Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932
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Compu e s and Ma hema ics wi h Applica ions
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Analysis o he hin ilm low in a ough domain illed wi h
mic opola luid
Igo Pažanina,∗, F ancisco Ja ie Suá ez-G aub
aDepa men o Ma hema ics, Facul y o Science, Uni e si y o Zag eb, Bijenička 30, 10000 Zag eb, C oa ia
bDepa amen o de Ecuaciones Di e enciales y Análisis Numé ico, Uni e sidad de Se illa, C/ Ta ia s/n, 41012 Se illa, Spain
a icle in o
A icle his o y:
Recei ed 24 Ma ch 2014
Recei ed in e ised o m 2 Oc obe 2014
Accep ed 3 Oc obe 2014
A ailable online 18 Oc obe 2014
Keywo ds:
Thin- ilm low
Mic opola luid
Rough bounda y
Di e en scales
Asymp o ic expansion
Two-scale con e gence
abs ac
Inspi ed by he lub ica ion amewo k, in his pape a mic opola luid low h ough a ough
hin domain is s udied. The domain’s hickness is conside ed as he small pa ame e ε,
while he oughness is de ined by a pe iodical unc ion wi h pe iod o o de ε2. S a ing
om h ee-dimensional mic opola equa ions and using asymp o ic analysis wi h espec
o ε, we o mally de i e he mac oscopic model clea ly de ec ing he e ec s o he speci ic
ugosi y p o ile and luid mic os uc u e. We p o ide he igo ous jus i ica ion o ou
o mally ob ained asymp o ic model by de i ing he e ec i e sys em by means o he
wo-scale con e gence.
©2014 Else ie L d. All igh s ese ed.
1. In oduc ion
The classical lub ica ion p oblem is mainly conce ned wi h he si ua ion in which wo solid su aces being in ela i e
mo ion a e sepa a ed by a hin laye o luid ac ing as a lub ican . Such si ua ion appea s na u ally in applica ions consis ing
o mo ing machine pa s, namely he jou nal bea ings. Fluid ilm bea ings a e machine elemen s whose unc ion is o
p omo e smoo h ela i e mo ion be ween wo su aces and a e c ucial ac o s in limi ing he dissipa ion o ene gy. The
ul ima e goal is ha luid ilm bea ing is well designed so ha he wea is no an issue ( wo su aces a e comple ely sepa a ed
by he lub ican ). Fo ha eason, i is essen ial o unde s and he beha io o he luid ilm in such machine elemen s.
The i s esul goes back o Reynolds and his celeb a ed wo k [1] published in 1886. He s udied he hin ilm low in a
a he heu is ic manne and did no p o ide any ela ion be ween his model and he Na ie –S okes equa ions. The o mal
ela ionship be ween Na ie –S okes equa ions and Reynolds equa ion in a hin domain was es ablished mo e han 60 yea s
la e in [2,3], while he igo ous ma hema ical jus i ica ion o he Reynolds equa ion o a New onian low be ween wo
plain su aces can be ound in [4].
I he gap be ween he mo ing su aces becomes e y small, he expe imen al esul s om he ibology li e a u e (see
e.g. [5–7]) sugges ha he luid’s in e nal s uc u e should be aken in o accoun as well. A possible way o acknowledge
such expe imen al indings is o employ he mic opola luid model. Being o iginally p oposed by E ingen [8] in he 60s,
he heo y o mic opola luids has gained much a en ion since i success ully desc ibes he e ec s o local s uc u e and
mic o-mo ions o he luid elemen s ha canno be cap u ed by he classical Na ie –S okes model. Physically, mic opola
luids ep esen luids consis ing o igid, sphe ical pa icles suspended in a iscous medium, whe e he de o ma ion o luid
∗Co esponding au ho .
E-mail add esses: [email p o ec ed],[email p o ec ed] (I. Pažanin), [email p o ec ed] (F.J. Suá ez-G au).
h p://dx.doi.o g/10.1016/j.camwa.2014.10.003
0898-1221/©2014 Else ie L d. All igh s ese ed.
1916 I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932
pa icles is igno ed. They a e, in ac , non-New onian luids wi h nonsymme ic s ess enso . In iew o ha , he ela ed
ma hema ical model in oduces a new ec o ield, he angula eloci y ield o o a ion o pa icles (mic o o a ion) and
one new ( ec o ) equa ion coming om he conse a ion o he angula momen um. As a esul , a complex coupled sys em
o PDEs is ob ained, ep esen ing a signi ican gene aliza ion o he Na ie –S okes equa ions. We e e he eade o he
monog aph [9] (and he e e ences he ein) p o iding a de ailed de i a ion o he mic opola equa ions om he gene al
cons i u i e laws oge he wi h an ex ensi e e iew o he ma hema ical heo y and he applica ions o his pa icula model.
Enginee ing p ac ice also indica es ha i is o g ea in e es o combine he lub ica ion phenomena wi h he analysis o
he oughness e ec s. Usually i means ha he lowe su ace is assumed o be pe ec ly smoo h, bu he uppe is ough and
desc ibed by a gi en unc ion. Exp essing he bounda y oughness using a pe iodic unc ion, hin- ilm low o New onian
luid has been ex ensi ely s udied o di e en ugosi y p o iles. The classical assump ion is ha he size o he oughness
is o he same o de as he ilm hickness, i.e.
hε(x)=εhx,x
ε,0< ε ≪1.(1)
In such se ing, he e ec i e model u ns ou o be he classical Reynolds equa ion (see e.g. [10,11]) and one needs o compu e
he co ec o s in o de o de ec he oughness-induced e ec s. Same esul is ob ained o hε(x)=εhx,x
εβwi h β < 1
(see [12]). In iew o ha , B esch and co-au ho s [13] in 2010 conside ed a new amewo k, namely
hε(x)=εhx,x
ε2.(2)
As a esul , hey de i ed he asymp o ic model in which an ex a e m (appea ing due o he bounda y oughness) modi ies
he s anda d Reynolds equa ion a he main o de . Whole asymp o ic expansion (a any o de ) o he solu ion has been
igo ously de i ed in [14] p o iding he op imali y wi h espec o he unca ion e o . I is impo an o emphasize ha ,
oughness pa e n desc ibed by hε(x)=εhx,x
εβwi h β > 1 is physically ele an and ealis ic (see e.g. [15]), and,
he e o e, has been s udied o di e en si ua ions in ecen yea s. Focusing on he wall laws, he e ec s o he abo e se ing
on he asymp o ic beha io o he Na ie –S okes sys em ha e been in es iga ed in [16]. Using he asymp o ic app oxima ion
om [13] de i ed o he hyd odynamic pa o he sys em, he oughness e ec s on he hea conduc ion in a hin ilm low
ha e been s udied in [17]. A semilinea pa abolic p oblem in a hin ough domain assuming di e en o de o he pe iod o
oscilla ions on he op and he bo om o he bounda y has been add essed in [18].
Ou goal is o ex end he analysis p esen ed in [13] o a case o lub ica ion wi h incomp essible mic opola luid. The e
a e no many pape s in he exis ing li e a u e dealing wi h he ma hema ical modeling o mic opola luid ilm lub ica ion.
In e es ing esul can be ound in [19] whe e he au ho s conside a speci ic slide - ype bea ing. A e w i ing he go e ning
p oblem in non-dimensional o m, hey o mally ob ain a gene alized e sion o he Reynolds equa ion in a c i ical case when
one o he non-New onian cha ac e is ic pa ame e s has speci ic (small) o de o magni ude. Rigo ous de i a ion o such
esul was b ough 14 yea s la e in [20] o wo-dimensional se ing (see also [21] o mic opola low in a cu ed channel).
The 3D lub ica ion p oblem was ecen ly add essed in [22] and new, second-o de B inkman- ype asymp o ic model has
been p oposed. In he abo e pape s, he oughness e ec s we e no aken in o accoun , i.e. he heigh o he channel is
assumed o be o he o m hε(x)=εh(x). To ou knowledge, he i s (and only) igo ous esul on he mic opola luid
ilm lub ica ion in a hin domain wi h ough bounda y can be ound in he ecen pape by Bouk ouche and Paoli [23]. They
conside a mic opola low in a wo-dimensional domain assuming ha he heigh o he channel is gi en by (1). Employing
wo-scale con e gence echnique, hey de i e he limi p oblem desc ibing he mac oscopic low. In he p esen pape , we
a e going o s udy a mic opola luid low in a h ee-dimensional domain gi en by
Ωε=(x,z)∈R2×R:x∈ω, 0<z<hε(x),(3)
whe e he heigh hεis de ined by (2). F om he poin o iew o asymp o ic analysis, we ind his amewo k mo e challenging
han he classical one (gi en by (1)) due o he echnical di icul ies caused by he speci ic heigh p o ile.
The main p oblem ela ed o a luid low h ough a domain wi h oughness is o deduce in which way he i egula
bounda ies a ec he low. This is especially impo an wi h ega d o nume ical compu a ions: indeed, oughness is in
gene al oo small o be cap u ed by he disc e iza ion g id o he simula ions. To o e come his di icul y, one can employ
he homogeniza ion heo y. In iew o ha , he idea is o eplace he i egula domain by a smoo h one, and hen desc ibe
he a e aged e ec o he oughness in he limi (homogenized) model. Fo ha eason, homogenized models ha e been
o p ac ical in e es in nume ical codes. In ou pa icula case, s a ing wi h o iginal p oblem (6)–(11) posed in hin ough
domain Ωε, we apply he sui able change o a iables, namely Z=z/hε(x), o ans o m Ωεin o Ωwhich is smoo h. Then
by means o a wo-scale con e gence echnique, we ob ain he simpli ied limi p oblem posed in Ω, in which he e ec s o
oughness can be clea ly obse ed (see Sec ion 2.3).
The pape is o ganized as ollows. A e o mula ing he p oblem in Sec ion 2, in Sec ion 3we pe o m a o mal asymp o ic
analysis wi h espec o he small pa ame e ε. In oducing a sui able change o a iables which akes in o accoun he ough
oscilla ions, we ew i e he go e ning p oblem in he ε-independen domain and employ wo-scale expansion echnique.
Since he p oblem is coupled, we cons uc he asymp o ic expansion o he solu ion by simul aneously ea ing bounda y-
alue p oblems o eloci y and o mic o o a ion. As a esul , we ob ain an e ec i e sys em desc ibing he mac oscopic
low and obse ing clea ly he e ec s o he ugosi y p o ile and luids mic os uc u e.
I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932 1917
Fig. 1. The di e en scales ela ed o he domain.
Finally, Sec ion 4is de o ed o a igo ous jus i ica ion o he o mally ob ained asymp o ic model. We apply a con enien
a ian o he wo-scale con e gence and e i y he e ec i e equa ions ob ained in a o mal way. To conclude, we belie e
ha he p esen ed esul could be ins umen al o unde s anding he e ec s o he ough bounda y and luid mic os uc u e
on he lub ica ion p ocess. In iew o ha , mo e e icien nume ical algo i hms could be de eloped imp o ing, hope ully,
he known enginee ing p ac ice.
2. Fo mula ion o he p oblem and he s a emen o he main esul
2.1. The domain
We conside he luid low in he ollowing h ee-dimensional domain
Ωε=(x,z)∈R2×R:x∈ω, 0<z<hε(x).(4)
He e we assume ha ωis a smoo h bounded subse o R2and
hε(x)=εh1(x)+ε2h2x
ε2.(5)
We also de ine Ω=ω× ⟨0,1⟩ ⊂ R2×R, and deno e by T2 he o us o dimension 2.
As we can see, lowe su ace is supposed o be plane, while he oughness o he uppe su ace is desc ibed by he gi en
unc ion hε. The unc ions h1,h2appea ing in (5) a e assumed o be egula : he posi i e unc ion h1∈H2(ω) ep esen s
he main o de pa o he oughness, while he T2-pe iodic unc ion h2∈H2(T2)(wi h 0 as a e age in T2) desc ibes he
oscilla ing pa (see Fig. 1).
2.2. The equa ions and bounda y condi ions
In iew o he applica ion we wan o model, we can assume a small Reynolds numbe and neglec he ine ial e ms in
he go e ning equa ions. Thus, we assume ha he low in Ωεis go e ned by he ollowing linea ized equa ions:
−(ν +ν )1uε+ ∇pε=2ν o wε,(6)
di uε=0,(7)
−(ca+cd)1wε−(c0+cd−ca)∇di wε+4ν wε=2ν o uε.(8)
The unknown unc ions a e uε,wεand pε ep esen ing he eloci y, he mic o o a ion and he p essu e o he luid
espec i ely. Posi i e cons an s ν, ν ,c0,ca,cda e he iscosi y coe icien s: νis he usual kinema ic New onian iscosi y,
while ν ,c0,ca,cda e new iscosi ies connec ed wi h he asymme y o he s ess enso and, consequen ly, wi h he
appea ance o he mic o o a ion ield wε. Fo he sake o no a ional simplici y, ex e nal o ces and momen s a e neglec ed
and luid densi y is assumed o be one.
The aim is o s udy he lub ica ion p ocess whe e wo igid su aces a e in ela i e mo ion and a e sepa a ed by a hin
laye o luid. The e o e, we impose he ollowing bounda y condi ions o he eloci y:
uε=0 o z=hε,uε=g o z=0.(9)
He e g∈R3is a gi en cons an co esponding o he imposed ho izon al eloci y o he plane wall. Ob iously, g·k=0
implying u3
ε|z=0=0. He e and in he sequel (i,j,k)deno es he s anda d Ca esian basis.
Along he la e al bounda y, se e al ypes o bounda y condi ions can be conside ed, depending on he pa icula de ice
o be conside ed. One can use s anda d Di ichle bounda y condi ion o he eloci y (see [11]), mixed (Di ichle –Neumann)
ype condi ion o he eloci y (see [13]) o e en combina ion wi h p essu e bounda y condi ion (see [22]), namely
uε×n=0,pε=qε o x∈∂ω, (10)
o gi en ou e p essu e qε=ε−2qand no mal uni ec o n.
1918 I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932
Finally, o close up he go e ning p oblem, we need o p esc ibe he bounda y condi ions o he mic o o a ion. Though,
ecen ly, some o he ypes o bounda y condi ions o he mic o o a ion can be ound in he ma hema ical li e a u e (see
e.g. [24]), using simple ze o bounda y condi ion s ill seems o be a common p ac ice. The e o e, we impose
wε=0 on ∂Ωε,(11)
meaning ha he luid mic oelemen s canno o a e on he solid su ace.
The exis ence and uniqueness o he solu ion (uε,pε,wε)∈H1(Ωε)3×L2(Ωε)/R×H1
0(Ωε)3 o he desc ibed bounda y-
alue p oblem can be es ablished using s anda d echniques (see e.g. [9,25]). Ou goal he e is o ind he mac oscopic law
desc ibing he e ec i e low in Ωε ia asymp o ic analysis wi h espec o he small pa ame e ε.
2.3. The main esul
We ind he low o be go e ned by he ollowing equa ions



−(ν +ν )∂2
Z +h2
1∇xp+(ν +ν )MZ∂Z =0,
∂Zp=0,
−(ca+cd)∂2
Zw−2ν h1(−∂Z 2i+∂Z 1j)+(c0+2cd)MZ∂Zw=0
wi h
di xh3
1
12(ν +ν )∇xp=di xh1
2CMg.
He e coe icien Mis gi en by
M=T2
|∇Xh2|2dX,
while CM=B
6Awi h
A=1
MeM/21
0
e−M 2/2dξ−1−1
M(eM/2−1)1
0s
0eM(s2− 2)/2dξds
1
0eMs2/2ds ,
B=1
M(eM/2−1)1
1
0eMs2/2ds.
No e ha coe icien s Mand CMa e bo h p o ided in he explici o m and ha hey depend only on he o m o he ugosi ies.
The mic opola na u e o he luid appea s h ough he iscosi y ν+ν , while gis he cons an co esponding o he imposed
ho izon al eloci y o he plane wall. The abo e equa ions ha e been i s o mally ob ained and hen igo ously con i med
ia wo-scale con e gence (see Theo em 1, Sec ion 4).
I is impo an o emphasize ha each unknown in he limi p oblem depends on xand Z, bu he p essu e depends only
on x. Fo ha eason, he abo e p oblem can be seen as linea second-o de ODE wi h espec o Z o and wleading o
explici exp essions o and w(see Sec ions 3.5 and 3.6). Since bo h and wdepend on he p essu e p, i is necessa y o
deduce an equa ion o pas well. We p oceed in a s anda d manne and as a esul ob ain he gene alized Reynolds equa ion
posed in ωsa is ied by p(see Sec ion 3.5). Finally, sol ing he Reynolds equa ion o pwe can deduce bo h he eloci y and
he mic o o a ion w.
To poin ou how he geome y and oughness o he hin domain a ec ou p oblem, le us ecall he e ec i e equa ions
in a hin domain wi hou oughness (see [22]). Using he supe sc ip · o he solu ions o such p oblem, he equa ions
ead 


−(ν +ν )∂2
Z
+h2
1∇x
p=0,
∂Z
p=0,
−(ca+cd)∂2
Z
w−2ν h1(−∂Z 2i+∂Z 1j)=0
wi h
di xh3
1
12(ν +ν )∇x
p=di xh1
2g.
Compa ing he abo e wo sys ems, no ice ha he oughness o he bounda y in oduces a new e m modi ying momen um
equa ions o he eloci y and mic o o a ion h ough he coe icien M, and gi ing a modi ied Reynolds equa ion h ough he
coe icien CM. As in [13], i s ill emains ue he ela ion p=CM
p. The e o e, o ou opinion, his con ibu ion ep esen s
an impo an gene aliza ion o he esul s p o ided in [13,22].
I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932 1919
3. Fo mal asymp o ic analysis
3.1. Rescaling
As a i s s ep, we need o ew i e he s a ing p oblem in he ix (ε-independen ) domain. To accomplish ha , we i s
in oduce a as a iable X=x
ε2cap u ing he oscilla ing phenomena o he hin domain. In iew o ha , he heigh hε
becomes
h(x,X)=εh1(x)+ε2h2(X). (12)
Nex we in oduce a new e ical a iable Z=z
h(x,X)and, co espondingly, he new unknown unc ions: eloci y uε(x,z)=
˜
u(x,X,Z), mic o o a ion wε(x,z)=˜
w(x,X,Z)and p essu e pε(x,z)=p(x,X,Z). In he sequel, we also adop he ollowing
no a ion
˜
u=( , 3)∈R2×R,˜
w=(w, w3)∈R2×R.
The bounda y condi ions sa is ied by eloci y and mic o o a ion a e pe o ming he change o a iables ead as ollows
˜
=0 o Z=1,˜
=g o Z=0,˜
w=0 o Z=0,1,
whe e g∈R3is gi en in (9). Mo eo e , mo i a ed by he pe iodic na u e o h2, we assume ha ˜
u,˜
wand pa e T2-pe iodic
unc ions in he a iable X, i.e.
˜
u(x,X+1,Z)=˜
u(x,X,Z), ˜
w(x,X+1,Z)=˜
w(x,X,Z), p(x,X+1,Z)=p(x,X,Z). (13)
Now we ha e o exp ess each di e en ial ope a o appea ing in Eqs. (6)–(8) acknowledging he abo e change o a iables.
The i s and second de i a i es o he unc ion θε(x,z)=θ(x,X,Z)can be w i en as
∇xθε= ∇xθ+1
ε2∇Xθ−1
h∇h·Z∂Zθ, ∂zθε=1
h∂Zθ,
∆xθε=∆xθ+2
ε2∇x· ∇Xθ+1
ε4∆Xθ−2
ε2
∇h
h·Z∇X∂Zθ−1h
hZ∂Zθ
+|∇h|2
h2Z∂Zθ−2∇h
h·Z∇x∂Zθ+|∇h|2
h2Z2∂2
Zθ+1
h2∂2
Zθ, ∂2
zθε(x,z)=1
h2∂2
Zθ,
whe e
∇h(x,X)=ε∇xh1(x)+ ∇Xh2(X)and 1h(x,X)=ε∆xh1(x)+1
ε2∆Xh2(X). (14)
The change o a iables applied o o a ion and di e gence yields
o ε=1
ε2 o X 3+(∂X1 2−∂X2 1)k+1
h−∂Z 2i+∂Z 1j+ o x 3+(∂x1 2−∂x2 1)k,
di ε=di x +1
ε2di X −1
h∇h·Z∂Z +1
h∂Z 3,
o a ec o unc ion ε(x,z)= (x,X,Z)and wi h
o x( 3)=∂x1 3i+∂x2 3j, o X( 3)=∂X1 3i+∂X2 3j,
di x( )=∂x1 1+∂x2 2,di X( )=∂X1 1+∂X2 2.
Using he abo e exp essions, we deduce
∇di wε= ∇x(di xw)+1
ε2∇X(di xw)−1
h∇h·Z∂Z(di xw)
+1
ε2∇x(di Xw)+1
ε4∇X(di Xw)−1
ε2h∇h·Z∂Z(di Xw)
−1
h1hZ∂Zw+|∇h|2
h2Z∂Zw−1
h∇h·Z∇x∂Zw−1
ε2h∇h·Z∇X∂Zw+|∇h|2
h2Z2∂2
Zw
−1
h2∇h·∂Zw3+1
h∇x∂Zw3+1
ε2h∇X∂Zw3−1
h2∇h·Z∂2
Zw3
+1
h∂Z(di xw)k+1
ε2h∂Z(di Xw)k−1
h∇h·∂Zw k −1
h2∇h·Z∂2
Zw k +1
h2∂2
Zw3k.

1920 I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932
In iew o he p eceding calcula ions, he momen um equa ion (6) has he ollowing o m in an ε-independen domain
Ω×R2= {(x,X,Z)∈R2×R2×R:x∈ω, 0<Z<1}:
(ν +ν )−h2∆x −2
ε2h2∇x· ∇X −1
ε4h2∆X +2
ε2h∇h·Z∇X∂Z +h1hZ∂Z − |∇h|2Z∂Z
+2h∇h·Z∇x∂Z − |∇h|2Z2∂2
Z −∂2
Z +h2∇xp+1
ε2h2∇Xp−h∇hZ∂Zp
=2ν
ε2h2 o Xw3+2ν h−∂Zw2i+∂Zw1j+2ν h2 o xw3,(15)
(ν +ν )−h2∆x 3−2
ε2h2∇x· ∇X 3−1
ε4h2∆X 3+2
ε2h∇h·Z∇X∂Z 3
+h1hZ∂Z 3− |∇h|2Z∂Z 3+2h∇h·Z∇x∂Z 3− |∇h|2Z2∂2
Z 3−∂2
Z 3+h∂Zp
=2ν
ε2h2∂X1w2−∂X2w1+2ν h2∂x1w2−∂x2w1.
The di e gence equa ion (7) in he escaled domain eads:
hdi x +1
ε2hdi X − ∇h·Z∂Z +∂Z 3=0.(16)
Finally, he angula momen um equa ion (8) can be ew i en as ollows:
(ca+cd)−h2∆xw−2
ε2h2∇x· ∇Xw−1
ε4h2∆Xw+2
ε2h∇h·Z∇X∂Zw
+h1hZ∂Zw− |∇h|2Z∂Zw+2h∇h·Z∇x∂Zw− |∇h|2Z2∂2
Zw−∂2
Zw
+(c0+cd−ca)−h2∇x(di xw)−1
ε2h2∇X(di xw)+h∇h·Z∂Z(di xw)
−1
ε2h2∇x(di Xw)−1
ε4h2∇X(di Xw)+1
ε2h∇h·Z∂Z(di Xw)
+h1hZ∂Zw− |∇h|2Z∂Zw+h∇h·Z∇x∂Zw+1
ε2h∇h·Z∇X∂Zw
− |∇h|2Z2∂2
Zw+ ∇h·∂Zw3−h∇x∂Zw3−1
ε2h∇X∂Zw3+ ∇h·Z∂2
Zw3+4ν h2w
=2ν
ε2h2 o X 3+2ν h−∂Z 2i+∂Z 1j+2ν h2 o x 3,(17)
(ca+cd)−h2∆xw3−2
ε2h2∇x· ∇Xw3−1
ε4h2∆Xw3+2
ε2h∇h·Z∇X∂Zw3
+h1hZ∂Zw3− |∇h|2Z∂Zw3+2h∇h·Z∇x∂Zw3− |∇h|2Z2∂2
Zw3−∂2
Zw3+4ν h2w3
+(c0+cd−ca)−h∂Z(di xw)−1
ε2h∂Z(di Xw)+h∇h·∂Zw+ ∇h·Z∂2
Zw−∂2
Zw3
=2ν
ε2h2∂X1 2−∂X2 1+2ν h2∂x1 2−∂x2 1.
3.2. Asymp o ic expansion
Now we o mally expand he unknowns:
(x,X,Z)= 0(x,X,Z)+ε 1(x,X,Z)+ε2 2(x,X,Z)+ · · · ,(18)
3(x,X,Z)= 0
3(x,X,Z)+ε 1
3(x,X,Z)+ε2 2
3(x,X,Z)+ · · · ,(19)
p(x,X,Z)=1
ε2p0(x,X,Z)+1
εp1(x,X,Z)+p2(x,X,Z)+ · · · ,(20)
w(x,X,Z)=w0(x,X,Z)+εw1(x,X,Z)+ε2w2(x,X,Z)+ · · · ,(21)
w3(x,X,Z)=w0
3(x,X,Z)+εw1
3(x,X,Z)+ε2w2
3(x,X,Z)+ · · · .(22)
F om condi ion (13), we assume ha he unc ions i, i
3,pi,wi, wi
3,i=1,2, . . . ,o (18)–(22), a e T2-pe iodic unc ions
in he a iable X.
I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932 1921
The p ocedu e is s anda d: we plug he abo e expansions in o he escaled equa ions (15)–(17) and collec he e ms
wi h equal powe s o ε. Fo ha pu pose, we need o de e mine he asymp o ic beha io o he e ms in ol ing unc ion h.
Taking in o accoun (12)–(14), we deduce
h2=ε2h2
1+2ε3h1h2+ε4h2
2∼O(ε2),
h∇h=εh1∇Xh2+ε2h1∇xh1+ε2h2∇Xh2+ε3h2∇xh1∼O(ε),
h1h=1
εh1∆Xh2+h2∆Xh2+ε2h1∆xh1+ε3h2∆xh1∼O1
ε,
|∇h|2= |∇Xh2|2+2ε∇xh1∇Xh2+ε2|∇xh1|2∼O(1).
3.3. Main o de e m
We s a by subs i u ing he expansions (18)–(22) in o momen um and di e gence equa ion (15)–(16). The leading o de
e ms a e gi en by
1
ε2: −(ν +ν )h2
1∆X 0+h2
1∇Xp0=0,
1
ε2: −(ν +ν )h2
1∆X 0
3=0,
1
ε:h1di X 0=0.
(23)
No e ha (23)1,(23)3is, in ac , a S okes sys em o ( 0,p0)wi h espec o X. On he o he hand, he hi d componen 0
3
sa is ies a simple Laplace equa ion (23)2, again wi h espec o X. The e o e, aking in o accoun he bounda y condi ions
wi h espec o X, we deduce
∇X 0=0,∇Xp0=0,∇X 0
3=0.(24)
The main o de e m om he angula momen um equa ion (17) yields
1
ε2: −(ca+cd)h2
1∆Xw0−(c0+cd−ca)h2
1∇X(di Xw0)=0,
1
ε2: −(ca+cd)h2
1∆Xw0
3=0.
(25)
Simila ly as abo e, we conclude
∇Xw0=0,∇Xw0
3=0.(26)
As we can see, main o de e ms led o a decoupled p oblem: (23) in ol es only he eloci y and p essu e, while (25) is
sa is ied only by he mic o o a ion. Consequen ly, we es ablished ha he leading o de e ms 0, 0
3,w0, w0
3do no depend
on he as a iable X.
3.4. Lowe o de e ms
We con inue he compu a ion and w i e he p oblems sa is ied by he lowe -o de e ms in he escaled equa ions. In
iew o (24) and (26), om (15)–(16) we ge
1
ε: −(ν +ν )h2
1∆X 1+h2
1∇Xp1+(ν +ν )h1∆Xh2Z∂Z 0−h1∇Xh2Z∂Zp0=0,
1
ε: −(ν +ν )h2
1∆X 1
3+(ν +ν )h1∆Xh2Z∂Z 0
3+h1∂Zp0=0,
1:h1di X 1− ∇Xh2·Z∂Z 0+∂Z 0
3=0.
(27)
Obse e ha he e is no con ibu ion o he e ms in ol ing mic o o a ion ield so we can p oceed simila ly as in [13].
We compu e he mean alue in Xo Eq. (27)3. Since h1, 0, 0
3do no depend on X, we ind ∂ 0
3=0. Using he bounda y
condi ions o 0
3a he op and he bo om o he domain, we deduce ha
0
3=0.
Then he ee-di e gence condi ion w i en a o de εgi es
h1di X 1= ∇Xh2·Z∂Z 0.(28)
1922 I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932
Taking he mean alue in Xo Eq. (27)2, we conclude ha
∂Zp0=0 and nex ∇X 1
3=0.(29)
Taking he cu lXope a o o he ho izon al componen in Eq. (27)1, we ge
h1cu lX∆X 1= ∇⊥
X∆Xh2·Z∂Z 0.
Since h1and 0do no depend on X,
h1cu lX 1= ∇⊥
Xh2·Z∂Z 0.(30)
We ha e ∆X 1= ∇Xdi X 1− ∇⊥
Xcu lX 1and ∇X∇Xh2− ∇⊥
X∇⊥
Xh2=(∆Xh2)Id, hus using exp essions (28) and (30), we
ob ain
h2
1∆X 1=h1∆Xh2Z∂Z 0.(31)
The e o e, Eq. (27) can be w i en as
∇Xp1=0.
The wo nex e ms om he mic o o a ion expansion a e gi en by (see (17)):
1
ε:(ca+cd)∆Xw1+(c0+cd−ca)∇X(di Xw1)=(c0+2cd)∆Xh2
h1
Z∂Zw0,
1
ε:(ca+cd)∆Xw1
3=(ca+cd)∆Xh2
h1
Z∂Zw0
3,
(32)
1:(ca+cd)h2∆Xh2Z∂Zw0− |∇Xh2|2Z∂Zw0− |∇Xh2|2Z2∂2
Zw0−∂2
Zw0
+(c0+cd−ca)h2∆Xh2Z∂Zw0− |∇Xh2|2Z∂Zw0− |∇Xh2|2Z2∂2
Zw0
+ ∇Xh2·∂Zw0
3+ ∇Xh2·Z∂2
Zw0
3+(ca+cd)−2h1h2∆Xw1+2h1∇Xh2Z∇X∂Zw1+h1∆Xh2Z∂Zw1
+(c0+cd−ca)−2h1h2∇X(di Xw1)+h1∇Xh2Z∂Z(di Xw1)+h1∆Xh2Z∂Zw1
+h1∇Xh2Z∇X∂Zw1−h1∇X∂Zw1
3−(ca+cd)h2
1∆Xw2−(c0+cd−ca)h2
1∇X(di Xw2)=0,
1:(ca+cd)h2∆Xh2Z∂Zw0
3− |∇Xh2|2Z∂Zw0
3− |∇Xh2|2Z2∂2
Zw0−∂2
Zw0
3
+(c0+cd−ca)∇Xh2·Z∂2
Zw0−∂2
Zw0
3
+(ca+cd)−2h1h2∆Xw1
3+2h1∇Xh2·Z∇X∂Zw1
3+h1∆Xh2Z∂Zw1
3
−(c0+cd−ca)h1∂Z(di Xw1)−(ca+cd)h2
1∆Xw2
3=0.
(33)
Le us p o e ha w0=0. Fo ha pu pose, we ake he mean alue wi h espec o Xin (33)1and ca e ully ea each e m
o his equa ion:
(i) Te ms in ol ing w0, w0
3: since w0, w0
3do no depend on X, we ha e
(ca+cd)T2h2∆Xh2Z∂Zw0− |∇Xh2|2Z∂Zw0− |∇Xh2|2Z2∂2
Zw0−∂2
Zw0dX
+(c0+cd−ca)T2−h2∆Xh2Z∂Zw0− |∇Xh2|2Z∂Zw0− |∇Xh2|2Z2∂2
Zw0+ ∇Xh2·∂Zw0
3+ ∇Xh2·Z∂2
Zw0
3dX
=(ca+cd)T2
h2∆Xh2dXZ∂Zw0−T2
|∇Xh2|2dXZ∂Zw0−T2
|∇Xh2|2dXZ2∂2
Zw0−∂2
Zw0
+(c0+cd−ca)−T2
h2∆Xh2dXZ∂Zw0−T2
|∇Xh2|2dXZ∂Zw0
−T2
|∇Xh2|2dXZ2∂2
Zw0+T2
∇Xh2·∂Zw0
3dX+T2
∇Xh2·Z∂2
Zw0
3dX
=(ca+cd)T2−T2
|∇Xh2|2dXZ∂Zw0−T2
|∇Xh2|2dXZ∂Zw0−T2
|∇Xh2|2dXZ2∂2
Zw0−∂2
Zw0
+(c0+cd−ca)−T2
|∇Xh2|2dXZ∂Zw0−T2
|∇Xh2|2dXZ∂Zw0−T2
|∇Xh2|2dXZ2∂2
Zw0dX
= −2(c0+2cd)MZ∂Zw0−(c0+2cd)MZ2∂2
Zw0−(ca+cd)∂2
Zw0.
I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932 1923
He e and in he sequel we in oduce
M=T2
|∇Xh2|2dX (34)
as a coe icien depending on he conside ed ugosi y p o ile.
(ii) Using (32)1and he ac ha w0does no depend on X, we ob ain
T2−2(ca+cd)h1h2∆Xw1−2(c0+cd−ca)h1h2∇X(di Xw1)dX
= −2T2
h1h2(ca+cd)∆Xw1+(c0+cd−ca)∇X(di Xw1)dX
= −2T2
(c0+2cd)h1h2
∆Xh2
h1
Z∂Zw0dX
= −2(c0+2cd)T2
h2∆Xh2dXZ∂Zw0
=2(c0+2cd)T2
|∇Xh2|2dXZ∂Zw0
=2(c0+2cd)MZ∂Zw0.
(iii) The emaining e ms in ol ing w1, w1
3: employing again (32)1and he ac ha h1and w0do no depend on X, we
ge
(ca+cd)T22h1∇Xh2·Z∇X∂Zw1+h1∆Xh2Z∂Zw1dX
+(c0+cd−ca)T2h1∇Xh2Z∂Z(di Xw1)+h1∆Xh2Z∂Zw1+h1∇Xh2Z∇X∂Zw1−h1∇X∂Zw1
3dX
=(ca+cd)T2−2h1h2·Z∂Z(∆Xw1)+h1h2Z∂Z(∆Xw1)dX
+(c0+cd−ca)T2−h1h2Z∂Z∇X(di Xw1)+h1h2Z∂Z(∆Xw1)−h1h2Z∂Z(∆Xw1)dX
= − T2
h1h2Z∂Z(ca+cd)∆Xw1+(c0+cd−ca)∇X(di Xw1)dX
= − T2
h1h2Z∂Z(c0+2cd)∆Xh2
h1
Z∂Zw0dX
= − T2
Z∂Z(c0+2cd)h2∆Xh2Z∂Zw0dX
= −(c0+2cd)T2
h2∆Xh2dXZ∂Zw0−(c0+2cd)T2
h2∆Xh2dXZ2∂2
Zw0
=(c0+2cd)T2
|∇Xh2|2dXZ∂Zw0+(c0+2cd)T2
|∇Xh2|2dXZ2∂2
Zw0
=(c0+2cd)MZ∂Zw0+(c0+2cd)MZ2∂2
Zw0.
(i ) Te ms in ol ing w2: in eg a ing by pa s, because he unc ion h1depends only on xand he unknown w2is pe iodic
in X, i ollows
−T2
(ca+cd)h2
1∆Xw2dX −T2
(c0+cd−ca)h2
1∇X(di Xw2)dX =0.
Adding all con ibu ions (i)–(i ), we easily ob ain
−(ca+cd)∂2
Zw0+(c0+2cd)MZ∂Zw0=0.
Combining he abo e equa ion wi h he co esponding bounda y condi ion, namely w0|Z=0,1=0 inally gi es
w0=0.
P oceeding analogously in (33)2, we de i e he equa ion sa is ied by w0
3:
−(c0+2cd)∂2
Zw0
3+(ca+cd)MZ∂Zw0
3=0
1930 I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932
+Ω
ε∆xh1+1
ε∆Xhε
2
h1+εhε
2
Z∂Zwε·φε−Ω
ε2|∇xh1+1
ε∇Xhε
2|2
(h1+εhε
2)2Z∂Zwεφε
−Ω
ε2∇xh1+1
ε∇Xhε
2
h1+εhε
2
· ∇xwε∂Z(Zφε)+Ω
ε2|∇xh1+1
ε∇Xhε
2|2
(h1+εhε
2)2∂wε·∂Z(Z2φε)
−Ω
∇xh1+1
ε∇Xhε
2
(h1+εhε
2)2·ε∂Zw3,εφε+Ω
1
(h1+εhε
2)2∇xw3,ε∂Zφε−Ω
∇xh1+1
ε∇Xhε
2
(h1+εhε
2)2ε∂Zw3,ε∂Z(Zφε)
=2ν Ω
1
h1+εhε
2
ε−∂Z 2,ε i+∂Z 1,ε jφε+2ν Ω
ε2 o x 3,εφε.
Passing o he limi , we ge
(ca+cd)ΩT2
∇Xw1· ∇Xφ+(ca+cd)ΩT2
∆Xh2
h1
Z∂Zw0φ
×(c0+cd−ca)ΩT2
di Xw1di Xφ+(c0+cd−ca)ΩT2
∆Xh2
h1
Z∂Zw0φ=0,
o any φ∈D(Ω;C1(T2)). This is equi alen o (61). We p oceed analogously wi h Eq. (53) in o de o deduce he ela ion
(62).
Finally, om Lemma 4 we can easily ge he addi ional esul o he p essu e:
Lemma 6. The p essu e is such ha limε→0Ωεpεdi X(φ) =0 o any φ∈L2(Ω;H1(T2)).
4.3.2. Passing o he limi
Now we a e in posi ion o pass o he limi in he escaled equa ions (49)–(53). Following same a gumen s as in
[13, Sec. 5.2 and 5.3], om di e gence and momen um equa ion i is s aigh o wa d o ob ain he weak o mula ions
co esponding o (57)–(59). I emains o e i y Eq. (60). Le us s a wi h he equa ion o w3,ε o con i m ha w1
3=0.
We employ εφ(x,Z),φ∈D(Ω)as a es unc ion in (53). Using he iden i y
1
h1h−1
h2|∇h|2=di 1
h∇h,
we ha e
(ca+cd)Ω
ε∇xw3,ε · ∇xφε−(ca+cd)Ω
ε
h1+εhε
2∇xh1+1
ε∇Xh2· ∇xw3,ε∂Z(Zφ)
+(ca+cd)Ω
ε|∇xh1+1
ε∇Xhε
2|2
(h1+εhε
2)2∂Zw3,ε ·∂Z(Z2φε)+(ca+cd)Ω
1
(h1+εhε
2)2
1
ε∂Zw3,ε ·∂Zφε
×4ν Ω
εw3,εφε+(c0+cd−ca)Ω
1
h1+εhε
2
di xwε∂Zφε−(c0+cd−ca)Ω
ε∇xh1+1
ε∇Xhε
2
h1+εhε
2
·wε∂Z(φε)
−(c0+cd−ca)Ω
ε∇xh1+1
ε∇Xhε
2
h1+εhε
2
·1
ε∂Zwε∂Z(Zφε)+(c0+cd−ca)Ω
1
(h1+εhε
2)2
1
ε∂Zw3,ε∂Zφε
=2ν Ω
ε∂x1 2,ε −∂x2 1,εφε.
Passing o he limi and aking in o accoun ha w1
3=w1
3(x,Z), we ob ain
−(ca+cd)ΩT2
1
h1
∇Xh2∇Xw2
3∂Z(Zφ) +(ca+cd)ΩT2
|∇Xh2|2
h2
1
∂Zw1
3∂Z(Z2φ)
+(ca+cd)ΩT2
1
h2
1
∂Zw1
3∂Zφ+(c0+cd−ca)ΩT2
1
h2
1
∂Zw1
3∂Zφ=0.
Using ela ion (62) om Lemma 5, i can be easily e i ied ha he abo e ela ion is, in ac , he ene gy o mula ion
co esponding o
−(ca+cd)T2
h2∆Xh2dX1
h2
1
Z∂Z(Z∂Zw1
3)−(ca+cd)T2
|∇Xh2|2dX1
h2
1
Z2∂Z(∂Zw1
3)
−(c0+2cd)T2
1
h2
1
∂2
Zw1=0.

I. Pažanin, F.J. Suá ez-G au / Compu e s and Ma hema ics wi h Applica ions 68 (2014) 1915–1932 1931
In eg a ing by pa s gi es
−(c0+2cd)T2
1
h2
1
∂2
Zw1+(ca+cd)T2
|∇Xh2|2dX1
h2
1
Z2∂Z(∂Zw1
3)=0.
Obse e ha his co esponds o Eq. (47) ob ained in a o mal way. In iew o he ze o bounda y condi ion o mic o o a ion,
we conclude w1
3=0.
Le us p oceed wi h equa ion o wε. Analogously, we mul iply Eq. (52) by εφ(x,Z),φ∈D(Ω) o ob ain
(ca+cd)Ω
ε∇xwε· ∇xφε−(ca+cd)Ω
ε
h1+εhε
2∇xh1+1
ε∇Xh2· ∇xwε∂Z(Zφ)
+(ca+cd)Ω
ε|∇xh1+1
ε∇Xhε
2|2
(h1+εhε
2)2∂Zwε·∂Z(Z2φε)+(ca+cd)Ω
1
(h1+εhε
2)2
1
ε∂Zwε·∂Zφε
×4ν Ω
εwεφε+(c0+cd−ca)Ω
di xwε(εdi xφε)−(c0+cd−ca)Ω
ε∇xh1+1
ε∇Xhε
2
h1+εhε
2
·di xwε∂Z(Zφε)
+(c0+cd−ca)Ω
ε|∇x+1
ε∇Xhε
2|2
(h1+εhε
2)2∂Zwε·∂Z(Z2φε)+(c0+cd−ca)Ω
ε∇x+1
ε∇Xh2
(h1+εhε
2)2
1
ε∂Zw3,εφ
+(c0+cd−ca)Ω
1
(h1+εhε
2)2
1
ε∂Zw3,ε∇xφ−(c0+cd−ca)Ω
∇x+1
ε∇Xh2
(h1+εhε
2)2
1
ε∂Zw3,ε∂Z(Zφ)
=2ν Ω
1
h1+εhε
2−∂Z 2,ε i+∂Z 1,ε jφε+2ν Ω
ε o x 3,εφε.
Passing o he limi and aking in o accoun ha w1=w1(x,Z)and w1
3=0 gi e
−(ca+cd)ΩT2
1
h1
∇Xh2· ∇Xw2∂Z(Zφ) +(ca+cd)ΩT2
|∇Xh2|2
(h1)2∂Zw1∂Z(Z2φ)
+(ca+cd)ΩT2
1
(h1)2∂Zw1·∂Zφ−(c0+cd−da)ΩT2
1
h1
∇Xh2·di Xw2∂Z(Zφ)
+(c0+cd−ca)ΩT2
|∇Xh2|2
(h1)2∂Zw1∂Z(Z2φ) =2ν ΩT2
1
h1
(−∂Z 0
2i+∂Z 0
1j)φ.
Using (61) om Lemma 5, i can be easily e i ied ha he la e is he ene gy o mula ion co esponding o
−(c0+2cd)T2
h2∆Xh21
h2
1
Z∂Z(Z∂Zw1)−(ca+cd)T2
|∇Xh2|2dX1
h2
1
Z2∂Z(∂Zw1)
−(c0+cd−ca)T2
|∇Xh2|21
h2
1
Z2∂Z(∂Zw1)−(ca+cd)
h2
1
∂2
Zw1=2ν
h1
(−∂Z 0
2i+∂Z 0
1j).
Finally, a e in eg a ing by pa s, we ge
−(ca+cd)∂2
Zw1+(c0+2cd)T2
|∇Xh2|2dXZ∂Zw1=2ν h1(−∂Z 0
2i+∂Z 0
1j)
which co esponds o (60).
Acknowledgmen s
The i s au ho o his wo k has been suppo ed by he C oa ian Science Founda ion (scien i ic p ojec 3955: Ma hema ical
modeling and nume ical simula ions o p ocesses in hin o po ous domains). The second au ho has been suppo ed by he
p ojec s MTM2011-24457 o he Minis e io de Economía y Compe i i idad and FQM309 o he Jun a de Andalucía.
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