On he Na ie bounda y condi ion o
iscous luids in a hin domain
co e ed by e y small aspe i ies
F.J. Su´
a ez-G au
Depa amen o de Ma em´
a icas, Uni e sidad de Huel a, Spain
[email p o ec ed]
1. In oduc ion
FOR a iscous luid in an open se o R3wi h a ugous bounda y, i is known ha i he
no mal eloci y anishes on he bounda y (Na ie condi ion), hen he luid beha es as
i he whole eloci y ec o anishes on he bounda y (adhe ence condi ion). This gi es a
ma hema ical explana ion o why i is usual o a iscous luid o impose he adhe ence con-
di ion. The equi alence be ween he Na ie and adhe ence condi ions was p o ed in [2] o a
pe iodic ough bounda y o small pe iod εand ampli ude ε. In [3] i was conside ed he case
o a weak oughness, namely he bounda y was desc ibed by a pe iodic unc ion o small
pe iod εand ampli ude δε, wi h δε/ε con e ging o ze o.
Ou aim in he p esen wo k is o s udy he ela ion be ween he Na ie and adhe ence con-
di ions in he case o a domain o small heigh ε. Namely, o a Lipschi z bounded open se
ω⊂R2and a unc ion Ψin W2,∞
loc (R2), pe iodic o pe iod Z0= (−1/2,1/2)2, we de ine Ωεby
Ωε=x= (x0, x3)∈ω×R:−δεΨx0
ε< x3< ε,(1)
whe e he pa ame e s ε,δεa e chosen non-nega i e and sa is ying
lim
ε→0
ε
ε= 0,lim
ε→0
δε
ε= 0, i.e. δε εε.
We conside a luid sa is ying he S okes sys em in Ωε, he Na ie condi ion on he ough
bounda y
Γε=x= (x0, x3)∈ω×R:x3=−δεΨx0
ε (2)
and ( o simpli y) he adhe ence condi ion on he es o he bounda y ∂Ωε Γε,









−µ∆uε+∇pε= in Ωε,di uε= 0 in Ωε,
uε= 0 on ∂Ωε Γε,
uε·ν= 0 on Γε, µ∂uε
∂ν pa allel o νon Γε.
(3)
He e = ( 0, 3)∈L2(ω)3,νdeno es he uni a y ou side no mal ec o o Ωεin Γεand µ > 0
co esponds o he iscosi y o he luid.
I is well known ha (3) has a unique solu ion (uε, pε)∈H1(Ωε)3×L2
0(Ωε)(L2
0(Ωε)deno es
he space o unc ions in L2(Ωε)whose in eg al in Ωεis ze o). Mo eo e , we can show he
ollowing es ima es
−
ZΩε|uε|2dx ≤Cε4,−
ZΩε|Duε|2dx ≤Cε2,−
ZΩε|pε|2dx ≤C. (4)
Ou pu pose is o s udy he asymp o ic beha io o his sys em when ε ends o ze o. We
show ha i depends on
λ= lim
ε→0
δε
3
2
ε
√ε∈[0,+∞].(5)
2. Changes o a iables
OUR aim is o s udy he asymp o ic beha io o uεand pεwhen ε ends o ze o. Fo his
pu pose, we use a sui able combina ion o wo changes o a iables:
(1) Fa o he ough bounda y Γεwe use a dila a ion in he a iable x3in o de o ha e he
unc ions de ined in an open se o ixed heigh . Namely, we ake Ω = ω×(0,1) and we de ine
˜
uε∈H1(Ω)3,˜
pε∈L2
0(Ω) by
˜
uε(y) = uε(y0, εy3),˜
pε(y) = pε(y0, εy3),a.e. y∈Ω.(6)
(2) Nea Γεwe use an o iginal adap a ion ([3]) o he Un olding Me hod ([1], [5]), which is
e y ela ed o he wo-scale con e gence me hod.
3. Main Resul
Le (uε, pε)∈H1(Ωε)3×L2
0(Ωε)be he solu ion o he S okes sys em (3) and le ˜
uε,˜
pεbe
de ined by (6). Then, he e exis 0∈H1(0,1; L2(ω))2,w∈H2(0,1; H−1(ω)) and p∈L2
0(ω),
whe e pdoes no depend on y3, such ha , up o a subsequence,
˜
u0
ε
ε2* 0in H1(0,1; L2(ω))2,˜
uε,3
ε3* w in H2(0,1; H−1(ω)),
˜
pε* p in L2(Ω).
Acco ding o he alue o λde ined by (5), we ob ain he di e en exp essions o 0and w
depending on pwhich sa is ies a Reynolds equa ion:
(i)I λ= +∞, hen deno ing by PW⊥ he o hogonal p ojec ion om R2 o he o hogonal o
he space W={∇z0Ψ(z0)∈R2:z0∈Z0}, we ha e ha 0and pa e gi en by
0(y) = (y3−1)
2µy3I+PW⊥∇y0p(y0)− 0(y0),a.e. y∈Ω,
−di y01
3I+PW⊥(∇y0p− 0)= 0 in ω, 1
3I+PW⊥(∇y0p− 0)·ν= 0 on ∂ω.
Mo eo e , he dis ibu ion wis gi en by w(y) = −Zy3
0di y0 (y0, s)ds, in Ω.(7)
(ii)I λ∈(0,+∞), hen de ining (b
φi,bqi),i= 1,2, as solu ions o he S okes sys ems









−µ∆zb
φi+∇zbqi= 0 in R2×(0,+∞),di zb
φi= 0 in R2×(0,+∞),
b
φi
3(z0,0) + ∂ziΨ(z0)=0, ∂z3(b
φi)0(z0,0) = 0,b
φi(., z3),bqi(., z3)pe iodic o pe iod Z0,
Dzb
φi∈L2(Z0×(0,+∞))3×3,bqi∈L2(Z0×(0,+∞)),
and R∈R2×2by Rij =µZZ0×(0,+∞)
Dzb
φi:Dzb
φjdz, ∀i, j ∈ {1,2},we ha e
0(y) = (y3−1)
2µ
y3I+ I+λ2
µR!−1
∇y0p(y0)− 0(y0),a.e. y∈Ω,
whe e psa is ies 














−di y0

1
3I+ I+λ2
µR!−1
(∇y0p− 0)
= 0 in ω,

1
3I+ I+λ2
µR!−1
(∇y0p− 0)·ν= 0 on ∂ω.
Mo eo e , he dis ibu ion wis gi en by (7).
(iii)I λ= 0, hen 0(y) = (y2
3−1)
2µ(∇y0p(y0)− 0(y0)),a.e. y∈Ω,
whe e psa is ies −∆y0p=−di y0 0in ω, ∂p
∂ν = 0·νon ∂ω.
Mo eo e , he dis ibu ion wis ze o.
4. Conclusions
•Fo λ= +∞, he main esul shows ha uε,pεbeha e as i in (3) we had assumed ha Γε
was he plane bounda y {x3= 0}and ha he bounda y condi ion on Γεwas
uε∈W⊥×{0}on Γε, ∂3u0
ε∈W. (8)
In pa icula , i Wag ees wi h R2we deduce ha he Na ie condi ion in (3) is equi alen o
he adhe ence condi ion uε= 0 on {x3= 0}.
•Fo λ∈(0,+∞), he esul shows ha he asymp o ic beha io o uεand pεis he same ha
i Γεwas he plane bounda y {x3= 0}and he bounda y condi ion on Γεwas
uε,3= 0 on Γε,−µ∂3u0
ε+λ2Ru0
ε= 0 on Γε,(9)
i.e. al hough he oughness is no s ong enough o deduce ha he Na ie condi ion on Γεis
equi alen o (8), i is su icien o p o ide he ic ion coe icien λ2Ru0
εin (9).
•Fo λ= 0, he oughness is so weak ha uεand pεbeha e as i Γεwas plane.
The c i ical size λ∈(0,+∞)can be conside ed as he gene al one. In ac , he cases λ= 0
and λ= +∞can be ob ained om his one by aking he limi when λ ends o ze o and
in ini y espec i ely.
Re e ences
[1]T. A bogas , J. Douglas, U. Ho nung, De i a ion o he double po osi y model o single phase low ia homogeniza ion heo y, SIAM J. Ma h. Anal. 21 (1990) 823-836.
[2]J. Casado-D´
ıaz, E. Fe n´
andez-Ca a, J. Simon, Why iscous luids adhe e o ugose walls: A ma hema ical explana ion, J. Di e en ial Equa ions 189 (2003) 526-537.
[3]J. Casado-D´
ıaz, M. Luna-Laynez, F.J. Su´
a ez-G au, Asymp o ic beha io o a iscous luid wi h slip bounda y condi ions on a sligh ly ough wall, M3AS 20 (2010) 121-156.
[4]J. Casado-D´
ıaz, M. Luna-Laynez, F.J. Su´
a ez-G au, A iscous luid in a hin domain sa is ying he slip condi ion on a sligh ly ough bounda y, C. R. Acad. Sci. Pa is, Se . I (2010).
[5]D. Cio anescu, A. Damlamian, G. G iso, Pe iodic un olding and homogeniza ion, C.R. Acad. Sci. Pa is, Se . I 335 (2002) 99-104.