upscaling heterogeneous media by asymptotic expansions

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Upscaling Heterogeneous Media by Asymptotic ExpansionsJ.-L. Auriault1

Abstract: Upscaling methods aim at representing the evolution of a given physical process in a given heterogeneous mediuequivalent macroscopic continuous behavior. In this paper, we recall the main features of the method of homogenization by masymptotic expansions. To illustrate the method, a few illustrative examples are revisited concerning heat transfer in composite~memory effects due to highly different conductivities of the components and the effect of contact thermal resistance in betwcomponents! and fluid flow through rigid porous media~transient flow and flow in noninertial porous media!.

DOI: 10.1061/~ASCE!0733-9399~2002!128:8~817!

CE Database keywords: Homogeneity; Scale effect; Porous materials; Composite materials; Seepage; Heat transfer; Acousti

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Introduction

Heterogeneous media with a large number of heterogeneitiesnot be described by considering each of the heterogeneities,would yield to intractable boundary value problems. The wknown method is to replace,if possible, the heterogeneous medby an homogeneous one, whose description is valid at a very lscale ~the macroscopic scale! with respect to the heterogeneitscale. As a continuous equivalent description, the derived mascopic behavior should be intrinsic to the medium and toexcitation, and should be independent of the macroscopic boary conditions. The macroscopic behavior is derived fromdescription at the heterogeneity scale that describes the phyprocess over a representative elementary volume~REV!. The ex-istence of a such a volume is required for any continuous mascopic representation of the physical system, and, as a coquence, is required for applying any upscaling technique.definition, the REV is1. sufficiently large for representing the heterogeneity sc

and2. small compared to the macroscopic volume.As a consequence, a condition of separation of scales is requThis fundamental condition can be expressed as

l

L5«!1 (1)

wherel andL are the characteristic lengths at the REV scale aat the macroscopic scale, respectively.

This definition intuitively conjures up a geometrical separatof scales, whereas this fundamental condition must also befied regarding the excitation~i.e., the physical process!. For in-stance, consider the propagation of a wave in an heterogen

1Professor, UJF, Laboratoire ‘‘Sols, Solides, Structures,’’ UJF, INPCNRS UMR 5521, BP 53X, 38041, Grenoble cedex 9, France. [email protected]

Note. Associate Editor: Franz-Josef Ulm. Discussion open until Jaary 1, 2003. Separate discussions must be submitted for individuapers. To extend the closing date by one month, a written request mufiled with the ASCE Managing Editor. The manuscript for this paper wsubmitted for review and possible publication on March 25, 2002;proved on April 1, 2002. This paper is part of theJournal of EngineeringMechanics, Vol. 128, No. 8, August 1, 2002. ©ASCE, ISSN 0733-9392002/8-817–822/$8.001$.50 per page.

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medium. The wavelength actually constitutes a third characterlength. Intuitively, we see that a continuous approach for descing this physical process will be possible only if the heterogenescale is small compared to the wavelength; a wavelength oforder of the heterogeneity length scale would lead to watrapping effects, which could not be completely described byequivalent continuous behavior at the macroscopic scale. Forflow in porous media as the medium is excited by a pressgradient, the characteristic length of the excitation is thus relato the pressure gradient.

Therefore, the fundamental condition of separation of scaleexpressed asl/L!1, whereL is the macroscopic characteristlength and is either geometrical or related to the excitation:existence of the REV and, as a consequence, the conditioseparation of scales, are not only constrained to geometricalsiderations but also related to the excitation~i.e., the physicalprocess!. The analysis should focus on what we call the physisystem, that consists of both the medium and the excitation.

There are two ways of deriving this macroscopic descriptiThe first one is a directly macroscopic approach, which is ofassociated with experiments and is called the phenomenologapproach. Many physical laws have been first derived by this kof approaches. It is the case of Darcy’s law in 1856. The seckind of continuous approach allows to derive the macroscobehavior from the local description. This is an upscaling tenique. Upscaling techniques are continuous approaches that athe derivation of an equivalent macroscopic continuous desction from the description at the REV scale. The equivalentscription is called the homogenized description. To apply thtechniques, the knowledge of the physical parameters and ogeometry is required over the REV only. Different techniquesavailable which address random as well as periodic heteroneous media. Among all, the most popular are the statistic meling, the self-consistent method, the volume averaging methand the homogenization for periodic structures.

The statistic modeling~Kroner 1986! has been designed tinvestigate elastic composites under static conditions and tosome information on the effective coefficients. The hypothesisseparation of scales is implicit. As it is assumed that the medis infinite, it gives«50, which means that a perfect separationscales is supposed. The method also imposes some prerequisthe structure of the macroscopic description. The ergodicitypothesis is imposed, which means that the ensemble avera

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supposed to be equal to the volume average. The effective cficients are defined by the local correlation functions. In theorydefine the effective coefficients, we may determine the correlafunctions at all orders. In practice, it is already difficult at tthird order. Authors like Matheron and Gehlar have applied silar ideas for fluid flow in porous media@see, for example,Matheron~1967! and Gehlar~1987!#.

Self-consistentmodels assume a state of perfect disorder~see,for example, Zaoui 1987!. It is the ideal case where there is nspace correlation. The hypothesis of perfect separation of scaimplicit ~«50!. The original idea of this model is to conceive thany heterogeneity sees the rest of the medium as being homneous. This bright idea is due to Hershey~1954!, Kroner ~1958!,and also Eshelby~1957, 1961!, in the framework of the study oelastic polycrystals. Actually, if we assume a perfect disordethe context of the statistic modeling, we get the self-consismodel. Both approaches, statistic modeling and self-consismethod, are part of what is called the ‘‘systematic theory of K¨-ner,’’ to model elastic composites.

There are many differentvolume averaging techniques. Themost popular for the purpose of fluid flow in porous media is tintroduced by Quintard and Whitaker~1987!. This method isbased on some techniques called space-convolution technielaborated by other authors, and which Quintard and Whitahave adapted to model fluid flow in porous media. The basic iis to substitute regular quantities to quantities that are varyvery quickly.

Materials with periodic structures are investigated by meanthe method of homogenization for periodic structures, which isalso called the method of double-scale asymptotic expansiThis method has been introduced by Sanchez-Palencia~1974,1980!, Keller ~1977!, and Bensoussan et al.~1978!. More re-cently, a more physical methodology based on dimensionanalysis has been introduced by Auriault~1991!. This approachhighlights the conditions under which homogenization can beplied. The presentation of this methodology is the purpose ofpresent paper. In this technique, a systematic use is made oseparation of scale parameter which, although small, is not gerally null ~«Þ0!. That procures some advantages to the tenique: ~1! avoiding prerequisites at the macroscopic scale:macroscopic equivalent description is obtained from the hetgeneity scale description plus the condition of separationscales, only;~2! modeling finite size macroscopic samples aphenomena with finite macroscopic characteristic lengths~«Þ0!;~3! modeling macroscopically nonhomogeneous media or pnomena;~4! modeling problems with several separations of scaby introducing several separation of scales parameter;~5! model-ing several simultaneous phenomena;~6! determining whether thesystem ‘‘medium1phenomena’’ is homogenizable or not, i.ewhether or not a continuous equivalent macroscopic descripexists; and~7! providing the domains of validity of the macroscopic models.

Evidently, real porous media are rarely periodic. Howeverwill be shown that there exists similitude in the behavior of peodic and random media, on the condition that a separationscales is present. Consider the parallelepipedic REV,VREV , of arandom medium. Construct a period,V, by introducing three successive plane symmetries with respect to three nonparallel fof VREV . The periodic medium of period,V, and the randommedium of REV,VREV , possess similar structure of their macrscopic description~although an eventual anisotropy could bmodified by the plane symmetries!. Therefore, we assume in thfollowing that the heterogeneous medium is periodic, without l

818 / JOURNAL OF ENGINEERING MECHANICS / AUGUST 2002

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of generality. In many applications, it was shown that this givreasonable results~Quintard and Whitaker 1993!.

Finally, after determining the equivalent macroscopic maematical model, the effective parameters herein can be invgated by using the variational formulation of the local boundavalue problem. Estimations~e.g., Mori and Tanaka 1973; PontCastan˜eda and Willis 1995! or bounds~e.g., Hashin and Shtrik-man 1963! of these effective parameters can be obtained bylowing this way.

The next section is devoted to a short presentation ofmethod of double-scale asymptotic expansions. Then, to illustthis technique, some available results are presented conceheterogeneous media subjected to different excitations: heat trfer in composite materials with strong discontinuities of the coponent conductivities or contact thermal resistance and fluid flin rigid porous media in the transient regime or in the quasistregime in noninertial matrices.

Homogenization Method

A periodic medium of periodV and a separation of scales,l/L5«!1 is assumed. Such a period of a porous medium is shin Fig. 1. Let X be the physical space variable of the mediuFrom the two characteristic lengthsl and L, two dimensionlessspace variables can be defined as

y5X/ l is the dimensionless ‘‘microscopic’’ space variable

x5X/L is the dimensionless ‘‘macroscopic’’ space variab

As a consequence of the separation of scales,y andx are twoseparated variables. Any space dependent quantityf required todescribe the physical process is,a priori, a function of these twoseparated variables:

f5f~y,x! (2)

Obviously, these space variables are related byx5«y. The mac-roscopic equivalent model is obtained from the description atheterogeneity scale by Auriault~1991!:1. assuming the medium to be periodic, without loss of gen

ality since the separation of scales condition is fulfilled. Thimplies that the differentf’s are periodic with respect tospace variabley. Note that the eventual dependence offwith respect tox shows the eventual macroscopic nonhomgeneity of the medium.

Fig. 1. Periodic cell of porous medium.Vp represents pores andVs

is matrix

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2. writing the local description in a dimensionless form,using l or L as the characteristic length~the choice ofl or Lis a technical point of no effect on the final result! and byusing characteristic values for the different quantities ening the local description.

3. deriving dimensionless numbers from the governing eqtions and evaluating these dimensionless numbers withspect to the scale ratio«. A dimensionless numberQ is saidto beO(«p) if

«p11!Q!«p21 (3)Once all dimensionless numbers have been estimatedgoverning equations can then be written as

(i50

`

«pi Ai50 (4)

whereAi5O(1), i.e., «!uAi u!«21, and thepi ’s are inte-gers. TheAi ’s are some dimensionless operators and, aconsequence, areO(1) with respect to«.

4. looking for the unknown fields in the form of asymptotexpansions in powers ofe. As a consequence of both thseparation of scales and the periodicity, all physical quaties can be looked for in the form of asymptotic expansioin power of« ~Bensoussan et al. 1978!:

f~y,x!5f0~y,x!1e1f1~y,x!1e2f2~y,x!1... (5)where thef i ’s arey periodic.

5. solving the successive boundary-value problems that aretained after introducing these expansions in the local dimsionless description.

6. obtaining the macroscopic equivalent model from compibility conditions which are the necessary conditions for texistence of solutions to the successive local boundary-vproblems. At a given order ofe, a local balance equation wilyield an equation of

“y•Fi 111“x•Fi5F (6)

whereF stands for ay-periodic vector andF is ay-periodicsource term. This latter equation will lead to the macroscodescription. It actually expresses the balance over the peof the quantityFi 11, in presence of the source termS5F2“x•Fi . Integrating it over the period gives what we cathe ‘‘compatibility condition’’. This compatibility conditionmust be checked~Fredholm alternative!, otherwise the origi-nal equation has no solution and the problem is not homenizable. It is written as

^“x•Fi&V2^F&V50 (7)

where^•&V denotes the volume average over the period ais defined by

^•&V51

uVu EVy

•dVy (8)

where the subscripty shows an integration with respect tvariabley.

7. checking if the aforementioned process yields resultsverify the estimations of dimensionless numbers in pointif not, then the system~medium1phenomena! under consid-eration is not homogenizable: There exists no equivamacroscopic description. Of course, an averaging procespossible in such a case, but it yields pseudoeffective cocients that depend on the boundary conditions of the mascopic boundary value problem and onL, thus leading toscale effects. Moreover, the mathematical model in use loits legitimacy.

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Some Examples of Upscaling

To illustrate the multiple scale asymptotic expansion method,revisit in this section a few results about heat transfer in compite materials and seepage in porous media. We limit ourselgeneral features. Details of the upscaling processes can be fin the hereafter given references for each example.

Heat Transfer in Composite Materials

Consider a two-component composite material. Heat transfecomponenta51, 2 is described at the heterogeneity scale by

Ca

]Ta

]t2

]

]XiS la i j

]Ta

]XjD50 (9)

where T5the temperature,C5the volume heat capacity anl5the thermal conductivity tensor. Successively addressedthe cases of perfect contact and the case of interfacial heat babetween the two components.

Perfect Interfacial ContactIn this case, both temperature and normal heat flux are continuon the component interface. For simplicity, we assumeC1

5O(C2). Therefore, we are left with two dimensionless numbein Eq. ~9!

Ql5

UC1

]T1

]t UU ]

]XiS l1i j

]T1

]XjD U , D5

ul2i j uul1i j u

(10)

where l is used as the characteristic length to estimateQ. uqustands for an evaluation ofq.

For the classical case, consider first the case whereD5O(1). Heat transfer is homogenizable only ifQl<O(«2), seeAuriault and Lewandowska~1993!, where a similar problem isinvestigated concerning diffusion. IfQl5O(«p), p.2, heattransfer is not homogenizable, i.e., there is no macroscoequivalent description. WhenQl5O(«2), the macroscopic equation is obtained from a compatibility condition that concerns tfirst termT0 of the asymptotic expansion ofT. The macroscopicequation is written in the form, when replacingT0 by T @seeAuriault ~1983! and Appendix I for details#

^C&V

]T

]t2

]

]XiFli j

eff]T

]XjG5O~«! (11)

where^C&V5the volume average of the heat capacity and stafor the effective heat capacity. The tensorleff5the effective heatconductivity, as defined in Auriault~1983!, see Appendix I. Notethat the mathematical macroscopic model is an approximatonly.

Memory effects appear when thermal conductivities are vdifferent, e.g.,D5O(«2), Q1l5O(«2), andQ2l5O(1) ~Auriault1983!. In this case, the macroscopic model can be put in the fo

^C&V

]T

]t2E

2`

t

M ~ t2t!]2T

]t2 ~t!dt2]

]XiFli j

eff]T

]XjG5O~e!

(12)

Memory effects are represented by the memory functionM andleff is an effective heat conductivity, different from the effectivconductivity in the classical case. As recalled herein, the valu« depends on the excitation. Therefore, changing the excitatio


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as to increaseL will decrease« and will change the evaluation oD. In the limit, model~12! reduces to classical model~11!. De-creasingL will increase« and thus will decrease the accuracythe model, and in fine will yield a nonhomogenizable situatio

Interfacial Heat BarrierNow the contact between the two components is not perfect. Aresult, at the boundary between the two components, the tempture, and the heat flux verify

l1i j

]T1

]Xjni5l2i j

]T2

]Xjni5h~T22T1! (13)

wheren5the outward normal to component 1 andh.05the in-terfacial thermal conductance. A new dimensionless numbeintroduced, the Biot’s number

Bl5uh~T22T1!u

Ul i j

]T1

]XjniU (14)

The other dimensionless numbers are assumed to be of theorder as in the classical case,Ql5O(«2), D5O(1). Dependingon the evaluation of the Biot’s number with respect to«, fivedifferent macroscopic models are obtained~Auriault and Ene1994!, see Fig. 2. Existing continuous passages from a modeanother are shown in Fig. 2. At a constant«, the contact resistancis increasing~decreasing conductanceh! from model I to modelV.

Models I, II, and III are one temperature models: local thermequilibrium is verified at the first order of approximation. Themodels are given by formulation~11!, but with different effectiveconductivities. TensorlI

eff is the effective conductivity in absencof thermal barrier, i.e., the conductivity in the classical casethus it ish independent. TensorlII

eff is h dependent, whereas tensor lIII

eff is h independent but different from TensorlIeff @see Auri-

ault and Ene~1994! for details#.Models IV and V are two-temperature models: local therm

equilibrium is not verified. Model IV is of the form

^C1&V

]T1

]t1H~T12T2!2

]

]XiFl1IVi j

eff]T1

]XjG5O~«! (15)

^C2&V

]T2

]t2H~T12T2!2

]

]XiFl2IVi j

eff]T2

]XjG5O~«! (16)

TemperaturesT1 and T2 are the temperatures in media 1 andrespectively. Effective coefficientH is the average ofh. Conduc-tivity tensors l1IV

eff and l2IVeff are the effective conductivities o

media 1 and 2, respectively, when these two components arefectly insulated and we havel1IV

eff 1l2IVeff 5lIII

eff . Model V is givenby Eqs.~15! and ~16! with H50. All the different effective con-ductivity tensorsleff can be shown to be symmetrical positivtensors.

Porous media can be classified in two overlapping categomedia withBl>1 and those which verifyBl<1. For each cat-egory, the richest models are models II and IV, respectively.

Fig. 2. Different models for heat transfer in composite materials wthermal heat barrier as function of Biot’s numberBl


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It is important to note that giving a composite material yielthe value of the Biot’s numberBl5hl/ul1u. On the one hand, thevalue of the separation of scales parameter«5 l /L depends onL,i.e., on the excitation. Therefore, changing the excitation coagain result in a change of the model to be used to describeheat transfer. This remark stands for many upscaling procesOn the other hand, changing the excitation in a way to decreaLrenders« larger and, in the limit, it suppresses the separationscales: the situation becomes nonhomogenizable.

Mass Transfer in Rigid Porous Media

Consider a rigid porous matrix which is saturated by an incopressible fluid. We present in this section some results concerthe transient seepage law in a galilean porous matrix andquasistatic flow in a noninertial porous matrix.

Transient Seepage Law in Galilean Porous MediaIt is convenient to consider fluid movements at constant pulsav. Then, a small perturbation of the fluid is described by NavieStokes equations where the nonlinear terms are negligible, byincompressibility condition and by the adherence condition onpore surface

r ivv i5m]2v i

]Xk]Xk2

]p

]Xi(17)

]v i

]Xi50 (18)

v i50 on the pore surface (19

The velocity5v and the pressure5p. Constantsm andr5the vis-cosity and the density of the fluid, respectively. Thei in front of vis defined byi 2521. The situation of interest~Levy 1979; Auri-ault 1980! is for equal weights at the pore scale of the inertial aviscous terms in the Navier–Stokes equation, the movembeing forced by a macroscopic gradient of pressure. That resin

Rl5ur ivv i u

Um ]2v i

]Xk]XkU 5O~1!, Pl5

U ]p

]XiU

Um ]2v i

]Xk]XkU 5O~«21!

(20)

Under these conditions, the monochromatic transient seepflow is described by the set

]^v i&V

]Xi5O~e!, ^v i&V52Ki j

]p

]Xi1O~«! (21)

TensorK is the transient monochromatic permeability tensor. Ia symmetrical, complex valued, and anv-dependent tensor. Returning to time space yields inertial memory effects. Note that tmacroscopic model was already introduced by Biot~1956!, byusing a phenomenological analysis. Asv goes to zero, we recovethe classical Darcy’s law: PermeabilityK becomes real valued@see Ene and Sanchez-Palencia~1975! for the Darcy’s law#.

Quasi-static Flow in Noninertial Porous MediaWhen the porous matrix is noninertial~Auriault et al. 2000,2002!, the quasistatic flow of the liquid is described with respeto the porous matrix frame by the incompressibility conditi~13!, the adherence condition~14!, and Navier–Stokes equatioin the form

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rgci1r« i jkv jvk5m]2v i

]Xk]Xk2

]p

]Xi(22)

In Eq. ~22!, gc5the convective acceleration,« i jk5the alternator,and« i jkv jvk5the Coriolis acceleration component, wherev5theangular velocity of the rigid porous medium. The fluid flowforced by a macroscopic gradient of pressure and the conveinertia and we assume an order 1 Ekman numberEk, which is atypical situation in filtration processes in centrifuge

Pl5O~«21!,urgciu

U ]p

]XiU 5O~1!, Ekl5

Um ]2v i

]Xk]XkU

ur« i jkv jvku5O~1!

(23)

By upscaling~Auriault et al. 2000, 2002!, the macroscopic flowmodel is as follows

]^v i&V

]Xi5O~«!, ^v i&V52Ki j S ]p

]Xi1rgciD1O~«! (24)

The second relation in Eq.~24! shows a similar structure to Darcy’s law. However, the permeability tensor is nowv dependentand it is generally not symmetrical. It verifies the filtration analof the Hall effect

Kpq~v!5Kqp~2v! (25)

The seepage law in Eq.~24! tends to the Darcy law whenv tendsto zero.

Final Remarks

As illustrated on several upscaling processes, the method ofmogenization by multiscale asymptotic expansions presentsportant advantages, particularly of giving the domains of validof the models and the domains of nonhomogenizability. Tanalysis is based on evaluations of dimensionless numberfunction of the small separation of scales parameter«. Moreover,a given heterogeneous material could obey different modeldifferent zones of a macroscopic boundary-value problem. Thfore, in order to determine these different zones and the cosponding models to be used, it is of prime importance to obtaina priori estimate of«5,/L, which needs ana priori estimate ofL. Unfortunately,L is obtained from the solution of the macroscopic boundary-value problem, only, by, e.g., in a ondimensional macroscopic problem, using the following relat~Boutin and Auriault 1990!

L'US ]

]X S f

f0D D 21U (26)

wheref0 is a typical value of the macroscopic quantityf. Tobreak out of this vicious circle, note that we do not needaccurate estimate of«, and, fortunately, the different models tdescribe a particular physical process do not generally givedifferent values ofL. Therefore,« can be estimated from thsimplest available model. After that, evaluating the dimensionnumbers involved in the problem yields the different zones toconsidered in the macroscopic volume. Ana posterioricontrol ofthese zones, after integration of the macroscopic boundary-vproblem, can be made to check their validity. An improvemcan also be obtained, if necessary, by iterating these zones.

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Appendix I. Heat Transfer in Composite Materials—Classical Case

WhenQl5O(«2), the dimensionless heterogeneity scale desction of heat transfer in the two-component medium,a51, 2, is ofthe form

«2Ca

]Ta

]t2

]

]yiS la i j

]Ta

]yjD50, in Va (27)

l1i j

]T1

]yjni5l2i j

]T2

]yjni , T15T2 , on G (28)

where for simplicity, similar notations for dimensional and dmensionless quantities are used, with the exception of the svariable. The interface between the two-component is denoteG.Introducing an asymptotic expansion like Eq.~5! for the tempera-tureT into the set Eqs.~27! and~28! yields boundary-value problems for the successive terms of the expansion. More detailsavailable in Auriault and Hene~1994!. At the lower order, weobtain the following boundary-value problem for theV-periodicunknownT0

]

]yiS la i j

]Ta0

]yjD 50, in Va (29)

l1i j

]T10

]yjni5l2i j

]T20

]yjni , T1

05T20, on G (30)

TemperatureT0 can be shown to be of the form

T105T2

05T0~x,t ! (31)

Then, theV-periodic temperatureT1 is the solution of the prob-lem

]

]yiS la i j S ]Ta

1

]yj1

]T0

]xjD D 50, in Va (32)

l1i j S ]T11

]yj1

]T0

]xjDni5l2i j S ]T2

1

]yj1

]T0

]xjDni , T1

15T21; on G

(33)

TemperatureT1 appears as an affine function of the gradient coponents]T0/]xj

T15t i

]T0

]xi1T1~x,t ! (34)

Componentt i(y), of zero average5the solution of zero averageof the aforementioned system when]T0/]xp5d ip and T1(x,t) isan arbitrary function ofx and t. Finally the problem for theV-periodic temperatureT2 is the following

Ca

]T0

]t2

]

]yiS la i j S ]Ta

2

]yj1

]Ta1

]xjD D 1

]

]xiS la i j S ]Ta

1

]yj1

]T0

]xjD D

50, in Va (35)

l1i j S ]T12

]yj1

]T11

]xjDni5l2i j S ]T2

2

]yj1

]T21

]xjDni , T1

25T22, on G

(36)

Eq. ~35! is the local balance of theV-periodic quantityla i j (]Ta

2/]yj1]Ta1/]xj) in presence of the source term

Ca]T0/]t2]/]xi@la i j (]Ta1/]yj1]T0/]xj)#. Therefore, the

source term should be of zero-volume average, that can beby integrating Eq.~35! over V. This result, which is related toFredholm alternative, gives


.128:817-822.

el

d

-

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,

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on-

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ri-

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e

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e

n

s.

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rary

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ight

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nly;

all

righ

ts r

eser

ved.

^C&V

]T0

]t2

]

]XiFli j

eff]T0

]XjG50 (37)

where

^C&V51

V EV

CadV (38)

and the effective conductivity is given by

li jeff5

1

V EV

la ikS ]t j

]yk1I jkDdV (39)

Finally, by replacingT0 by T, we obtain the macroscopic modin the form

^C&V

]T

]t2

]

]XiFli j

eff]T

]XjG5O~«! (40)

Notation

The following symbols are used in the paper:Ai 5 dimensionless differential operator;Bl 5 Biot number;C 5 heat capacity;D 5 dimensionless number;

Ek 5 Ekman number;H 5 effective interfacial thermal conductance;h 5 interfacial thermal conductance;K 5 permeability tensor;L 5 characteristic macroscopic length;l 5 heterogeneity characteristic length;

M 5 thermal memory function;Pl 5 dimensionless number;p 5 pressure;

Q,Ql 5 dimensionless numbers;R 5 dimensionless number;T 5 temperature;v 5 velocity;X 5 physical space variable;

x, y 5 dimensionless space variables;G 5 component interface;

gc 5 convective acceleration;« 5 scale ratio;l 5 thermal conductivity tensor;

leff 5 effective thermal conductivity tensor;m 5 dynamic viscosity;r 5 density;V 5 period of the heterogeneous medium;v 5 angular velocity; andv 5 pulsation.

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upscaling heterogeneous media by asymptotic expansions

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