ONTHEASYMPTOTICBEHAVIOROFTHERMALTAYLOR DISPERSIONPROCESSESINPERIODICMEDIA ClaudiaTimofte DepartmentofMathematics,FacultyofPhysics,University… [610108]

JOURNALOFTHEORETICAL
ANDAPPLIEDMECHANICS
41,1,pp.75-88,Warsaw2003
ONTHEASYMPTOTICBEHAVIOROFTHERMALTAYLOR
DISPERSIONPROCESSESINPERIODICMEDIA
ClaudiaTimofte
DepartmentofMathematics,FacultyofPhysics,University ofBucharest,Romania
e-mail:[anonimizat]
Usingthegeneralizedmethodofmomentsandacentrallimitt heorem,we
shalldescribealargeclassofthermaldispersionphenomen aoccurringin
somemacrohomogeneoussystems.Weshallbeinterestedinco mputing
themacroscalecoefficientsintermsofthemicroscalecoeffici entsand
thesystemgeometry.Also,thefunctionaldependenceofthe effective
coefficientsonthevelocityandthespatialscaleparameters isanalyzed.
Key words:macrotransport equation, macroscale coefficients, scale
parameters
1. Introduction
Thefieldofmacrotransportprocessesconstitutesanatural extensionof
classicalTaylordispersiontheoryforunidirectional,re ctilinearflows(seeTay-
lor,1953)toalargeclassofflowanddispersionproblems.It iswell-knownby
nowthatG.I.Taylorusedratherintuitivesemi-analytical argumentstoprove
theFickiannatureofthemeanaxialdispersionofadiffusing soluteinjected
intoaviscousfluidflowingthroughacircularcylindricaltu be.Heshowedthat,
asymptotically,inthelong-timelimit,suchadispersionp rocessisdescribedby
aone-dimensionalconvective-diffusiveequationandthedi spersioncoefficient
characterizingthisaxialmacrotransportequationgovern ingthecross-sectional
meansoluteconcentrationis
D∗=D+a2U2
48D(1.1)

76 C.Timofte
whereaistheradiusofthetube, Uisthemeanvelocityofflowand Dis
thecoefficientofmoleculardiffusion.Thisnewcoefficientcom binesthemi-
croscopicallydistincteffectsofradialmoleculardiffusio nandaxialconvective
soluteflow.Dispersioniscausedbytheradialinhomogeneit yofthePoiseuil-
levelocityfieldwhichinteractswiththelateraldiffusiono fsolutemolecules.
Infact,allmacrotransportprocessescombinesuchaBrowni an(stochastic)
diffusivetransportmechanismwithaninhomogeneous,conve ctive(determini-
stic)transportmechanism.Thestochasticdispersionisas sumedtoactover
aperiodoftimelargeenoughtoallowthesamplingofallsuch velocitiesby
moleculardiffusionacrossthestreamlinesoftheflow.
Taylor‘stechnique,laterextendedtothecaseofsolutedis persioninturbu-
lentflows,providedthefirstframeworkforthegenericpheno menonreferredto
intheliteratureas”Taylordispersion”.In1956,Aris(see Aris,1956)extends
theseresultsforcylindersofnoncircularcross-sectiona nddevelopsarigorous
theory,basedona methodofmoments scheme.Also,heanalyzestheeffects
oftime-periodicconvectionondispersion(seeAris,1960) .In1971Horn(see
Horn,1971)madeanotherimportantsteptoputthefoundatio nsoftheso-
calledgeneralizedTaylordispersiontheory .Heextendedtheclassicaltheory
tomultidimensionalphasespaces.Adecisivestepwasdonet henbyBrenner
(seeBrenner,1980)whodevelopedaparadigmaticdispersio ntheoryforvery
generalandcomplexsystems.In1993,recognizingtheanalo gyexistingbe-
tweenthemasstransportandothermodesoftransportphenom ena,Brenner
andEdwardsextendedthistheorytonon-materialtransport processes(see
BrennerandEdwards,1993).
Hence,baseduponarigorousdescriptionofmicrotransport processesoc-
curringinheterogeneoussystems,macrotransporttheory, alternativelyknown
intheliteratureasthegeneralizedTaylordispersiontheo ry,allowsustodescri-
bealargeclassofmaterialandnonmaterialdispersivephen omenaoccurring
inmacrohomogeneoussystems.
Applicationsofthemacrotransporttheoryarepresentlyre cognizedinma-
nyfieldsofscientificandengineeringresearch.Variousoth ermethodshave
beendevelopedforobtainingthemacroscalebehaviorandpr opertiesofso-
meheterogeneouscomplexsystems.Theseincludehomogeniz ationtechniques
(seeBensoussanetal.,1978,Sanchez-Palencia,1980),statisticalandvolume-
averagingmethods(seeKochandBrady,1985)andprobabilis ticmethods
basedoncentrallimittheorems(seeBhattacharya etal.,1989).
Inthispaper,weshallbeespeciallyinterestedingettinga macrotran-
sportparadigmforaclassofthermaltransportphenomenaoc curringinsome
complexmultidimensionaladiabaticsystems.Weshalldedu cesimplemacro-

Ontheasymptoticbehaviorofthermal… 77
transportequations(andeffectivecoefficientsappearingth erein)whichapply,
forlongtimes,atacoarse-grainedlevelofdescriptionofo ursystems.
Ouranalysisisbasedontwoalternativemethods:theabovem entioned
generalizedmethodofmomentsandaprobabilisticmethodba sedonacentral
limittheoremforMarkovprocesses.
Wearealsointerestedingettingtheasymptoticbehaviorof themacrodi-
spersioncoefficientsasfunctionsofthevelocityandthespa tialscaleparame-
terswhichcharacterizeourtransportprocesses.
SpecificexamplesaregiveninSection3toillustratethecom putationof
thesemacrotransportcoefficientsasfunctionsoftheprescr ibedmicroscale
data.
2. Amacrotransportparadigmforthermaldispersionphenom ena
inadiabaticsystems
Letusintroducetwodistinctlydifferentclassesofindepen dentcoordinate
variableswhichcharacterizeagenericmicrotransportpro cess.Thesewillbe
designatedasglobalandlocalvariablesanddenotedby Qandq
Q=[Q1,Q2,…,Qr]q=[q1,q2,…,qs] (2.1)
Together,thevectors( Q,q)defineamultidimensionalphasespace Q∞⊕
q0withinwhichourtransportprocessesoccur.Theglobalsubs paceQ∞,
representingthedomainofthevaluestakenby Q,willbeunbounded,while
thesubspaceq0(q∈q0)will,generally,bebounded.Theglobalcoordinate Q
properlycorrespondstoalong-timescale,whilethelocalv ariablecorresponds
toashort-timescale.Theyarealsocalled slowand,respectively,fastvariables
(seeSanchez-Palencia,1980).
Thegenericmicrotransportequationgoverningtheevoluti onofthetempe-
raturefieldT=T(Q,q,t)incontinuousadiabaticsystemsmayberepresented
as
ρCp∂T
∂t+∇Q·J+∇q·j=0 (2.2)
wheretheconstitutiveequationsfortheglobalandtheloca linternalenergy
flux-densityvectorsare
J=ρ(q)Cp(q)U(q)T−KT(q)·∇Q(T)
(2.3)
j=ρ(q)Cp(q)u(q)T−kT(q)·∇q(T)

78 C.Timofte
Here, (KT,kT)denotetheglobalandthelocal-spacethermalconductivit ies
and (U,u)thecomparablevelocityvectors.Here, ρandCparepositive
functionsandKTandkTarepositivedefinitetensors.
Thissystemofequationsissubjectedtothefollowingcondi tions
n·j=0n·KT·∇qT=0 on∂q0
|Q|m{T,J}→{0,0}as|Q|→∞∀(q,t/x}∇eate∇equal0)
m=0,1,…(2.4)
T(Q,q,0)=T0(Q,q)
withtheright-handsideaprescribedfunction.
Notethattheenergydissipationandkineticenergycontrib utionarene-
glectedinthemicrotransportequation.
Weshalllimitourselvestotheclassofproblemsforwhich n·u=0on
∂q0andu·∇qCp=0.Also,itwillbesupposedthatthethermalproperties
areeverywherenonnegativedefinite.
It proves useful to reformulate linear microscale problem ( 2.2) in
terms of a Green’s function. In this context, let us define a qu antity
P=P(Q,q,t|Q′,q′,t′)suchthatρCpPtobeinterpretedastheconditio-
nalprobabilitydensityofhavingthetemperature T(Q,q,t)attheposition
(Q,q)atthemomenttifwehadthetemperature T(Q′,q′,t′)at(Q′,q′)at
theearliermoment t′.Pwillgenerallydependonlyonthedifferences Q−Q′
andt−t′andso,choosingQ′=0andt′=0,wecanconsider,withoutloss
ofgenerality,thatP=P(Q,q,t|q′).
Sincewearemodelingthetransportofconservedentities,w ehave
/integraldisplay
Q∞/integraldisplay
q0ρCpPdqdQ=1t/x}∇eate∇equal0
and
P=0t<0
Also,therelationshipbetweenthetemperaturefield TandtheGreenfunction
Pis
T(Q,q,t)=/integraldisplay
Q′∞/integraldisplay
q′
0ρ(q′)Cp(q′)P(Q,q,t|q′)T(Q′,q′,0)dq′dQ′(2.5)
andthemicrotransportequationofenergydispersionincon tinuoussystems
mayberepresentedas
ρCp∂P
∂t+∇Q·JP+∇q·jP=δ(Q)δ(q−q′)δ(t)

Ontheasymptoticbehaviorofthermal… 79
where
JP=ρ(q)Cp(q)U(q)P−KT(q)·∇Q(P)
jP=ρ(q)Cp(q)u(q)P−kT(q)·∇q(P)
DefiningthemacroscaleGreen’sfunction
P(Q,t|q′)=1
ρCp∗/integraldisplay
q0ρ(q)Cp(q)P(Q,q,t|q′)dq(2.6)
thiswillbecomeasymptoticallyindependentof q′
P(Q,t|q′)∼=P(Q,t)
and,hence,followingamomentanalysis(seeBrennerandEdw ards,1993;
Timofte,1996),weareledtothefollowingmacrotransporte quation
ρCp∗/parenleftBig∂P
∂t+U∗·∇QP/parenrightBig
=kT∗:∇Q∇QP+δ(Q)δ(t) (2.7)
subjectedtotheconditions
P=0t<0
P→0 when|Q|→∞(2.8)
Here,themacroscalecoefficients ρCp∗andU∗aregivenby
ρCp∗=1
τ0/integraldisplay
q0ρCpdq (2.9)
where
τ0=/integraldisplay
q0dq (2.10)
andby
U∗=/integraldisplay
q0ρCpP∞
0Udq (2.11)
Theeffectivethermalconductivitydyadichastheexpressio n:
kT∗=kM+kC(2.12)

80 C.Timofte
where
kM=1
τ0sim/integraldisplay
q0KTdq (2.13)
isthe”molecular”contributionand
kC=sim/integraldisplay
q0/bracketleftBig/parenleftBig
P∞
0−1
τ0ρCp∗/parenrightBig
KT+ρCpP∞
0B(U−U∗)/bracketrightBig
dq(2.14)
istheconvectivecontribution.
Thesephenomenologicalcoefficientsaretobeobtainedafter solvingthe
associatedlocalproblemsfor P∞
0(q)andB(q)
∇q·j∞
0=0
j∞
0=ρCpuP∞
0−kT·∇qP∞
0(2.15)
n·KT·∇qP∞
0=0 on∂q0
/integraldisplay
q0ρCpP∞
0dq=1
and
j∞
0·∇qB−∇q·(P∞
0kT·∇qB)=ρCpP∞
0(U−U∗)
(2.16)
ρCpP∞
0n·kT·∇qB=0 on∂q0
More,weshallrequirethat P∞
0isnonnegativeforall q∈q0.Also,P∞
0
andBmustbesingle-valuedforall q∈q0.
So,wecanexpressthemacrotransportcoefficients ρCp∗,U∗andkT∗
intermsoftheprescribedmicroscaledataandthesystemgeo metry.Itis
worthwhiletonoticethat,infact, ρCp,whichisinhomogeneousin q,actslike
abiasingpotential.Thiscausesaredistributionoftheint ernalenergyinthe
localspace,whichisfinallyreflectedinthemagnitudeofthe macrotransport
coefficientsU∗andkT∗.
Thecoarse-grainedmacroscaletemperaturefield
T(Q,t)=1
τ0/integraldisplay
Q′∞/integraldisplay
q′
0ρ(q′)Cp(q′)P(Q−Q′,t|q′)T(Q′,q′,0)dq′dQ′(2.17)

Ontheasymptoticbehaviorofthermal… 81
willsatisfythemacrotransportequation
ρCp∗/parenleftBig∂T
∂t+U∗·∇QT/parenrightBig
=kT∗:∇Q∇QT(2.18)
subjectedtoappropriateinitialandboundaryconditions( seeBrennerand
Edwards,1993;Timofte,1996).
Asimilaranalysiscanbedonefortheproblemofthermaldisp ersionin
discontinuousadiabaticsystems.
Thegeometricalstructureofthediscontinuousmediumisid ealizedasbeing
spatiallyperiodic(porousmedia,compositematerials,la minatedmedia).The
periodicmediumisrepresentedasaspatiallyperiodicarra yinR3,composed
oftopologicallyindistinguishableunitcellsofperiodic ity,havingthesame
shape,orientation,volumeand”content”.
Ifwedenotebyτ0thevolumeofsuchanelementarycellandby τpthe
volumeofthesolidpart,wehave
τp=τ0−τf
whereτfistheinterstitialfluidvolumewithinsuchacell.
Arbitrarilydesignatingoneoftheelementarycellsasbein gthezerothcell,
itisconvenienttomeasurethepositionvector Rofanypointinspacerelative
tothecentroidofthiscell.Denotingby Rnthepositionvectorofthecentroid
ofthen-thcellrelativetothecentroidofthezerothcellandby r∈τ0{n}
thelocalpositionvectorforanypointwithinthe n-thcellrelativetoanorigin
atitscenter,wehave
R=Rn+r (2.19)
Ifwesupposethatattheparticle-fluidinterface Spwehavealocalequ-
ilibriumdescribedbyalinearpartitioningrelationship, usingsimilarnota-
tionsasinthecontinuouscaseandintroducingmicroscaleG reen’sfunction
P=P(Rn,r,t|r′),thiswillobeythefollowingsystemofcellular-levelequ –
ations(seeBrennerandEdwards,1993;Timofte,1996)
ρ(r)Cp(r)∂P
∂t+∇·J=δnn′δ(r−r′)δ(t)
J=ρ(r)Cp(r)U(r)P−KT(r)·∇P
(2.20)
ν·∆SpJ=0 onSp
|Rn−Rn′|mP→0
|Rn−Rn′|mJ→0/bracerightBigg
as/braceleftBigg
{n−n′}→∞
m=0,1,…

82 C.Timofte
In(2.20)3,foranarbitrarytensorfield f,∆Spdefinesthe”jump”of f,
acrossthediscontinuousphasesurface Spandνisaunitvectorwhichis
normaltothissurface.
Thethermophysicalproperties ρ,Cp,KTandthelocalfluidvelocityare
regardedasbeingspatiallyperiodic.
Followingamoment-matchingschemeandconsideringmacros caleGreen’s
functionP
P(Rn,t|r′)=1
ρCp∗/integraldisplay
τ0ρ(r)Cp(r)P(Rn,r,t|r′)d3r∼=P(R,t) (2.21)
wegetthefollowing macrotransportequation
ρCp∗/parenleftBig∂P
∂t+U∗·∇P/parenrightBig
=kT∗:∇∇P+δ(R)δ(t) (2.22)
with
P→0 as|R|→∞ (2.23)
Here,Risthemacroscale(Darcy)positionvectorofalatticepoint relative
toanoriginOarbitrarilychosenintheunitperiodiccell.
Themacroscalecoefficients ρCp∗,U∗andKT∗aregivenbythefollowing
formulas
ρCp∗=1
τ0/integraldisplay
τ0ρ(r)Cp(r)d3r
U∗=/integraldisplay
τ0J∞
0(r)d3r (2.24)
α∗=KT∗
ρCp∗=/integraldisplay
τ0P∞
0(r)(∇B)⊤(r)·simKT(r)·∇B(r)d3r
ThefieldsP∞
0(r)andB(r)satisfy,forr∈τ0,thefollowingboundary-
valueproblems
∇·J∞
0=0
J∞
0=ρCpUP∞
0−KT·∇P∞
0
ν·∆SpJ∞
0=0∆SpP∞
0=0 onSp(2.25)
/ba∇dblP∞
0/ba∇dbl=0/ba∇dbl∇P∞
0/ba∇dbl=0 on∂τ0
/integraldisplay
τ0ρ(r)Cp(r)P∞
0(r)d3r=1

Ontheasymptoticbehaviorofthermal… 83
and
∇·(P∞
0KT·∇B)−J∞
0·∇B=ρCpP∞
0U∗
Biscontinuousacross Sp
(2.26)
ν·∆Sp(KT·∇B)=0 onSp
/ba∇dblB/ba∇dbl=−/ba∇dblr/ba∇dbl /ba∇dbl∇B/ba∇dbl=0 on∂τ0
Here,foranytensor-valuedfield F,/ba∇dblF/ba∇dbldefinesthe”jump”inthevalueof F
betweentheequivalentpointslyingonoppositepairsofthe cellfaces.
Inthismanner,wecanobtainamacrotransportparadigmfora classof
thermaldispersionphenomenaoccurringinperiodicmedia.
Moreover,forthiscase,introducingtwopositivescalars U0anda,wecan
expressthefluidvelocity Uintheform
U(r)=U0V(r/a) (2.27)
U0andawillbeinterpretedasbeingthevelocityandthespatialsca lepara-
meterswhichcharacterizeourtransportprocesses.
Weareinterestedingettingthefunctionaldependenceofth easymptotic
dispersioncoefficients α∗intermsofthesetwoparameters.
UsingacentrallimittheoremforMarkovprocesses,itcanbe proved(see
Timofte,1999;Bhattacharya etal.,1989)thatforaspecialcaseofthermal
dispersionphenomenainperiodicmedia,themacroscalecoe fficientsα∗
ijde-
pendonlyontheproduct aU0,theresultbeinginaccordancewithallthe
experimentalstudiesthathavebeendone.Infact
α∗(a,U0)=α∗(U0,a)=α∗(aU0,1) (2.28)
Thisinterchangeabilityofthevelocityandspatialscalep arametersinthe
large-scaledispersionmatrixenablesustoconsider,ifne eded,thatthespatial
scaleparameteraisheldfixedata=1,whilethevelocityparameter U0is
allowedtovary.
Amorepreciseanalysisoftheasymptoticbehaviorofthedis persioncoeffi-
cientsα∗
ijcanbedoneifthethermophysicalproperties ρ,Cp,KTaresupposed
tobeconstantandifwemakemorerestrictiveassumptionsab outthevelocity
field(seeTimofte,1999).

84 C.Timofte
3. Applications
Asafirstexample,weshallconsidertheproblemofinternale nergydi-
spersioninanincompressibleviscousfluidmovingunderlam inarflowcondi-
tionsbetweentwoparallel,insulatedporousplatessepara tedbyadistanceh.
Theupperplatemovesatavelocity U0paralleltoitinthe x-direction.Si-
multaneously,thereexistsauniformflowacrossthechannel (inthenegative
y-direction)ataconstantvelocity v0.Inthiscase,thefluidvelocityfield U
isgivenby
U=U0y
hi−v0j (3.1)
Att=0,anamountofheatisinstantaneouslyaddedintooursyste mover
someregionoftheinfinitedomainbetweentheplatesinthefo rmofsome
initialtemperaturedistribution T0(x).
Assumingthatthethermophysicalproperties ρ,CpandKTareconstant,
theevolutionofthetemperature T(t,x)willbegovernedbythefollowing
equation
∂T
∂t+U0y
h∂T
∂x−v0∂T
∂y=α∆T (3.2)
withtheinitialcondition T(0,x)=T0(x)andwithα=KT/(ρCp).
Introducingthedimensionlessparameter
β=v0h
α(3.3)
andconsideringtheincompletegammafunction
γ(n+1,β)=β/integraldisplay
0ξnexp(−ξ)dξ n=0,1,2,…(3.4)
themacroscalethermalvelocity Uisgivenby
U∗=U∗i (3.5)
where
U∗=U0γ(2,β)
βγ(1,β)(3.6)
Ifweconsiderthemeanaxialfluidvelocity
V=U0
2(3.7)

Ontheasymptoticbehaviorofthermal… 85
weget
U∗
V=2γ(2,β)
βγ(1,β)(3.8)
So,thethermalvelocity U∗isdifferentfromthemeanaxialfluidvelocity V.
Asthecrossflowvelocity v0→0,correspondingto β→0,itiseasyto
seethat
lim
β→0U∗
V=1
Ifv0→∞,thenβ→∞and
lim
β→∞U∗
V=0
Usingthegeneralformulasgivenbytheabovemethodofmomen ts,wesee
thattheonlycomponentoftheeffectivethermaldispersivit ydyadicα∗which
isdifferentfromzerois
α∗
11=α+k(β)h2V2
α(3.9)
with
k(β)=4
β4/bracketleftBigγ(2,β)
γ(1,β)/bracketrightBig2/bracketleftBig
2γ(2,β)
γ(1,β)+3γ(3,β)
γ(2,β)/bracketrightBig
(3.10)
Wenoticethatifv0=0wegetaformulawhichissimilartoclassicalformula
(1.1)forthecaseofTaylor’ssolutedispersion.
Asasecondexample,letusconsidertheproblemofinternale nergydisper-
sioninalayeredperiodicporousmedium,saturatedwithavi scousincompres-
siblefluid.Weshallchooseasaperiodiccelltheparallelep ipedτ0havingthe
sideslx,lyandlz.Letussupposethatthethermophysicalproperties ρ,Cp,
KTareconstantandthevelocityfield Uisperiodic,withtheperiod lz
U=/bracketleftBig
U0/parenleftBig
1+sin2πz
lz/parenrightBig
,U0sin2πz
lz,U0ω/bracketrightBig
(3.11)
Here,U0andωaregivenrealparameters(seeTimofte,1996).
Initially,themediumhasanuniformtemperature T0(wecanchoose
T0=0).Att=0,anamountofheat Qisinstantaneouslyintroducedin-
tothesystemastheinitialdistributionoftemperature T(0,x)=T0(x).
Withα=KT/(ρCp),theevolutionofthetemperature T(t,x)willbe
governedbythefollowingequation
∂T
∂t=α∆T−U0/parenleftBig
1+sin2πz
lz/parenrightBig∂T
∂x−U0sin2πz
lz∂T
∂y−U0ω∂T
∂z(3.12)

86 C.Timofte
subjectedtotheinitialcondition T(0,x)=T0(x).
Obviously
U∗=(U0,0,U0ω) (3.13)
Followingthegeneralschemeofferedbytheabovemethodofmo ments,we
cancomputethemacroscalecoefficients α∗
ij
α∗
11=α∗
22=α+αl2
zU2
0
2[(2πα)2+(U0lzω)2]
α∗
33=α
(3.14)
α∗
12=α∗
21=αl2
zU2
0
2[(2πα)2+(U0lzω)2]
α∗
13=α∗
31=α∗
23=α∗
32=0
Itissimpletoseethatforsmallvaluesof lzU0,α∗
ijdependquadraticallyon
lzU0.However,aslzU0→∞,eachα∗
ijbecomesasymptoticallyconstant.
Asthefinalexample,weshallconsidertheproblemofinterna lenergy
dispersioninaperiodicporousmedium,saturatedwithanin compressible
viscousfluidhavingthevelocityfield U(x)=U0V(x)givenby
V=[0,2+sin2πx,2+cos2πxcos2πy] (3.15)
Weassumethatthespatialscaleparameter aisfixedata=1andthe
phenomenologicalcoefficients ρ,,CpandKTarestrictlypositiveconstants.
Obviously
U∗=[0,2U0,2U0] (3.16)
Itiseasytoseethatinthiscase
α∗
11=α
(3.17)
α∗
12=α∗
21=α∗
13=α∗
31=α∗
23=α∗
32=0
Forthisexample,closed-formsolutionsofthemacrotransp ortcoefficientsα∗
22
andα∗
33cannotbeobtained.However,theanalyticaltheorydevelop edby
Timofte(1999)andBhattacharya etal.(1989)showsthat,as U0→ ∞,
α∗
22=α+O(U2
0)andα∗
33=α+O(1).
Thisexamplereflectstheinfluenceofthegeometryoftheflowc urveson
theasymptoticbehaviorofthemacrotransportcoefficients.

Ontheasymptoticbehaviorofthermal… 87
References
1.ArisR.,1956,Onthedispersionofasoluteinafluidflowingthrougha tube,
Proc.Roy.Soc.,Ser.A,235,67-78
2.ArisR.,1960,Onthedispersionofasoluteinpulsatingflowthrough atube,
Proc.Roy.Soc.,Ser.A,259,370-376
3.BensoussanA.,LionsJ.L.,PapanicolaouG. ,1978,AsymptoticAnalysis
forPeriodicStructures ,Amsterdam,North-Holland
4.BhattacharyaR.N.,GuptaV.K.,WalkerH. ,1989,Asymptoticsofsolute
dispersioninperiodicporousmedia, SIAMJ.Appl.Math. ,49,86-98
5.BrennerH.,1980,Dispersionresultingfromflowthroughspatiallyper iodic
porousmedia,Phil.Trans.Roy.Soc.London ,A,297,81-133
6.BrennerH.,EdwardsD.A. ,1993,MacrotransportProcesses ,Butterworth-
Heinemann,Boston
7.HornF.J.M.,1971,Calculationofdispersioncoefficientsbymeansofmo-
ments,AIChEJ.,613-620
8.KochD.L.,BradyJ.F. ,1985,Dispersioninfixedbeds, J.FluidMech.,154,
399-427
9.Sanchez-PalenciaE. ,1980,Non-HomogeneousMediaandVibrationTheory ,
Lect.NotesPhys.,127,Springer-Verlag,Berlin
10.TaylorG.I.,1956,Dispersionofsolublematterinasolventflowingslow ly
throughatube,Proc.Roy.Soc.,Ser.A,219,186-203
11.TimofteC.,1996,StochasticProcessesandtheirApplicationstoFlui dMe-
chanics,Ph.D.Thesis,InstituteofMathematicsoftheRoma nianAcademy,
Bucharest(inRomanian)
12.Timofte C.,1999,Asymptotics ofthermaldispersionin periodic media ,
JournalofTheoreticalandAppliedMechanics ,37,1,95-108
OasymptotycznejnaturzeprocesutermicznejdyspersjiTay lora
wośrodkachperiodycznych
Streszczenie
Wpracydokonanoprzegląduszerokiejklasyzjawiskzwiązan ychztermicznądys-
persjąwwybranychukładachjednorodnychwskalimakro.Wop isiewykorzystano
uogólnionąmetodęmomentówitwierdzenieogranicycentral nej.Jakoszczególnie

88 C.Timofte
interesującyprzedstawionoproblemobliczaniawspółczyn nikówmakroskaliwfunkcji
współczynnikówmikroskaliigeometriibadanegoukładu.Po nadtoprzeanalizowano
funkcjonalnązależnośćwspółczynnikówefektywnychodpol aprędkościiprzestrzen-
nychparametrówskali.
ManuscriptreceivedOctober16,2001;acceptedforprintOc tober22,2002