Modelling Fluid Flow and Heat Transfer in a
Saturated Porous Medium
D.A.NIELD [email protected]
DepartmentofEngineeringScience,UniversityofAuckland,PrivateBag92019,Auckland,New
Zealand
(DedicatedtoA. McNabbontheoccasionofhis 70thBirthday)
Abstract. Since the daysof Darcy,manyrenements havebeen madeto the equations used
to model single-phase uidow and heattransfer ina saturatedporousmedium, to allow for
suchbasicthingsasinertialeects,boundaryfrictionandviscousdissipation,andalsoadditional
eectssuchasthoseduetorotationoramagneticeld.Thesedevelopmentsarereviewed.
Keywords:porousmedium,uidow,heattransfer
1. Introduction
Shenoy(1994)givesatwo-pagelistofapplicationsofthepresentsubjectunderthe
headingsBiomechanics,Ceramicengineering,Chemicalengineering,Foodtechnol-
ogy,Geophysics,Groundwaterhydrology,Industrialengineering,Mechanicalengi-
neering,Petroleum engineering,Soilmechanics.
Aporousmediumisaxed(oralmostxed)solidmatrixwithaconnectedvoid
space throughwhich a uid canow. Thefraction of voidspace to total volume
is called theporosity. Mostnaturallyoccurring porousmedia haveporositiesless
than 0.6 (an exception is hair), but man-made materials, such as metallic foam,
canhaveporositiesupto 0.99.
The observations of Henry Darcy (1856) of the water supply at Dijon, and ex-
periments on steadystate unidirectional ow suggestedDarcy'slaw, which in its
renedmodern formcanbeexpressed as
@p=@x=(=K)v; (1)
where @p=@x is the pressure gradient, v is the ltration velocity, is the uid
viscosityandK isthepermeability(units lengthsquared). Theltration velocity
v(velocityaveragedoverthemedium)isrelatedtotheintrinsicvelocityV (velocity
averagedovertheporespace)byv=V,whereistheporosity. Thepermeability
Kdependsontheporesize(orparticlediameter)D
p
,theporosity,andalsoonthe
detailed geometry. AusefulestimateisgiventheCarman-Kozenyrelationship
K= D
2
p
3
180(1 ) 2
: (2)
Darcy's law meansthat the dragis proportionalto thevelocity. This holds for
breaks down for larger velocities. Dupuit (1863), and Forchheimer (1901) found
empiricallythatforlargervelocitiesthedragisaquadraticfunctionofthevelocity.
(Isimplifythematter;adetailedhistoricalaccounthasbeengivenbyLage(1998).)
2. The BrinkmanForchheimer equation
A modern renement (see e.g. Hsu & Cheng(1990), Vafai &Kim (1990))is the
equation
1
@v
@t +
1
2
(vr)v
= rp+
e r
2
v
K v
c
F
K 1
2
vv : (3)
This applies to an incompressible uid of density . Here v denotes jv j, the
magnitude of the Darcy velocity, while
e
is an eective viscosity and c
F is a
dimensionless Forchheimer coeÆcient. The inertial terms (onthe left-hand side)
resultfrom formal averaging. The rst viscous termis the Brinkman term. The
lasttermistheForchheimerterm. Wenowconsiderthesignicanceofthevarious
termsinEq. (3).
1. The local time-derivative inertialterm.
This is derived on theassumption that a spatial averagingprocess commutes
with a derivative with respect to time. This breaks down when the porous
medium has macroscopic structure such as a system of tubes. The decay of
a transient is more rapid in narrow tubes than in wide tubes. Nield (1991)
suggestedthatinthiscase(1=)@v =@tbereplacedbyc
a
@v =@t,wherec
a isa
constanttensor(thatisdeterminedmainlybythenatureoftheporesoflargest
cross-sections). Inanycase,theratioofthetime-derivativetermtotheDarcy
resistance is c
a
K =T, where T is acharacteristictime of the process being
investigated,andthisratioisnormallyverysmall.
2. Advectiveinertial term.
Joseph,NieldandPapanicolaou(1982)arguedthat,whenmodellingdenseme-
dia,theadvectiveterminvolving(vr)vshouldbeomittedbecausetheinertial
eectsarealreadyaccountedforinthequadraticdragterminvolvingvv . This
arisesasaresultof formdrag onthe solidparticles. Thedrag isindependent
oftheviscosityandactsin adirectionoppositetov . Nield(1991)arguedthat
theinclusionofthe(vr)vtermleadstothepredictionthatlongitudinalmo-
mentumcan,unimpededbythexedsolidmatrix,betransmittedtransversely,
inconict withexpectationbasedonbasicphysics.
This isrelatedto thediÆculty ofspin-up (by just rotatingasolid boundary),
andtheabsense oftruemacroscopicturbulence(involvingacascadeofenergy
fromlargeeddiestosmallereddies),inadenseporousmedium. Theaveraging
processleadstomisleadingresultsbecauseitleadstoalossofvitalinformation
about the way in which the geometry of the solid matrix aects the ow by
Onehasavectoridentity(vr)v =r(v 2
=2)+v(rv ). Itwasnotedby
Nield(1994)thatatleasttheirrotationalpart,r(v 2
=2),of(vr)vneedstobe
retainedin order to accountfor thephenomenonof chokingin thehigh speed
owofacompressibleuid,buthesuggestedthattherotationalpart,involving
theintrinsicvorticity,bedeleted. His argumentwasbasedontheexpectation
that a medium of low porositywill allow scalarentities likeuid speedto be
freelyadvected,butwillinhibittheadvectionofvectorquantitieslikevorticity.
Nield and Bejan (1999) went a step further, and suggested that even when
vorticityis beingcontinuously produced(e.g. bybuoyancy) onewould expect
thatitwouldbedestroyedbyamomentum dispersionprocessduetothesolid
obstructions.
An argument providing further support for this point of view was presented
by Nield (2000a). There are some subtleties about the eect of the inertial
termsonmotion in aporous medium. Thepowerof thetotal dragforce (per
unitvolume)isequaltotherateofviscousdissipation(perunit volume);fora
detailed discussion see Nield (1999). TheForchheimer drag term, although it
appearstobeindependentoftheviscosity,contributestotheviscousdissipation.
Theeect of inertiais mediatedviaachange in thepressuredistribution and
the velocity distribution. The ip side of the coin is that when one closes
thesystemofequationsbyintroducingaForchheimerdragtermoneshouldnot
assumethattheconvectiveinertiatermthatremainsinthemomentumequation
isidentical withthat obtainedbyformal volume-averaging. After integration,
itshould lead tothe correctexpression fortheaveragedkineticenergy,which
involvesthemagnitudebutnotthedirectionofthevelocity,andthismeansthat
the irrotational partof thevolume-averagedconvectiveinertial term must be
unchanged,buttherotationalpartisnotdeterminedbytheaveragingprocess,
andthereisnoinconsistencyinsettingittozeroaspartoftheclosureprocess.
In the process of performing the closure after volume-averaging, it has been
traditionaltoadjustforthecontributiontotheoveralldragforce,thatincludes
aquadraticdragforcethathasaspecicdirection(paralleltotheDarcyvelocity
inthecaseofananisotropicmedium),buttoignorethefactthatonealsoneeds
toadjustforthefactthattheoverallmomentoftheforcesystemhastobezero.
Nield(2000a)suggestedthatanappropriateadjustmentissimplytosettozero
theirrotationalpartofthevolume-averagedconvectiveinertialterm.
Ithassometimesbeenclaimedthattheretentionoftheconvectiveinertialterm
is necessaryin order to account forthe formationof hydrodynamic boundary
layersinchannel ow,andinordertoestimatetheentrancelength,butthisis
notcorrect. Theformationofsuchlayersisprimarilyduetotheactionofviscous
diusion, and the entrance length canbeestimated using thetime-derivative
inertialterm.
In many practical cases it does not matter computationally whether the ad-
term,itisoforderofmagnitudeK 1
2
=c
F
2
L(whereLisacharacteristiclength
scale),andthisisnormallysmall. [CompareLage(1992).]
Thistopicisrelatedtothequestionofhowbesttomodelturbulenceinaporous
medium. Thisiscurrentlyacontroversialtopic(Nield, 2000b).
3. Brinkmanviscousterm
Brinkman(1947)introducedtheLaplacianviscousterminarestrictedcontext.
Its global use is due to other authors. Theglobal treatment may fail to deal
adequatelywiththedistinctivefeaturesofowin aporousmedium. Theratio
of theBrinkman term to theDarcy term isof order Da=K =L 2
(where L is
theappropriatemacroscopic lengthscale), soDa!1correspondsto auid
clear of solid material. In most practical cases Da will be very small, and
the Brinkman termwill havea signicant eect only in thin layers(within a
dimensionaldistance oforder K 1
2
ofasolidwall. Inmanycasesthereduction
in velocity in this thin layer will be masked by an increase in velocity (the
channelingeect)duetoincreaseinporositynearthewall(wheresolidparticles
cannotpackastightlyastheycanintheinterior).
TheBrinkmanequationcannotberigorouslyjustiedexceptwhentheporosity
isclosetounity. TheselfconsistentformulationofBrinkmanbreaksdownwhen
< 0:6. There is an uncertaintyaboutthe eective viscosity
e
. Brinkman
took
e
=u. Formalaveraging(Bear&Bachmat, 1990)leadsto
e
==T,
where T is the tortuosity . Whitaker (1999, p. 173) ignores the tortuosity.
He emphasizesthat theBrinkman correction essentiallyinvolvesthe intrinsic
velocity, sothatwhen thecorrection iswritten in termsof theDarcy velocity
thisimmediatelyleadsto
e
==. Untilrecentlyithadnotbeenpossibleto
checkthealternativesagainstexperimentbecausealltheavailableexperimental
data pertained to media whose porosity was outside the range for which the
theoretical results are valid. Givler & Altobelli (1994), using NMR, found
e
= 8 approximately for water owing through a rigid foam material (
=0.972). Itisclearthataveragingisinadequatein thiscase.
4. Dupuit-Forchheimerterm.
Theterm(c
F
=K 1
2
)vv is in theform recommendedbyJoseph,Nield and Pa-
panicolaou(1982). Thescalar form isdue to Ward(1964), whothought that
c
F
mightbeauniversalconstant, 0.55. Subsequentexperimenters found that
c
F
isapproximatelyconstantforaparticularfamilyof materials,e.g c
F
=0:1
forfoamedmetalbres. A semi-empiricalderivationofanestimateforc
F was
reportedbyJoseph etal. (1982). Theyemphasizedthat thedrag isquadratic
andinadirectionopposedtov(soonecannotwritedownuncoupledequations
for the x- and y-components of v ). In fact, acubic drag term arises in two
circumstances. First,Lage etal. (1997)pointedoutthat, whencomplications
resulting from transition to turbulence are takeninto account, the coeÆcient
ofthequadratictermvariesslowlywith velocity,andso thedragiseectively
(1991)and others havepointedout that in the weak inertialregime (one for
which thepore Reynoldsnumberislessthanunity,thevariationfromthelin-
eartermisin factcubic, ratherthanquadratic. Antohe andLage(1999)have
argued that thefactorK 1
2
in the Dupuit-Forchheimer termis better replaced
byanotherquantityhavingthedimensionsoflength,namelyatypicalparticle
diameter.
3. Modelling a porous-medium/clear-uidinterface
Since the dierential equations for the two regions are of second order in spatial
derivatives,four matching conditionsare neededifthe Brinkmanequation is em-
ployed. Theseinvolvethecontinuityoftangentialvelocity,normalvelocity,tangen-
tialstressandnormalstress. Thevelocitymatchingcausesnoproblems,but with
thestressmatchingitisdierent. Considerthematchingoftangentialstress. Over
theporeportionoftheinterfacethevelocityshear,andhencethetangentialstress,
is continuous. Overthe solidportion thetangentialshear is notcontinuous; it is
obviously zeroin thesolid, buthas someindeterminantnon-zerovaluein thead-
jacentclearuid. Authorswhohavematchedvelocityshearshaveoverdetermined
thesystemofequations.
When one uses the Darcy equation (instead of the Brinkman equation) in the
porousmediumthediÆcultycanbeside-stepped. Nowoneneedsonlythreematch-
ing conditions; twoof these are the continuity of tangential velocity and normal
velocity,andtheBeavers-Joseph(1967)boundarycondition:
u
f
y
=
BJ
K 1
2 (u
f u
m
): (4)
Heretheclearuidoccupiestheregion(y>0), andu
f
istheuidvelocity,and
u
f
and @u
f
=@y are evaluated at y =0+. TheDarcy velocityu
m
is evaluated at
somesmalldistancefromy=0. TheBeavers-Josephconstant
BJ
isdimensionless
andindependentoftheuidviscosity,butitdependsonthestructureoftheporous
materialwithintheboundaryregion. Sahraoui&Kaviany(1992)haveshownthat
the value of
BJ
depends on the ow direction at the interface, the Reynolds
number, the extent of the clearuid, and nonuniformities in the arrangement of
solidmaterialatthesurface. Itseemsbesttoregard
BJ
asanempiricalconstant,
to bedetermined experimentally. Its presencein theboundaryconditionprovides
theneededexibilityin modelling thetangentialstressrequirement.
Thesituation with respect to thenormalstress is similar, but there isan addi-
tional factorinvolved. Thenormalstressis thesumofthepressureandaviscous
stressterm. Someauthorshavearguedthatthepressure,beinganintrinsicquan-
tity,hastobecontinuousacrosstheinterface. Theyhavefailedtorealizethatthe
interfaceisanidealizationofathinlayerinwhichthepressurecanchangesubstan-
tiallybecauseofthepresenceofthesolidmaterial. Inpractice,theviscoustermin
thenormal stress may be smallcompared with the pressure,and in this casethe
Also, for an incompressible uid, the continuity of normal stress does reduce to
continuity of pressure if one takes the eective Brinkman viscosity equal to the
uidviscosity,asshownbyChenandChen(1992). Authorswhohaveformulated
a problem in termsof stream function and vorticity have failed to deal properly
withthenormalstressboundarycondition(Nield,1997). Foramoresoundlybased
procedurefornumericalsimulation,andforafurtherdiscussionofthismatter,the
readerisreferredtoGartlinget al. (1996).
Ochoa-Tapia and Whitaker (1995a,b) have expressly matched the Darcy and
Stokesequationsusingthevolume-averagingprocedure. Thisapproachproducesa
jump in the stress(but notin thevelocity)andinvolvesaparameterto betted
experimentally. Theyalsoexploredtheuseofavariableporositymodelasasubsti-
tute forthejump condition,andconcludedthatthelatterapproachdoesnotlead
to asuccessful representation ofall theexperimental databut it providesinsight
into the complexity of the interface region. Kuznetsov (1996) applied the jump
conditionto owsin parallel-plate and cylindricalchannelspartially lled witha
porousmedium. Kuznetsov(1997)reportedananalyticalsolutionforownearan
interface.
Salingeretal. (1994)foundthataDarcy-slipnite-elementformulationproduced
solutionswhich were moreaccurate and moreeconomicalto compute than those
obtainedusing aBrinkmanformulation.
4. Non-Newtonianuid
Shenoy(1994) reviewed studies of ow of non-Newtonian uids in porous media.
Attention has been concentrated on power-lawuids. Shenoy suggested, on the
basisofvolumetricaveraging,thattheDarcy termbereplacedby(
=K
)v n 1
v ,
theBrinkmantermby(
= n
)rfj p
[ 1
2 :]j
n 1
rgforanOstwald-deWaeleuid,
and the Forchheimer term be left unchanged (because it is independent of the
viscosity). Here nis thepower-lawindex,
reectsthe consistencyofthe uid,
K
isamodiedpermeability,andisthedeformationtensor. Theauthoragrees
withShenoy'ssuggestion,butin theBrinkmantermhewouldreplace(
= n
)by
anequivalentcoeÆcient.
5. Eectofrotation
Theeectofrotationistoaddextrabody-forcetermstothemomentumequation,
reectingthecentrifugalandCorioliseects. Inthecontextofnaturalconvection,
thetopichasbeendiscussedin papersreviewedbyVadasz(1997,1998). Theleft-
handsideofEquation(3)isreplacedby
v
+ 1
2
(vr)v+ 2
!v+!(!x )
(5)
where ! is the angular velocity of the rotating frame of reference and x is the
positionvectorrelativeto thatframe. TheratiooftheCoriolistermto theDarcy
termisoforderE 1
,wheretheEkman-DarcynumberE isgivenby
E=Ek=Da; Ek==2!L 2
; Da=K =L 2
: (6)
Here L is a characteristic length. In most practical situations E is large, gener-
ally because the Darcy number Da is small, so the Coriolis term is usually not
important. However, Vadasz points out that in the case of heterogenous media
the Coriolis accelerationdistorts the direction of any existing ow and generates
vorticesin aplaneperpendicular totheow. Forisothermaluidsthecentrifugal
term,beingirrotational, merelyaectsareducedpressure,but forfreeconvection
thistermmaybeimportant.
ManyauthorshavewronglyomittedthefactorfromtheCoriolisterm. AsNield
(1999)pointed out,they failed to realizethat thepressurein Darcy'sequation is
anintrinsicquantity,andhenceallthevelocityappearingin aninertialtermmust
alsobeanintrinsicquantity. Infact,Nieldhassuggestedthat itwouldbesimpler
and less confusing to rewrite the momentum equation in terms of the intrinsic
velocity and with the permeability K replaced by the \retardability" R , dened
by R = =K. This proposed change has other advantages. The new \eective
viscosity"becomescloseto theuidviscosityandthenewForchheimercoeÆcient
becomesclosertobeingauniversalconstant.
6. Eectofa magnetic eld
Thetechniqueofvolume-averagingleadstothepredictionthattheeectofamag-
neticeld isto add a body-forceterm (vB)B=to the right-hand side of
Equation(2). Here istheelecticalconductivityoftheuidandBistheapplied
magnetic induction. (See,for example, Raptis&Perdikis (1987).) Inthe caseof
two-dimensional ow and with the magnetic induction in the plane of that ow,
theextrabody forcereduces to( B 2
)v =. Thus theeectofthemagnetic eld
isthen simplytoadd anadditionaldragforce. The ratioof themagneticdrag to
the Darcy drag is B 2
K =. In most practical cases this Chandrasekhar-Darcy
numberisverysmall,sotheeecttothemagneticeld isnegligible. Again,many
authorshaveomittedthefactorinerror.
7. Viscousdissipation
For convectionproblemsonemustsupplementthemomentumequationbyather-
malenergyequation,whichinsteadystateformis
c
p
vrT =r(k
e
rT)+ (7)
where k
e
is theeectivethermalconductivityof theporousmedium and isthe
givenbythe powerof thedragforce (perunit volume),i. e. vFwhere Fisthe
dragforce. InthecasewhereEq.(3) applieswehave
=
K vv+
c
F
K 1
2
jv jvv : (8)
The remarkablething is that the last term does not involve the viscosity asa
factor, despite the fact that it contributesto the viscous dissipation term. This
paradoxwasresolvedby Nield(2000a). Theshort explanationis thattheinertial
eects aremediatedbythepressuredistribution andthis aectsthevelocityeld
and hence the drag in a complex fashion. Boundary layer separation and wake
formationareinvolvedintheexplanation.
References
1. B.V.Antohe. and J. L.Lage, Darcy's experiments and the transition to nonlinear ow
regime. Proceedings ofthe 33rd NationalHeatTransferConference, August15-17, 1999,
Albuquerque,NewMexico,NHTC99-180,pp.1-7,1999.
2. J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous
Media.KluwerAcademic,Dordrecht,1990.
3. G.S.BeaversandD.D.Joseph,Boundaryconditionsatanaturallypermeablewall.J.Fluid
Mech.30:197-207,1967.
4. H.C.Brinkman, Acalculation of theviscousforceexerted bya owing uidona dense
swarmof particles. Appl.Sci.Res.A,1:27-34,1947.
5. F.ChenandC.F.Chen, Convectioninsuperposed uidandporouslayers. J.FluidMech.
234:97-119,1992.
6. H.P.C.Darcy,LesFontsinesPubliquesdelaVilledeDijon. VictorDalmont,Paris,1856.
7. J.Dupuit,EtudesTheoriquesetPratiquessurleMouvementdesEaux.Dunod,Paris,1863.
8. P.Forchheimer, WasserbewegungdurchBoden. VDIZ.,45:1782-1788,1901.
9. D.K.Gartling, C.E. HickoxandR.C.Givler, Simulation ofcoupled viscousand porous
owproblems. Comp.FluidDyn.7:23-48,1996.
10. R.C.GivlerandS.A.Altobelli,AdeterminationoftheeectiveviscosityfortheBrinkman-
Forchheimerowmodel. J.FluidMech.258;355-370,1994.
11. C.T.HsuandP.Cheng,Thermaldispersioninaporousmedium.Int.J.HeatMassTransfer,
33:1587-1597,1990.
12. D.D. Joseph,D. A.Nield andG.Papanicolaou, Nonlinear equation governing owina
saturated porousmedium. WaterResourcesResearch,18:1049-1052and19:591,1982.
13. M.Kaviany,PrinciplesofHeatTransferinPorousMedia.Springer,NewYork,1991.
14. A.V.Kuznetsov, Analyticalinvestigationof theuid owin theinterfaceregionbetween
aporousmediumandaclearuidinchannelspartiallylledwithaporousmedium. Appl.
Sci.Res..56:53-571997.
15. A.V.Kuznetsov, Inuenceof thestress jumpconditionattheporous-medium/clear-uid
interfaceonaowataporouswall. Int.Comm.HeatMassTransfer24:401-410,1997.
16. J.L.Lage, EectoftheconvectiveinertiatermonBenardconvectioninaporousmedium.
Num.HeatTransferA,22:469-485,1992.
17. J.L.Lage, The fundamentaltheory of owthrough permeablemedia from Darcyto tur-
bulence. TransportPhenomena inPorousMedia(D. B.Ingham& I.Pop, Eds.)Elsevier
Science,Oxford,pp.1-30,1998.
18. J.L.Lage,B.V.AntoheandD.A.Nield,Twotypesofnonlinearpressure-dropvesrusow-
rate relationobserved forsaturated porous media. ASMEJ. FluidsEngng.119:701-706,
1997.
19. C.C.MeiandJ.L.Auriault,Theeectofweakinertiaonowthroughaporousmedium.
20. D.A.Nield,ThelimitationsoftheBrinkman-Forchheimerequationinmodellingowina
saturated porousmediumandataninterface.Int.J.HeatFluidFlow12:269-272,1991.
21. D. A. Nield, Modelling high speed ow of anincompressible uid in a saturated porous
medium. TransportinPorousMedia14:85-88,1994.
22. D.A.Nield,DiscussionofaDiscussionbyF.ChenandC.F.Chen.ASMEJ.HeatTransfer
119:193-194,1997.
23. D.A.Nield,Modelingtheeectsofamagneticeldorrotationonowinaporousmedium:
momentumequationand anisotropic permeabilityanalogy. Int. J.Heat MassTransfer42:
3715-3718,1999.
24. D.A.Nield, Resolutionofa paradox involvingviscousdissipation andnonlineardragina
porousmedium.TransportinPorousMedia.(inpress),2000a.
25. D. A.Nield, Alternative models of turbulence in a porous medium, and related matters.
SubmittedtoASMEJ.FluidsEngng.2000b.
26. D.A.NieldandA.Bejan,ConvectioninPorousMedia.2ndEd.,Springer,NewYork,1999.
27. J.A.Ochoa-TapiaandS.Whitaker, Momentumtransferattheboundarybetweenaporous
mediumand ahomogeneousuid? I.Theoreticaldevelopment. Int.J.HeatMassTransfer
38:2635-2646,1995a.
28. J.A.Ochoa-TapiaandS.Whitaker, Momentumtransferattheboundarybetweenaporous
medium and a homogeneous uid? II. Comparison with experiment. Int. J. Heat Mass
Transfer38:2647-2655,1995b.
29. A.Raptisand C.Perdikis, Hydromagneticfree-convectiveowthroughporousmedia. En-
cyclopediaofFluidMechanicsandModeling.(N.P.Cheremisino,Editor),GulfPublishing
Co.,Houston,Chapter8,pp.239-262,1987.
30. M.SahraouiandM.Kaviany, Slipandno-slipvelocityboundaryconditionsattheinterface
ofporous,plainmedia.Int.J.HeatMassTransfer,35:927-943,1992.
31. A.G.Salinger,R.ArisandJ.J.Derby,Finiteelementformulationsforlarge-scalecoupled
owsinadjacentporousandopenuiddomains.Int.J.Numer.Meth.Fluids18:1185-1209,
1994.
32. A.V.Shenoy,Non-Newtonianuidheattransferinporousmedia.AdvancesinHeatTransfer,
24:101-190,1994.
33. P. Vadasz, Flow in rotating porous media. FluidTransport inPorousMedia (ed. P.du
Plessis).ComputationalMechanicspublications,Southampton,Chapter4,1999.
34. P.Vadasz,Freeconvectioninrotaingporousmedia.TransportPhenomenainPorousmedia
(eds.D.B.InghamandI.Pop),Elsevier,Oxford,1998.
35. K.Vafaiand S.J.Kim, Fluidmechanicsof aninterfaceregionbetween aporousmedium
andauidlayer|anexactsolution. Int.J.HeatFluidFlow11:254-256,1990.
36. J.C.Ward, Turbulentowinporousmedia.J.Hydaul.Div.Am.Soc.Civ.Eng.90(HY5):
1-12,1964.
37. S.Whitaker, TheMethodofVolumeAveraging. Kluwer,Dordrect,1999.
