. Thetimerateof changeof
momentumofbothtrains,therefore,readsF12= 2 m12mfpvx,F21=
F12.Applyingthistrainmodeltotherandommotionofthemoleculesofasemiidealgastheshovelingofcoalfromtrain1totrain2canberegardedasthenumberofparticlesmovinginpositivex-directionnx.
Similarly, nxcorrespondstotheshovelingof coalfromtrain2totrain1.
mfpisinterpretedasthemeanfreepathofthemolecules,i.e.theaveragedistancebetweentwosubsequentcollisions.
Thus,adoptingthederivationofthepressureinidealcaseswehave
m12=16nm..=Aum.Hence,weobtainthetimerateofchangeofmomentumperunitarea,thatistheshearstress(1)yx
,
whicharisesfromaowiny-directionandfromthemoleculesmovinginx-direction,(1)yx=F12A=13ummfpvx.
(40)In accordance with (1)yxa time rate of change of y-momentum per
unit area results
fromaowinx-directionandfromthemoleculesmovinginy-direction[2](2)yx=16um_u(yAy+
mfp) u(yAy mfp)_. (41)This is a preprint of an article accepted for
publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean? 17Taylor series
expansionof thelast twoterms inequation(41)
andneglectinghigherordertermsgives(2)yx=13ummfpuy.
(42)Thusthetotaltimerateofchangeofmomentumreadsyx= (1)yx+
(2)yx=13ummfp_uy+vx_. (43)Itiscommontodene =13ummfp,
(44)wheredenotes themolecular viscosity. FromFigure9it
canbededucedthat yxcontributestofmy.
Notethatthevelocitygradientiscommonlydenotedasshearratesinceithasthedimensionsofs1.uid
yx!VE =!x!y!z!Ax!Ay!xx!yx!yy!xyFigure9. Normal andtangential
constraintsduetoowini-directionarisingfrommolecularmomentumdiusioninj-direction(i,
j x, y. Thesecanbetakenintoaccountbynormal andshearstresses(normal
andtangential forceperarea)actingonthe Euleriancontrol volume
VE=xyz. Inthe two-dimensional case z isequivalenttoz 1.An
additional contribution to fmyis caused by a ow in y-direction and
the moleculesmovinginy-direction. Thatis[11]yy= 2vy, (45)which can
be obtained from equation (43) by replacing x by yand u by v. Note
that incase of vanishing velocity gradients the pressure forces
remain the only force
componentresultingfromrandommotionofmolecules.Byfollowingthederivationoffpinsection5.2aandfromFigure9wendfmy=yxx+yyy.
(46)Substitutingequations(43)and(45)intoequation(46)yieldsfmy=
x_uy+vx_ + 22vy2. (47)This is a preprint of an article accepted for
publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean?
18ManipulatingaboveequationandapplyingSchwarztheoremyieldsfmy=
_yux+2vx2+ 22vy2_.
(48)Bysubstitutingthecontinuityequationforincompressibleuids(28)weobtainfmy=
_2vx2+2vy2_. (49)Bysimilarconsiderationsfmxiscalculatedasfmx=
_2ux2+2uy2_. (50)Note that the above derivation shows that xy=
yxholds. Fluids, where isproportional tothevelocitygradient(i.e.
=const.), arereferredtoasNewtonianuids.
Forexample,airandwatercanbeconsideredasNewtonianuids.
Incontrast,uids,wheretheviscosityitselfisafunctionofthevelocitygradient,arereferredtoasnon-Newtonianuids.
Forinstance,ketchupshowsnon-Newtonianbehavior.5.3.
SummaryofthegoverningequationsforincompressibleuidsCombiningtheresultsof
sections5.1and5.2yieldsthesetof
incompressibleNavier-Stokesequations, whichisasetof partial
dierential equations. Theseconsistof
thecontinuityequationandthemomentumequations.
Sincewehaverestrictedourselvestoincompressibleowsgravityfgcanbeincludedinthepressuretermyielding
p = p + gy. (51)Thequantity
pissometimescalledtotalhydrostaticpressure[14].
Thus,theNavier-Stokesequationsforanincompressibleuidreadux+vy= 0,
(52)ut+ uux+ vuy= 1 px+_2ux2+2uy2_, (53)vt+ uvx+ vvy= 1
py+_2vx2+2vy2_. (54)The mathematical properties of solutions of the
Navier-Stokes equations is one concernof the millennium problem
referred to as The Navier-Stokes existence and smoothnessproblem
[3]. While numerical solutions of the Navier-Stokes equations are
widelyestablishedinscienceandapplications,thetheoreticalunderstandingofitssolutionsisquite
incomplete [15]. Exact solutions are mostly restricted to special
cases as discussedinthenextsection.This is a preprint of an article
accepted for publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean?
19NoteonvectorialnotationoftheNavier-Stokesequations: Equations
(52)(54) can bepresentedinamoreconcisevectorialnotation,whichreads
u = 0,ut+uu = 1 p +u +1fb,where u=(u, v),=(/x, /y), uu=(uu, uv), p
= p gand =2/x2+2/y2. In this notation denotes the dot product and
fbdescribes additional bodyforces, as for example, magnetic or
electric forces. Thisnotationallows easilythe generalizationof
equations (52)(54) tothree dimensionsbydeningu=(u, v, w), =(/x, /y,
/z), uu=(uu, uv, uw), = 2/x2+ 2/y2+ 2/z2.Noteoninitial
andboundaryconditions: Solutionsoftheinitial
valueproblem(52)(54)aredeterminedbyinitial andboundaryconditions.
Theinitial
conditionsforthegoverningequationsforincompressibleuids(52)(54)aregivenbyu(x,
t0) = u0(x), (55)p(x, t0) = p0(x), (56)whereu0andp0aretheinitial
velocityandpressureelds.
Theboundaryconditionsforuandpforagivendomaincanbeexpressedindierentways:(i)
Dirichlet boundaryconditions:
Auidproperty,thatisuorp,isprescribedattheboundaryofasfollows(x, t)
= D(x, t), (57)withx andwhereD(x, t)isagivenfunction.(ii) Von
Neumann boundary conditions: The gradient in normal direction to
theboundaryofauidpropertyisspeciedas(x, t)n(x, t) = vN(x, t),
(58)withx andwherevN(x, t)isagivenfunction.
ndenotestheoutwardunitsurfacenormaland isdenedas = (/x, /y).(iii)
TheRobinboundaryconditionsrepresentalinearcombinationoftheabove,i.e.a(x,
t) + b(x, t)n = R(x, t), (59)withx ,a ,= 0,b ,= 0andR(x,
t)given.Physical boundaryconditions for
pressureandvelocityareusuallyacombinationofDirichlet-
andvonNeumann-type boundaryconditions. At aninlet
withspeciedvelocityor a wall (Dirichlet boundarycondition) a
vonNeumann(zero-gradient innormal
direction)boundaryconditionmustbesuppliedforpressure.
Foraconstantpressure outlet, a zero-normal-gradient boundary
condition for velocity must bespecied.
Atsymmetryplaneszero-normalgradientboundaryconditionsmustbeusedThis
is a preprint of an article accepted for publication in the
European Journal of PhysicsWhatdotheNavier-Stokesequationsmean?
20for all ow quantities. Robin-type boundary conditions are hardly
ever used for pressureandvelocitybut can, for example, occur for
temperatureboundaryconditions
whensolvingadditionallytheenergyequation.Noteoncompressibleuids:
Inthecaseofcompressibleuidsadditional
termsariseinthemomentumequations(53)and(54).
Theseresultfromthetimerateofchangeof thevolumeof thecontrol
uidelement movingwiththeow. Whenthevolumedilates (increases or
decreases) the momentum of the control volume changes due to
therandommotionofthemolecules.Additionally,thesetofcompressibleNavier-Stokesequationsisnotclosed.
Thereare3partialdierentialequationsbut4unknowns(u,v,pand).
Therefore,weneedanadditionalequationtoclosethesystemofequations.
Itiscommontointroduceanequationof state, forexampletheisothermal
ideal gaslaw, whichrelatesthedensitywiththepressure,thatis = (p).6.
ExactsolutionsoftheincompressibleNavier-StokesEquationsIn this
section we discuss three simple examples of exact solutions of the
Navier-Stokesequations,
whichshowtheinuenceoftheviscosityoftheuidonthevelocitydistribution.
Theseexamplesmayalsobesubjectofin-depthphysiccoursesatseniorhighschools.6.1.
SimpleshearowIntherstexampleweconsidertheowbetweentwoinniteparallel
conningwallsparallel tothex-axis. Theowis drivenbytheupper
movingwall andaconstantpressure is assumed. In literature such a
owis referred to a simple shear ow.Furthermore,
insteadystatetheowdoesnotvaryinx-direction. Thus,
thevelocityandpressuregradientsreadux= 0,vx= 0, px= 0. (60)Sinceu/x
= 0,itfollowsfromthecontinuityequation(equation(52))thatvy= 0.
(61)Integrationyieldsthatvisconstant.
Itisclearthatvhastobezerobecausev
,=0wouldimplyaowthroughtheconningwalls, whichisnotpossible. Thus,
forthesteadystate,whenthevelocitiesdonotvarywithtime,wehaveut=
0,ux= 0, px= 0,vt= 0,vx= 0,vy= 0. (62)This is a preprint of an
article accepted for publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean?
21Substitutingequation(62)intoequations(53)and(54)yields2uy2= 0,
(63) py= 0. (64)Integrationof equation(64) andusingthedenitionof
pyields p(y) = gy.
Thepressurevarieslinearlywithyinordertocounteractthegravitationalforcebuthasnoinuenceonthevelocityeld.
Hence,equation(54)isidenticallyzerosincev(x, y) =
0anduissolelyafunctionofy.
Integrationoftheu-momentumequationgivesu(y) = ay + b, (65)where a
and b are constants of integration. These can be calculated by
substituting theboundaryconditionsu(0) = 0andu(h) = uhu(0)= 0 = b,
(66)u(h) = uh= ah + b.
(67)Theseboundaryconditionsassumethattherelativevelocitybetweenthewallandtheadjacent
uid particles is zero (no-slip condition). Solving the linear
system of equationsyieldsa = uhy/handb = 0.
Thus,wenallyhavealinearvelocityproleu(y) =uhhy with y [0, h].
(68)The solution is shown in Figure 10a. It is noteworthy that the
steady solution (t )forudoesnotdependontheviscosity,
whereastheshearstressesactingonthewallsarexy=uyy=0=uyy=h=uhh.
(69)Thus, increasing the viscosity of the uid increases the force
required to move the
upperconstrainingwallwithuh,whichisF=uhAh,whereAistheareaofthewall.
However, forinvisciduids, i.e. =0andthereforenomomentumdiusion,
wewouldobtainu(y)=0sincethereisnophysical
process,whichtransfersthemomentumoftheconstrainingwallstotheuid.6.2.
Two-dimensional
shearowwithpressuregradientNowthepreviousexamplewillbemodiedbydroppingtheassumptionp/x
= 0,sothat the uid will not only be driven by the wall movement,
but also due to the pressuregradient. Weconsideratwo-dimensional
channel of lengthl andheighth,
whereweapplyattheinletapressurepinandattheoutletapressurepout.
Thus, thepressuregradientsread px=poutpinl, py= 0. (70)This is a
preprint of an article accepted for publication in the European
Journal of PhysicsWhatdotheNavier-Stokesequationsmean? 22(a)
(b)Figure 10. a) Simple steady shear ow (Couette-ow) between two
constraining wallsatadistanceh. Theupperwall
moveswithavelocityuhinpositivex-direction;
b)Steadyowthroughatwo-dimensional channel of lengthl
andheighth(Poiseuille-ow).
Attheinletapressurepinandattheoutletapressurepoutareapplied.Byusingthedenitionof
p(gravityacts innegativey-direction) equation(63)
nowbecomes1pl=2uy2(71)byintroducingp = poutpin.
Integrationgivesu(y) =p2ly2+ by + c,
(72)wherebandcareconstantsofintegration.
Applyingtheboundaryconditionsu(0) = 0andu(h) = uhyieldsu(0)= c = 0,
(73)u(h) = uh=p2lh2+ bh.
(74)Theconstantbcaneasilybecalculatedfromthesecondequationandwenallyobtainu(y)
= p2ly(h y) +uhhy. (75)The result is a superposition of a quadratic
velocity prole due to the pressure gradientand the linear prole due
to the wall movement already known from equation (68).
Thetwospecial cases, whenoneofthetwovelocitycomponentsvanishes,
arebothnamedafterfrenchphysicists. Thelinearprolewhenp =
0iscalledCouette-ow.For uh=0 andp,=0 the velocityprole is of a
parabolic shape andis calledPoiseuille-ow(comparewithFigure10b).
Itsmaximumvelocityinthecenterofthechannelisumax=u(h/2)= (p)h2/8l.
Increasingtheviscosity(forexampleusinghoneyinsteadofwater)leadstoadecreaseofthemaximumvelocityandtoadecreaseThis
is a preprint of an article accepted for publication in the
European Journal of PhysicsWhatdotheNavier-Stokesequationsmean?
23ofthemassowthroughthechannel,thatisq0= z_h0u(y)dy=
zp2l_y2h2y33_h0= z12plh3.
(76)zdenotesthewidthofthechannelinthethirdspatialdirectiontoreceivetherightdimensions
of themass ow(kg s1). This behavior sounds
plausiblesinceviscosityiscommonlydenotedasinternal friction.
Finally, notethatif pout1thevelocityinthechannel
exceedsthewallvelocity. ForP<
1thestrongadversepressuregradientleadstolocalbackownearthelowerwall.
Theviscousforcebetweentheuidlayersisnotstrongenoughtoovercometheadversepressuregradient.In
pratice, simple Couette ows are very important in rheology,
viscosimetry
andrheometry,thestudiesoftheviscousshearbehaviourofliquidsandtheirexperimentalinvestigation,
respectively. The main advantage of Couette viscometers is linear
relationbetweenthewall forceandtheuidviscosity(seeequation(69)),
whichallowsforasimpledeterminationoftheviscositybyforceormomentmeasurements.Poiseuille-Couette
ows are an important aspect in lubrication theory, or,
moregenerally, for ows through small gaps and channels with moving
boundaries,encounterede.g.inthecoatingprocessofsurfaces,non-hermeticsealing,etc.6.3.
Poiseuilleowwithwall suction/injectionA simple generalization of
the Poiseuille ow is possible, if we consider the ow
throughachannel madeof porouswalls. Fluidcanenterorexitthechannel
viatheseporouswalls. We consider the same setting as before, but
with the upper wall at rest (uh=
0).Fluidisnowinjectedthroughthelowerwall
(y=0)withaconstantvelocityvwandalsoexitsattheupperwall
(y=h)withthesameconstantvelocity. If
thechannelisassumedtobeinnitelylong, wecanstill
chosethevelocityindependentofx.
Thecontinuityequationthenreadsux+vy=vy= 0v(y) = vw= const. (79)This
is a preprint of an article accepted for publication in the
European Journal of PhysicsWhatdotheNavier-Stokesequationsmean?
24(a) (b)Figure 11. a) Dierent solutions for a
Poiseuille-Couette-ow between twoconstraining walls at a distance
h. Pis the dimensionless pressure gradient according
toequation(78);b)Velocityprolesofasteadyowthroughatwo-dimensionalchannelheight
hwithuniformcrossowat dierent crossowReynolds numbers Revfor
agivenpressuregradientinx-direction.Hence, thereisaconstantvertical
uidmotioninthechannel. Theinuenceof thisuniformcrossowonthe
owinx-directionwill be calculatedinthe following. Incontrast tothe
previous examples the termvuyinequation(53) is nolonger
zero.Hence,thex-momentumequationbecomesvwuy= 1pl+2uy2.
(80)Thus,wehavetodealwithaninhomogeneousordinarydierentialequationofsecondorder
with constant coecients. First we solve the homogeneous part of the
dierentialequation2uy2 auy= 0,witha = vw/.
Thecharacteristicpolynomialofthisequationis2a = 0withthesolutions
1=0and2=a. Thesolutionof thehomogeneous
equation,therefore,isuhom(y) = c1 e1y+ c2 e2y= c1 + c2
eay.Sincetheinhomogeneous pressuretermis constant and1=0,
weassumealinearparticularsolutionypar= c3 y.
Substitutionintothedierentialequationyieldsc3= 1 vwplupar(y) = y
vwpl.This is a preprint of an article accepted for publication in
the European Journal of PhysicsWhatdotheNavier-Stokesequationsmean?
25Superpositionofthehomogeneousandtheparticularsolutionsleadstou(y)
= c1 + c2 evwyy vwpl.The constants c1andc2canbe determinedfromthe
no-slipconditionat the wallsu(0) = u(h) = 0.
Thesolutionnallybecomesu(y) =h vwpl1 evwy1 evwhy vwpl. (81)If we
introduce dimensionless quantities this solution can be written in
a more simple andgeneral form. By relating the x-velocity to the
maximum velocity of the Poiseuille-ow,thatisu0,max= (p)h2/8l
andbyintroducingadimensionlesscrossowReynoldsnumberRev= vw h,
(82)thesolutionreadsu(y)u0,max=8Rev_yh 1 eRev y/h1 eRev_.
(83)CalculatingthemassowandrelatingittotheresultforthePoiseuille-owq0fromequation(76)yieldsqq0=12Rev_12
1Rev11 eRev_. (84)Several
solutionsforthevelocityprolesatdierentcross-owReynoldsnumbersareshowninFigure
11b. The velocityprole is shiftedupwards, awayfromthe
wallwithuidinjection.
Themaximumvelocityandthemassowdecreasewithhigherinjection/suction
rates. At higher injection/suction rates, i.e. higher crossow
Reynoldsnumbers, the prole becomes nearlylinear throughout most of
the channel height.Applicationsof channel
owswithsuperposedcrossowarecommoninthechemicalindustry,forlterapplications,etc.6.4.
FurtheranalyticsolutionsAwiderangeof otherproblemsexist,
whereithasbeenpossibletosolvetheNavier-Stokes-equations
analytically. Coveringthemhere exceeds the scope of this
paper,especiallysincetheseproblemsarewellcoveredinexistingliterature[2,16,17,14,18,19].
Someexamplesarethestagnationpointowinthevicinityof awall
positionednormal to the oncoming ow (Hiemenz-ow), the ow through
converging and divergingchannels (Jerey-Hamel-ow), the diusionof
avortexinaviscous uidover time(Lamb-Oseenvortex),etc.7. SummaryIn
this article we have attempted to make the Navier-Stokes available
to a widerreadership, especially teachers and undergraduate
students, by outlining the
underlyingphysicalprinciplesandassumptions.This is a preprint of an
article accepted for publication in the European Journal of
PhysicsWhatdotheNavier-Stokesequationsmean? 26AcknowledgmentsThe
authors wouldlike tothankProfessors UrbaanTitulaer, ErichSteinbauer
andStefanPirkerformanyusefuldiscussionsandhelpfulcomments.References[1]
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Physics
