Longwave Instabilities and Patterns in Fluids

461 Pages·2017·8.618 MB·English

by  Sergey Shklyaev,Alexander Nepomnyashchy (auth.)

Advances in Mathematical Fluid Mechanics
Sergey Shklyaev
Alexander Nepomnyashchy
Longwave
Instabilities
and Patterns in
Fluids

Advances in Mathematical Fluid Mechanics
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JohnG.Heywood,Vancouver,Canada
RolfRannacher,Heidelberg,Germany
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•
Sergey Shklyaev Alexander Nepomnyashchy
Longwave Instabilities
and Patterns in Fluids
SergeyShklyaev AlexanderNepomnyashchy
InstituteofContinuousMediaMechanics Technion
UBRAS DepartmentofMathematics
Perm,Russia Haifa,Israel
ISSN2297-0320 ISSN2297-0339 (electronic)
AdvancesinMathematicalFluidMechanics
ISBN978-1-4939-7588-4 ISBN978-1-4939-7590-7 (eBook)
https://doi.org/10.1007/978-1-4939-7590-7
LibraryofCongressControlNumber:2017957643
MathematicsSubjectClassification:76E06,76E17,76E30,35B36,35Q35
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Preface
The longwave instabilities are inherent to a variety of systems in fluid dynamics
(fromdynamicsofatmosphereandoceanto motionin microfluidicdevices),geo-
physics(mantledynamics,surges,avalanches),electrodynamics(electronicbeams,
plasma), biophysics (wetting and lubrication in biophysical applications, such as
teardynamicsandflowsintherespiratorysystem),andmanyothers.
The characteristic feature of all these systems is the existence of two signifi-
cantlydifferentlengthscales,a“shortscale”(e.g.,thicknessofathinliquidfilm,a
reactionzone,aphasetransitionfront)anda“longscale”ofpatternsgeneratedby
instabilities.Inmanycases,suchadifferencemakesitpossibletoapplyasymptotic
methods(e.g.,averagingoverthe“fast”coordinateorothermultiscaleapproaches),
whichallowtosignificantlysimplifytheproblemandreduceitsdimension.
Theabovementioneddifferenceinlengthscalesgivesrisetoseveralclearlydis-
tinguishedtimescales. Someappropriatemathematicaltools(e.g.,averagingtech-
nique or disregardingfast and/or slow processes) are efficient in many cases, and
theydrasticallysimplifytheproblemunderconsideration.Forinstance,withinthe
lubricationapproximationinfluidmechanics,onecansafelyneglectsuchprocesses
astransversaltransferofmomentumandotherphysicalquantities(heat,mass,etc.),
becausetheyimmediatelyadjusttotheinstantaneousdistributionofcorresponding
forcesandthermodynamicforces.Incontrast,therelaxationoflongitudinalfluxes
isaslowprocess,whichhastobeafocusofattention.Suchasimplificationmakes
the lubrication approximation a very powerful and frequently used approach (see
usefulsurveysonthatsubjectbyA.Oron,S.H.Davis,andS.G.Bankoff,Rev.Mod.
Phys.69,931(1997),andR.V.CrasterandO.K.Matar,Rev.Mod.Phys.81,1131
(2009)).It should be emphasized thatthe abovementionedlubricationapproxima-
tion,whichusuallyleadstothederivationofageneralizedCahn–Hilliardequation,
is a specific example of a more general approach, which results in a much wider
classofreducedequations.Thisapproachisthesubjectofthepresentbook.
Historically, shortwave instabilities characterized (near their thresholds) by a
well-definedcriticalwavelengthhavebeenextensivelyinvestigatedduringthelast
ix
x Preface
decades (see S. Fauve, Pattern forming instabilities, in Hydrodynamics and Non-
linear Instabilities, edited by C. Godre`cheand P. Manneville(CambridgeUniver-
sity Press, 1998); L.M. Pismen, Patterns and Interfaces in Dissipative Dynamics
(Springer,2006);R.Hoyle,PatternFormation,AnIntroductiontoMethods(Cam-
bridgeUniversityPress,2006);andM.CrossandH.Greenside,PatternFormation
andDynamicsinNonequilibriumSystems(CambridgeUniversityPress,2009)).The
analysis of the nonlineardevelopmentof that kind of instabilities is simplified by
theexistenceofagenericnonlinearequation,theso-calledGinzburg–Landauequa-
tion,whichisvalidnearthestabilitythresholdindependentlyofthephysicalnature
oftheproblemunderconsideration.Anextensivereviewonthatequationhasbeen
writtenbyI.S.AransonandL.Kramer,Rev.Mod.Phys.74,99(2002).
In contrast, the longwave instabilities are much more diverse. A number of
prominent equations (Fisher–Kolmogorovequation, Allen–Cahn equation, Cahn–
Hilliardequation,Kuramoto–Sivashinskyequation,andmanyothers)havebeenob-
tainedandinvestigatedindifferentphysicalcontexts.Moreover,duringthelasttwo
decades,thereappearedalotofnewpapersdevotedtononlinearlongwaveregimes.
In particular,an outstandingprogresshas been made in studying oscillatorylong-
wave patterns. Furthermore,the numberof applications, where the longwave per-
turbations are crucial, grows fast. Such modern fields as micro- and nanofluidics,
materialsscience,chemicalengineering,andbiophysicscanbementionedtoname
butafew.
On one hand, this diversity of applications leads to a situation when the same
resultsarerediscoveredindifferentcontexts.Therefore,webelievethatabookcon-
taining a description of main advances in that field from a general point of view
would be sought for. On the other hand, in our opinion, the scientific community
wouldneeda handbookwith receipts,methods,andapproachescollectedinorder
tousethisexperienceinnewinvestigations.Wehopethatboththesefactorswould
makethesuggestedbookpotentiallyuseful.
In our book, which is addressed to both students and experts, we describe the
techniquesofthederivationoflongwaveamplitudeequations,aswellasthe anal-
ysis of numerousnonlinearequationswhich governthe developmentof longwave
instabilities.Thebookisbasedonthefollowinggeneralprinciple:weconsiderphys-
icalproblemsthatleadtodefinitemathematicalproblems,ratherthanmathematical
problemsper se. Each equationis introducedin a contextof a particularphysical
problem, and it is derived by means of multiscale asymptotic methods. Thus, we
present a sequence of paradigmatic physical problems, rather than a sequence of
equations.Mostofthebasicexamplesbelongtothefieldoffluidmechanics,though
applications in other fields of science are also indicated. Longwave patterns cre-
atedbybuoyancyandsurface-tension-drivenconvectionaremostoftenconsidered
inorderto presenta physicalinterpretationofequations.Thereasonofdoingthat
istwofold.First,thefluiddynamicsandconvectionasitssubdomainareintuitively
clearformostreadersindependentlyoftheirbackground.Thepatternscanbeeasily
imagined and described. Moreover,most of the known nonlinearamplitude equa-
tions can be derived in applications to fluid dynamics. Second, the buoyancy and
Preface xi
surface-tension-driven convection is the authors’ field of expertise. However, we
believe that the focus on pattern formationin convectiondoes not make the book
too specific in that subject;the main ideas and results can be easily transferredto
otherfields.
Thebookconsistsofninechapters.
Chapter1presentsbasicphysicalexamplesandbasicclassificationofthetypesof
instabilitiesandcorrespondingnonlinearamplitudeequations,whicharethesubject
ofthebook.
The physical problem of convection, discussed in Chapters 2–4, gives the op-
portunitytointroducebasicamplitudeequations.Westartwiththeproblemsofthe
cylindricalgeometryinChapter2thatallowsasimplederivationofone-dimensional
equationsforarealamplitudefunction,whichisagoodstartingpointforthedemon-
stration of the main approachand presentation of transparentphysicalresults that
makeitpossibleto discusssuchbasic notionsasLyapunovfunctional,stabilityof
stationary patterns, wavenumberselection, defect dynamics, influence of inhomo-
geneities, pinning of defects, influence of distant boundaries on the patterns, and
higher-ordercorrectionsto the amplitude equation derived at the leading order of
asymptoticexpansions.InChapter3,werevisittheclassicalproblemofbuoyancy
convectioninahorizontallayerheatedfrombelow,whichisaparadigmaticexam-
pleofthecellularpatternformation,butfromadifferentpointofview:wediscuss
the case where the instability is a longwave one; hence, it is not characterized by
aselectedcriticalwavenumber.Also,thesurface-tension-drivenconvectioniscon-
sidered.
WhiletheRayleigh–Be´nardconvectioninapureliquidservesasaparadigmatic
example of stationary patterns generated by monotonic instability, convection in
binary liquids (Chapter 4) provides a basic example of an oscillatory instability
generatingwavepatterns.Therefore,withinthischapter,wemainlyfocusontime-
periodicpatterns;thestationaryconvectionisdiscussedonlywhenitisqualitatively
differentfromthatinapureliquid.Thedescriptionoflongwaveoscillatorypatterns
is strongly different from both longwave stationary patterns and shortwave oscil-
latory patterns. Partial differential amplitude equations, which are local in space,
cannot be derived in this case, and a different approach, based on the analysis of
resonances,isdeveloped.
InChapter5,weconsiderinstabilitiesofthesimplestkindofflows,theparallel
flows, which are characterized by a number of new effects; specifically, the prob-
lem is anisotropic.From a large set of problems,we have selected two problems,
where the progress in understanding the nonlinear development of instabilities is
significant,namely,theflowofaliquidfilmoveraninclinedplaneandthefamous
Kolmogorovflow,whichprovidesanunderstandingofthegenerationoflarge-scale
structuresbyinstabilitiesofshort-scalestructures.
Within Chapter6, wediscussinstabilitiesofpropagatingfronts.Themainsub-
jectofChapter7 is thedescriptionof longwavemodulationsofspatially periodic,
stationary,andoscillatory patterns. We considerintrinsic longwaveinstabilitiesof
short-scalepatterns,whicharecrucialforthepatternselectionanddevelopmentof
xii Preface
the spatiotemporalchaos. In Chapter 8, we considerthe ways to controlthe long-
waveinstabilitiesbyvariationofparameters.Bothopenloopandclosedloop(feed-
back) approaches are discussed. Chapter 9 contains a review of directions for a
furtherdevelopmentinthefield.Sometechnicalstuffisputintotheappendices.
Inouropinion,thisbookwillbeusefulforexpertsinfluidmechanics,heatand
mass transfer theory, nonlinear dynamics, and applied mathematics as well as for
physicists and chemical engineers interested in the investigation of the nonlinear
evolutionandpatternformationcausedbylongwaveinstabilities.Thebookcanbe
usedalsobygraduatestudents.
Thescientificresultsincludedinthisbookhavebeenobtainedduetocollabora-
tion and fruitful discussions with our colleagues and friends: S.I. Abarzhi, A.A.
Alabuzhev, D.E. Bar, A. Bayliss, S.H. Davis, I.S. Fayzrakhmanova, E.A. Glas-
man, A.A. Golovin, V. Gubareva, Y. Kanevsky, M. Khenner, B.J. Matkowsky,
A. Mikishev, M. Morozov, B.I. Myznikova, A. Oron, A. Pikovsky, L.M. Pismen,
A. Podolny, B.I. Rubinstein, A.E. Samoilova, T. Savina, I.B. Simanovskii, B.L.
Smorodin,A.V. Straube,M.G. Velarde,V.A. Volpert, andM.A. Zaks. The helpof
I.S.Fayzrakhmanovainthemanuscriptpreparationishighlyappreciated.Thebook
wouldnotbewrittenwithouttheinspiringsuggestionofU.Frisch.