Fluid-Structure Interactions in Low-Reynolds-Number Flows

496 Pages·2015·160.128 MB·English

by  Camille Duprat, Howard A Stone (eds.)

Fluid–StructureInteractionsinLow-Reynolds-NumberFlows
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Fluid–Structure Interactions
in Low-Reynolds-Number Flows
Editedby
CamilleDuprat
E´colepolytechnique,Palaiseau,France
Email:[e-mail]
and
HowardA.Stone
PrincetonUniversity,Princeton,NewJersey,USA
Email:[e-mail]
RSCSoftMatterNo.4
PrintISBN:978-1-84973-813-2
PDFeISBN:978-1-78262-849-1
ISSN:2048-7681
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(cid:13)C TheRoyalSocietyofChemistry2016
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Preface
CAMILLEDUPRATa ANDHOWARDA.STONEb
aE´colepolytechnique,91128PalaiseauCedex,France;bDepartment
ofMechanicalandAerospaceEngineering,PrincetonUniversity,
Princeton,NJ08544,USA
The topics in this book fall into the domain of fluid-structure interactions
where the fluid flow is characterized by motions at low Reynolds num-
bers.Ingeneral,theterm“fluid-structureinteraction”referstotheclassof
mechanics problems where a flow field affects the position, orientation,
and/orshapeofanobject,and,correspondingly,thestructuremodifiesthe
flow field. Therefore, the rigid-body dynamics and/or elasticity problems
associatedwithanobjectorwithsoftboundaries,andthedynamicsproblem
ofthemotionofthefluidarecoupled.Notsurprisingly,thesekindsofprob-
lemsintroducemanyvariablesandlinktwosetsofdynamicsquestionsthat,
individually,aretypicallyrathercomplicated.Onefeaturethatwewillseein
everychapterofthisbookisthatsimplifiedmodels,oftenonedimensional,
and the use of dimensional analysis can go very far in providing physical
insightandquantitativeestimatesformanyproblems.
Thesubjectoffluid-structureinteractionsoccursinmanyguises,suchas
thesloshingofliquidinatankrestingonanelasticfoundation,aerodynamic
flutter of wings, hydroplaning of tires, acoustical effects triggered by flow-
induced vibrations (think of a telephone line “singing” in the wind), and
manybiomechanicsproblemsinvolvingflowinandaroundsoftboundaries
(e.g.elasticvalvesandwavepatternsinfieldsofwheat).Thegeneraltopichas
been the subject of books,1,2 and even specific topics such as aeroelasticity
RSCSoftMatterNo.4
Fluid–StructureInteractionsinLow-Reynolds-NumberFlows
EditedbyCamilleDupratandHowardA.Stone
(cid:13)C TheRoyalSocietyofChemistry2016
PublishedbytheRoyalSocietyofChemistry,www.rsc.org
v
vi Preface
havetheirownextensivebooks.3,4 Thegreatmajorityofthesepresentations
focus on high-Reynolds-number motion and highlight the various models
andapproximationssuitableforinertia-dominatedfluidflow.
Althoughthesetopicshavebeenstudiedformanyyears,twoareaswhich
havereceivedmuchlessattentionconcernfluid–structureinteractionprob-
lems where fluid-fluid interfaces are important (e.g. surface tension is
capableofdeformingsoftsurfaces),andflowsituationswheretheReynolds
number is less than unity, so that viscous effects are expected to dominate
thefluidstresses.Thesetwotopicsformthemainthemesofthisbook.We
hopetohighlightthemanydifferentkindsofproblemsthatfallunderthis
umbrellaoffluid-structureinteractionsatlowReynoldsnumbers,including
materialssciencequestions(e.g.theshapeoffibersdeformedbythesurface
tensionexertedbyaliquiddrop,andthewrappingofliquidsheetsaround
liquiddroplets),rheologyquestionsinspiredbytheslowflowofasuspension
offlexiblefibers,andbiophysicsquestionssuchasthemotionofflagellaand
thecorrespondingswimmingofmicroorganisms.Indeed,basicquestionsin
embryology involve fluid that is stirred by the movement of flexible biofil-
aments. Moreover, cells are soft and so, not surprisingly, the motion and
deformationofcellsandvesiclesareimportantquestionsinphysiologythat
involvetheinterplayofelasticityandfluiddynamics.Wehavebeenfortunate
to get outstanding contributions from colleagues with expertise spanning
thesedifferentsubjects.
The chapters of the book have been written with a student or interested
professional in mind, and we are hopeful that some if not all of the chap-
terswillproveusefulasasourceofexamplesforuseincoursesorevenfor
sectionsofacourse.Inparticular,webeginwithtwointroductorychapters
highlightingimportantideasfromthetheoryofelasticityandlow-Reynolds-
numberhydrodynamics.Thefirstofthese,Chapter1,waswrittenbyBasile
Audoly,whointroducestheelasticamodelofafilament,whichplaysanim-
portantroleinmanyofthelaterchapters.WehavewritteninChapter2about
low-Reynolds-numberflows,highlightingseveralthemessuchaselementary
channel flows, mathematical properties, particle motion, and slender-body
theory,whichareallthemesthatrecurinlaterchapters.
One of the great models of elasticity is the elastica, as mentioned
above. The coupling of the elastica to a simplified form of viscous fluid
motion, called resistive force theory or slender-body theory, forms the
basisofChapter3,whichconsistsofasetofmodelproblems.Hereweguide
the reader through various steps, including scaling of equations, identify-
ingcharacteristicquantitiesofbeamdeformation,andsimilaritysolutions.
Wehopethatthischapterwillprovideareadysourceofclassroomexamples
orhomeworkproblemsthatcombinetheideaofbuildingphysicalintuition
with the important steps of solving some fun but nontrivial fluid-structure
interactionproblems.Thiscouplingoffluiddynamicswiththeelasticapro-
videsanintroductiontothenexttwochapters.First,inChapter4,OnShun
Pak and Eric Lauga provide an introduction to swimming microorganisms
in a comprehensive survey that includes many important theoretical ideas
Preface vii
fromlow-Reynolds-numberhydrodynamics.Then,inChapter5,AnkeLind-
ner and Mike Shelley describe fluid flows in the presence of flexible fibers
fromtheperspectiveofmaterialsscienceandrheology,includingtheresults
oftheoryandexperimentsonmodelflows.
A widely applicable topic that combines elasticity and fluid mechanics
is elastocapillarity, which brings in the topic of surface tension and is de-
scribed in Chapter 6. We wrote this chapter with the idea of including a
largeassortmentofgeometriesandconfigurations,sincethebreadthofthe
rangeofapplicationsisinspiring.Also,inChapter7,LudovicPauchardand
Fr´ed´eriqueGiorgiuttiprovideafascinatingdescriptionofthedryingof(soft)
suspensions and polymer solutions. Again, the important ideas of elastic-
ity and capillarity occur throughout, and the authors even link these ideas
to paintings, including the Mona Lisa! Although some of the problems in
thesetwochaptersinvolvestaticsonly,othersincludethedynamicsofavis-
cous fluid flow. We think the reader will be amazed by the wide range of
elastocapillaryproblemsandhowfarelementaryphysicalbalancescangoin
illuminatinginterestingphysicalproblems.
Biophysical ideas provide the motivation for three other chapters in the
book, which deal with topics including internal flows, where the motions
are typically within soft boundaries, and external flows, where the motions
arearoundsoftobjects,suchasvesiclesandcells.Theformertopicofflows
inflexiblechannelsispresentedinChapter8byMatthiasHeilandAndrew
Hazel, who make stimulating connections to applications of elasticity and
fluid dynamics in physiology. In Chapter 9, Petia Vlahovska describes the
theoryandmathematicalmodelingofthelow-Reynolds-numbermotionand
deformation of vesicles and polymer capsules in simple model flows and
in applied electric fields. Vesicles are liquid drops bound by a lipid mem-
brane and so have similarities to biological cells. Finally, in Chapter 10,
Manouk Abkarian and Annie Viallat discuss the motion and deformation
ofredbloodcellsinlow-Reynolds-numberflows.Theycharacterizetheme-
chanicalpropertiesofthecells,highlightingmechanicalfeaturesatdifferent
lengthscales,andsummarizeexperimentalandtheoreticalunderstanding.
TherearemanysynergiesbetweenthethemesdiscussedinChapters8–10.
Inordertotrytohelpthereader,wehaveencouragedalloftheauthorsto
adoptaconsistentschemeofnotation,andthedefinitionsaresummarized
in the following pages. Given the wide range of physical problems that ap-
pearandthelargenumberofphysicalvariables,however,thereareinevitable
conflictsinthechoiceofnotationinsomeofthechapters,andinthosecases
wehavetriedtoindicatetherelevantdefinitionsclearlyineachchapter.
We hope that you enjoy this tour of fluid-structure interactions in low-
Reynolds-numberflows.
We thank our many colleagues, at our home institutions and elsewhere,
fortheircollaborationandintellectualsupport.Weappreciatetheuniversal-
ityoftheideasthatwehavebeenabletopursue.
CamilleDupratandHowardA.Stone
viii Preface
References
1. M.Pa¨ıdoussis,S.PriceandE.deLangre,Fluid-StructureInteractions:Cross-
flow-inducedInstabilities,CambridgeUniversityPress,NewYork,2011.
2. M. Pa¨ıdoussis, Fluid-Structure Interactions: Slender Structures and Axial
Flow,AcademicPress,SanDiego,2ndedn,2014.
3. R.Bisplinghoff,H.AshleyandH.Halfman,Aeroelasticity,Dover,1996.
4. AModernCourseinAeroelasticity,ed.E.Dowell,KluwerAcademicPublish-
ers,4thedn,2004.
Nomenclature
Quantities
B Bendingmodulus(beam)
C Capillarynumber
a
D Deborahnumber
E Youngmodulus
E Freeenergy
(cid:51)(cid:51)(cid:51) Straintensor
f Forceperunitlength
F Force
G Shearmodulus
(cid:103) Surfacetension
(cid:103)˙ Shearrate
K Bulkmodulus
(cid:107) Meancurvature
(cid:96) (cid:90)((cid:103)/(cid:114)g)1/2 Capillarylength
c
(cid:109) Dynamicviscosity
n Unitnormalvector
(cid:110)(cid:90)(cid:109)/(cid:114) Kinematicviscosity
(cid:110) Poisson’sratio
p
(cid:117)(cid:117)(cid:117) Vorticity
p Pressure
rorx(cid:90)(x,y,z) Positionvector
(cid:114) Density
Re Reynoldsnumber
(cid:115)(cid:115)(cid:115) Stresstensor
RSCSoftMatterNo.4
Fluid–StructureInteractionsinLow-Reynolds-NumberFlows
EditedbyCamilleDupratandHowardA.Stone
(cid:13)C TheRoyalSocietyofChemistry2016
PublishedbytheRoyalSocietyofChemistry,www.rsc.org
ix
x Nomenclature
t Time
T Temperature
t Unittangentvector
Acronyms
2D two-dimensional
3D three-dimensional
BC boundarycondition(s)
Operators
∧ crossproduct
(cid:86) gradient
(cid:86)·f divergenceofavectorfieldf
(cid:86)∧f curlofavectorfieldf
(cid:86)2 Laplacian