Free surface microflows and particle transport [Elektronische Ressource] / von Michael Schindler

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Free SurfaceMicroflowsandParticleTransportDissertationzurErlangungdesakademischenGradeseinesDoktorsderNaturwissenschaften,vorgelegtderMathematisch–NaturwissenschaftlichenFakultätderUniversitätAugsburgvonMichaelSchindlerausBaselAugsburg,imApril2006ErsterGutachter: Prof.Dr.PeterHänggiZweiter Prof.Dr.AchimWixforthDritterGutachter: Prof.Dr.RolandNetzTagdermündlichenPrüfung: 29.Mai2006ImAndenkenanmeinenVater,dermiralsMenschundWissenschaftlereinVorbildist.ivContents1. Introduction 11.1. Theflowinmicrofluidicsetups . . . . . . . . . . . . . . . . . . . . . . 41.1.1. TheStokesequations . . . . . . . . . . . . . . . . . . . . . . . 41.1.2. Flowswithfreesurfaces . . . . . . . . . . . . . . . . . . . . . 61.1.3. Themicrofluidicparameterregimeofsurfacedeformations . . . 71.1.4. Limitofinfinitesurfacetension . . . . . . . . . . . . . . . . . 81.2. Acousticstreaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3. Transportofsmallparticles . . . . . . . . . . . . . . . . . . . . . . . . 121.3.1. Particletransportinclosedchannels: thedriftratchet . . . . . . 121.3.2. Particleinopenchannels . . . . . . . . . . . . . . . . 142. Computing stationary free-surface flows 172.1. Basicfinite elementdiscretisation . . . . . . . . . . . . . . . . . . . . 17Imposingboundaryconditionsthenaturalway . . . . . . . . . . . . . . 19Previousworkandotherapproachestofree surfacediscretisations . . . 202.2. Variationalformulationofthestressbalanceatafreesurface . . . .
Publié le : dimanche 1 janvier 2006
Lecture(s) : 16
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Source : WWW.OPUS-BAYERN.DE/UNI-AUGSBURG/VOLLTEXTE/2006/357/PDF/SCHINDLER_DISS.PDF
Nombre de pages : 129
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Free SurfaceMicroflows
andParticleTransport
Dissertation
zurErlangungdesakademischenGradeseines
DoktorsderNaturwissenschaften,
vorgelegtder
Mathematisch–NaturwissenschaftlichenFakultät
derUniversitätAugsburg
von
MichaelSchindler
ausBasel
Augsburg,imApril2006ErsterGutachter: Prof.Dr.PeterHänggi
Zweiter Prof.Dr.AchimWixforth
DritterGutachter: Prof.Dr.RolandNetz
TagdermündlichenPrüfung: 29.Mai2006ImAndenkenanmeinenVater,
dermiralsMenschundWissenschaftler
einVorbildist.ivContents
1. Introduction 1
1.1. Theflowinmicrofluidicsetups . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1. TheStokesequations . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2. Flowswithfreesurfaces . . . . . . . . . . . . . . . . . . . . . 6
1.1.3. Themicrofluidicparameterregimeofsurfacedeformations . . . 7
1.1.4. Limitofinfinitesurfacetension . . . . . . . . . . . . . . . . . 8
1.2. Acousticstreaming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3. Transportofsmallparticles . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1. Particletransportinclosedchannels: thedriftratchet . . . . . . 12
1.3.2. Particleinopenchannels . . . . . . . . . . . . . . . . 14
2. Computing stationary free-surface flows 17
2.1. Basicfinite elementdiscretisation . . . . . . . . . . . . . . . . . . . . 17
Imposingboundaryconditionsthenaturalway . . . . . . . . . . . . . . 19
Previousworkandotherapproachestofree surfacediscretisations . . . 20
2.2. Variationalformulationofthestressbalanceatafreesurface . . . . . . 22
2.2.1. Differentialgeometryofasurface . . . . . . . . . . . . . . . . 23
2.2.2. Thethermodynamicstressterms: withoutflow . . . . . . . . . 24
2.2.3. Thedynamicstressterms: viscousflow . . . . . . . . . . . . . 26
BothStokesequationsinvariationalform . . . . . . . . . . . . 28
Consequencesforthediscretisation . . . . . . . . . . . . . . . 28
2.2.4. Completelydynamicformulation . . . . . . . . . . . . . . . . 29
2.2.5. Secondvariations . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.3. Finite elementdiscretisationintwodimensions . . . . . . . . . . . . . 34
2.3.1. Separatingthealgorithmintotwosteps . . . . . . . . . . . . . 36
Thefluidicsystem . . . . . . . . . . . . . . . . . . . . . . . . 38
Thegeometricsystem . . . . . . . . . . . . . . . . . . . . . . 41
2.3.2. Similarityoffreesurfacesandrubberbands . . . . . . . . . . . 43
2.3.3. Summaryofthealgorithm . . . . . . . . . . . . . . . . . . . . 45
2.3.4. Accuracytests . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.3.5. Stabilisationtests . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4. Pinneddropletsintwodimensions . . . . . . . . . . . . . . . . . . . . 52
Conclusionsforthedrivingbythe SAW . . . . . . . . . . . . . . . . . 57
2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
vContents
3. Particle accumulation 61
3.1. Transportofanextendedparticleinaflow . . . . . . . . . . . . . . . . 61
Point likeparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Extended . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.1. Zero orderforces . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.1.2. First order . . . . . . . . . . . . . . . . . . . . . . . . . 63
Faxén’stheoremoftranslationalmotion . . . . . . . . . . . . . 64
3.1.3. Higher orderforces . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.4. Randomforces . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2. Particleaccumulationatboundaries . . . . . . . . . . . . . . . . . . . 71
Volumeeffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Boundaryeffect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.1. Channelswithno slipboundaries . . . . . . . . . . . . . . . . 74
3.2.2. Comparisonofslipandno slipboundaries . . . . . . . . . . . 81
3.3. ParticleaccumulationinSAW driveneddies . . . . . . . . . . . . . . . 86
3.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4. Summary and outlook 91
Appendices
A. Free-surface flow in a half cylinder: an analytical solution 95
B. Calculus of variations for the differential geometry of a surface 99
C. Animplementationoftheperfect-slipboundaryconditions: aconstraint
method 103
D. Green functions for stationary Stokes flow 107
D.1. GreenfunctionsfortheinhomogeneousStokesequations . . . . . . . . 107
D.2.forthehomogeneousStokes . . . . . . . . . 108
Bibliography 111
Acknowledgements 119
Publications 121
Curriculum vitae 123
vi1. Introduction
For theoretical and mathematical physics, the description of fluid flow has always been
animportantsourceofinspiration. GoingbacktothetimesofMaxwell,ithasledtothe
field concept and thus made the description of electromagnetism possible. Then, in the
twentieth century, the nonlinearity of the Navier–Stokes equation has given physicists
and mathematicians a very hard time – which lasts until today. Still, a “mathematical
1theory which will unlock the secrets hidden in the Navier–Stokes equations” is one of
the famous Millennium Problems. During the past decades, complex fluids such as non
Newtonian polymer solutions have gained more and more interest, but this is by far not
thelasttopicthattheNavier–Stokesequationhasinspired.
In recent years, microfluidic setups have fascinated many scientists (Squires and Quake,
2005; Stone and Kim, 2001; Stone et al., 2004). Microfluidics opens a door towards
the physical and mathematical treatment of biological systems, ranging from details of
polymerdynamicsinflows(SchlagbergerandNetz,2005;DoiandEdwards,1986)over
themechanicsandflowinthinbloodvesselsandcells(Sackmann,2004)uptocreatures
livingandwalkingonwatersurfaces(Huetal.,2003;BushandHu,2006),tonameonly
a few. Its potential use for miniaturising standard chemistry setups seems tremendous
(seeStoneandKim2001andreferencestherein). Usingandunderstandingmicrofluidic
systems in practice, however, often implies a change of paradigm. On typical length
scales up to 100μm, water does not behave as we are used to from everyday life. It
appears to be viscous like honey, its interface with the surrounding air is stiff and hard
such that one could walk on it like water a strider. And, last but not least, all movement
issubjecttofluctuations,leadingtowell knownphenomenasuchasBrownianmotion.
A promising new technique for manipulating small amounts of water and other liquids
has been developed recently by Wixforth et al. (2004). They actuate droplets of water
with so called surface acoustic waves (SAWs) which are waves of mechanical elonga
tionatthesurfaceofacrystallinematerial,illustrativelyunderstoodasatinyearthquake.
When such a wave comes into contact with water, it gets damped and gives rise to a
streaming in the fluid. The photographs in Figures 1.1 and 1.2 give an immediate im
pression of the resulting flows. The driving force, which is caused by the SAW, gives
risetoaninternalstreamingpattern,asismadevisibleinFigure1.1. Astrong SAWmay
deformandmovethedropletasawhole,asisdepictedin1.2.
1seee.g.http://www.claymath.org/millennium/Navier Stokes_Equations/
11. Introduction
a b c d
Figure 1.1. A resting water droplet, which is mixed by a weakSAW. The snapshots are taken at consec
utivetimesfrom(a)to(d). TheSAWapproachesthedropletfromtheleftandisdampedbytheinfluence
oftheviscousfluid. Thisgivesrisetotheinternalstreamingpattern. Aspotofcolourdyeonthesubstrate
isslowlydissolvedandmakestheflowvisible. (PicturesfromC.StroblandA.Wixforth)
a b c
d e f
Figure 1.2. A “jumping” droplet containing 50nl of water, on a flat substrate. The photographs are taken
at consecutive times from a film. Thus, several snapshots overlap in each picture. In the middle pictures
(b)and(e),apowerfulpulsedSAWarrivesfromtheleftanddeformsitstrongly. WhentheSAWisturned
off,thedropletrelaxesintoitsinitialsphericalshapeataslightlyshiftedposition. (PicturesfromC.Strobl
andA.Wixforth,seealsoWixforthetal.2004)
This new manipulation method opens a wide range of applications and fundamental
problems. Several lab on a chip applications become realisable, such as the ability to
transport, mix and separate droplets of some nanoliters volume, as well as the replica
tion of DNA in small droplets, and more (see Guttenberg et al. 2004; Rathgeber et al.
2005; Sritharan et al. 2006; Strobl et al. 2004;g et al. 2005; and http://www.
.advalytix.com). The experimental setup is accessible from many sides, which is con
venientregardingboththeinjectionofparticlesintothefluidandtheobservationoftheir
motion. From a theoretical side, SAW driven microflows imply several aspects raising
fundamentalquestions,someofwhichremainunsolvedalreadyforseveraldecades. The
jumping motion of the droplet, as in Figure 1.2, might allow a closer look on the prob
lem of a moving contact line. What boundary condition is imposed on the flow at the
2substrate in the vicinity of the contact line? Does the droplet in Figure 1.2 glide, roll
or rather jump – or is it a combination of all three motions? The driving mechanism by
the SAW itself is not fully understood either. It is currently under investigation by the
experimentalphysicsgroupofA.WixforthandthemathematicsgroupofK.Siebert,see
Gantner et al. (2006) for recent achievements. Another fundamental question concerns
the thermodynamics of a fluid in motion, for which an equilibrium description is not ap
plicable anymore. Such a fluid comprises fluctuating velocities and forces. Describing
theirstatisticalpropertiesinthepresenceofboundaries,suchasrigidimmersedparticles,
is a demanding challenge. Altogether, the present thesis will have to describe and to
combineseveralinterestingandimportantaspectsofthemicroflows.
The focus of the thesis is on the description of stationary SAW driven microflows with
free surfaces and the motion of particles therein. Different geometries of the fluid will
be considered, such as droplets and water channels, confined partly by the flat substrate
at the bottom, partly by free surfaces. For the actuation mechanism we will assume a
qualitativecharacterisationoftheacousticstreamingeffectbytheSAW.Theverydetails
of the SAW will not be treated here. The typical length scales of the systems is between
tenanhundredmicrometers. Insuchsmallsystems,theboundariestendtodominatethe
behaviour of the flow. Above all, we will confront ourselves with free surfaces, which
introduce a nonlinearity and thus are considerably more difficult to treat than ordinary
boundaryconditionsforviscousflow.
Fortheextensivespecificationoftheflowsinthementionedgeometries,inChapter2we
will derive a numerical algorithm which is capable of calculating the shape of a curved
free surface together withthe surroundingflow causingthe deformation. A keyelement
of our analysis is the variational formulation of the equations describing both the flow
and the geometry of the curved free surface. This formulation allows the derivation of
an algorithm which is more stable than previously existing ones, due to the fact that
the discrete representation of the free surface adapts some properties of a rubber band.
AhighlightofChapter2willbethenumericalsolutionoftheflowinsideasmalldroplet.
Thequalitativepropertiesofitsdeformationyieldanimprovedinsightintothenatureof
thedrivingforcesbythe SAW.
In Chapter 3wewillusethenumericallyobtainedfree surfaceflowstoinvestigatetheir
potential application concerning particle accumulation. The benefit of the free surface
numerics here is twofold. On one hand, the calculated velocity fields are utilised for
the combined advective and diffusive transport of small particles. Here, we will focus
on the effects by the boundaries on the accumulation of On the other hand,
what we have learned about the SAW from the droplet deformations, will prove useful
tounderstandanaccumulationeffectinthevolume. Thiseffectisbasedonthedifferent
impactsofthevelocityandthepressurefieldsondifferentlysizedparticles.
In the following sections of this chapter we will introduce those aspects of the SAW-
driven microflows which are of most interest here. We start with the description of the
flow, governed by free surfaces, and then qualitatively characterise the driving mechan
ismbytheSAW.Thethirdsectionofthischaptermotivatesacloserlookontheabilityof
31. Introduction
such microflows to generate directed transport of particles. The following two chapters
thencontainthedetailedtreatmentsofthefreesurfaceandoftheparticletransport.
1.1. The flow in microfluidic setups
Throughout this thesis, we will use tensor notation for fields in arbitrary curvilinear co
ordinates. This considerably simplifies the differential geometric notation in Chapter 2,
where the flow around curved free surfaces is described. For the formulation of the
Navier–Stokes equations in curvilinear coordinates we refer to Aris (1989). Indices that
are preceded by a comma denote covariant derivatives, and a repeated index that occurs
bothinco andcontravariantpositionissummedover. Themetrictensoroftheunderly
ingcoordinatesysteminspaceisdenotedbyg .ij
1.1.1. The Stokes equations
The equation that describes a general flow of an incompressible fluid is known as the
Navier–Stokesequation. Itisanonlinearequationinthevelocitycomponents,reading
i j i ij iρ∂v +ρv v =σ +f , (1.1)t ,j ,j
iwherev is the velocity field of the flow andρ the mass density of the fluid. The stress
ijthat a fluid element experiences from its surroundings is given by the stress tensorσ ,
iandf is an externally applied body force that drives the flow. The stress tensor may be
splitintotheisotropiccontributionofthelocal equilibriumpressure pandintoaviscous
part,containingthesymmetrisedderivativesofthevelocityfield,
σ =−pg +2ηe with (1.2)ij ij ij
1
e = (v +v ). (1.3)ij i,j j,i
2
The tensor e is called the rate of strain tensor, and η denotes the viscosity. The con-ij
dition for the fluid being incompressible is that the velocity is a solenoidal vector field,
i.e.withzerodivergence,
iv = 0. (1.4),i
Inmicrofluidics,thenonlineartermintheNavier–Stokesequationisoftenmuchsmaller
thantheviscoustermandcanthereforebeneglected. Thevalidityofthisapproximation
isbetterwhentheratioofinertialforcesandviscousforces,whichiscalledtheReynolds
number
ρv¯x¯
Re := , (1.5)
η
is smaller. Here, v¯ andx¯ denote the typical scales of the velocity and of the length on
whichthevelocitychanges. Botharesmallinmicrofluidics. Thelengthscaleofvelocity
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