J. Di e en ial Equa ions 189 (2003) 526–537
Why iscous fluids adhe e o ugose walls:
A ma hema ical explana ion
Juan Casado-D!
ıaz,
a
En ique Fe na
´ndez-Ca a,
a,
* and
Jacques Simon
b
a
Dp o. E.D.A.N., Uni . de Se illa, Ap do. 1160, 41080 Se illa, Espan˜a, Spain
b
CNRS, Labo a oi e de Ma he
´ma iques Applique
´es, Uni . Blaise Pascal (Cle mon -Fe and 2),
63177 Aubie
` e cedex, F ance
Recei ed Oc obe 12 2001; e ised Ap il 29 2002
Abs ac
The main pu pose o his pape is o jus i y igo ously he ollowing asse ion: A iscous
fluid canno slip on a wall co e ed by mic oscopic aspe i ies because, due o he iscous
dissipa ion, he su ace i egula i ies b ing o es he fluid pa icles in con ac wi h he wall. In
ma hema ical e ms, his co esponds o an asymp o ic p ope y es ablished in his pape o
any amily o fields ha slip on oscilla ing bounda ies and emain uni o mly bounded in he
H1-no m.
2002 Else ie Science (USA). All igh s ese ed.
1. In oduc ion
This pape is de o ed o jus i y igo ously he ac ha , asymp o ically, a fluid
canno slip on a wall co e ed by mic oscopic aspe i ies: he slip condi ion, i.e. he
equi emen
un¼0 on he wall;
whe e uis he eloci y and n¼nðxÞis a no mal ec o a a bounda y poin x;which
exp esses he ac ha he wall is no pe meable o he fluid pa icles, p o ides
su ficien in o ma ion o ensu e ha , as he size o aspe i ies goes o 0, he fluid
*Co esponding au ho . Fax: +34-954-55-28-98.
E-mail add esses: [email p o ec ed] (J. Casado-D!
ıaz), [email p o ec ed] (E. Fe n!
andez-Ca a),
jacques.simon@ma h.uni -bpcle mon . (J. Simon).
0022-0396/03/$ - see on ma e 2002 Else ie Science (USA). All igh s ese ed.
PII: S 0 0 2 2 - 0396(02)00115-8
sa isfies he no-slip condi ion, i.e.
u¼0 on he wall:
This was no iced and jus ified o a 2D pe iodic S okes flow in [11] and was
ma hema ically p o ed o a 3D pe iodic Na ie –S okes flow in [1]. Howe e , he
pe iodici y o he flow a he mic oscopic scale assumed in hese pape s is e y
es ic i e. Indeed, i p e en s any o ex o any o he s uc u e la ge han aspe i ies
o occu and i implies ha he mean eloci y o e a pe iod is a Coue e flow ( his
enables a sa is ac o y analysis in his case wi h a pa icula p oo based on scaling
a gumen s, see [1]).
In he p esen pape , we will gi e a ma hema ical p oo o he p e ious asse ion
o any 3D flow wha e e he go e ning equa ion (in ac , no equa ion is p esc ibed).
This can be iewed as a p ope y o he limi u0o a amily o ec o fields ue ha slip
on a bounda y co e ed by aspe i ies o size e;wi h an ens ophy Rj uej2dx ha
emains bounded as e-0 (Theo em 1).
Roughly speaking, his is due o he ac ha sliping wi h a non-ze o eloci y
dissipa es ene gy on aspe i ies because he di ec ion o eloci y suddenly a ies as he
slope does. Fo ins ance, in a 2D domain wi h a se a ed bounda y whose slope is
al e na ely þ1 and 1;i he ho izon al eloci y is ; hen he e ical eloci y is
al e na ely þ and  :When he size eo aspe i ies goes o 0, he ene gy dissipa ed
by each aspe i y goes o 0 bu no as enough o compensa e he ac ha he e a e
many o hem. The e o e, he o al dissipa ion g ows o infini y and he unique
possibili y o ens ophy o be uni o mly bounded is ha he limi eloci y anishes
on he wall. A igo ous o mula ion o his asse ion will be gi en in (8).
We will also p o e ha ou gene al esul applies o a flow go e ned by he
Na ie –S okes equa ions oge he wi h Na ie ’s law
un¼0;ðsnÞ an þku¼0;
whe e sdeno es he s ess enso and he subsc ip an deno es he angen ial
componen , i.e. an ¼ ð nÞn o any ec o field :O cou se, he second
p e ious equali y means ha he ic ion o ces on he wall a e p opo ional o he
angen ial eloci y. Indeed, in his si ua ion he ens ophy emains bounded as e-0
and, he e o e, he limi eloci y u0 anishes on he limi bounda y wha e e he
ic ion coe ficien k(see Theo em 2). This gene alizes, o non-pe iodic flows, he
abo e-men ioned esul s o [1,11].
I is wo h men ioning ha his esul is in con adic ion wi h a s a emen in [8],
bu he a gumen used in ha e e ence is alse, as we will explain in Rema k 5, a he
end o Sec ion 4.
Ou a gumen elies on he in e nal iscous dissipa ion in he fluid and he
geome y o he domain only. I does no equi e any dissipa ion o ene gy due o he
ic ion (o molecula in e ac ion) o he fluid pa icles in con ac wi h he solid
walls.
J. Casado-Dı´az e al. / J. Di e en ial Equa ions 189 (2003) 526–537 527
The e ec i e ela i e impo ance o su ace oughness and fluid/solid molecula
in e ac ions is discussed in [14]. The e, he au ho s show ha oughness domina es
excep o e y smoo h walls. The eade is e e ed o [5,7] o an analysis o
molecula in e ac ion by molecula dynamics simula ion and o [3] o a simila
analysis in he case o a wo-componen fluid.
The flow a he su ace o a po ous medium is ex ensi ely discussed in [6] and
e e ences he ein. In his case ou a gumen does no apply, since he slip condi ion
un¼0 is no imposed. In pa icula , we do no find in he limi he no-slip
condi ion when a ugose in e ace is modeled by Fou ie ’s law
snþku¼0 on he wall
(see [2], whe e a homogeneized ic ion coe ficien k0is ob ained in he limi ).
Le us finally men ion ha many physical and nume ical expe imen s ha e shown
ha , when a fluid flows be ween wo pla es, he occu ence o aspe i ies on he walls
is no i ele an . In pa icula , i is known ha small ible s ( iny aspe i ies pa allel o
he flow) can be used o educe conside ably he d ag expe ienced by he fluid; see
[4,12] and e e ences he ein.
This pape is o ganized as ollows. The main esul (Theo em 1) is s a ed and
commen ed in Sec ion 2. I is p o ed in Sec ion 3. Finally, Sec ion 4 is conce ned
wi h he applica ion o Theo em 1 o a iscous fluid nea a wall wi h aspe i ies.
2. Main esul
Le us now p esen ou main esul wi h p ecision. Le SCR2be a bounded open
se and assume ha , o each ewi h 0oepe0; he unc ion eis gi en by
eðx0Þ¼ 0ðx0ÞþeZ x0
e

;
whe e 0AC1ð%
SÞ; 0ðx0ÞXa>0 and ZAC1ðR2Þis a pe iodic unc ion o pe iod ðc1;c2Þ
in he a iable y0¼x0=e:Le Gebe he open se
Ge¼ xAR3:x0AS;0ox3o eðx0Þg
and le us pu
Re¼ xAR3:x0AS;x3¼ eðx0Þg
( he oscilla ing piece o bounda y). We also se
G0¼ xAR3:x0AS;0ox3o 0ðx0Þg
( he limi domain) and
R0¼ xAR3:x0AS;x3¼ 0ðx0Þg:
J. Casado-Dı´az e al. / J. Di e en ial Equa ions 189 (2003) 526–537528
Assume ha o each ewe ha e ueAðH1ðGeÞÞ3;wi h
ZGe
j uej2dxpb;ð1Þ
whe e bis independen o e:Also, assume ha u0is a dis ibu ion on G0such ha , as
e-0;one has o all c>0
ue-u0in ðL2ðocÞÞ3;ð2Þ
whe e oc¼ xAR3:x0AS;0ox3o 0ðx0Þcg:Finally, assume ha Z a ies in any
di ec ion y0;a leas a one poin z0; ha is
8y0AR2;y0a0; he e exis s z0AR2and cARsuch ha Zðz0þcy0ÞaZðz0Þ:ð3Þ
Then he ollowing holds:
Theo em 1. I , o e e y e>0;we ha e
uene¼0on Re;ð4Þ
hen
u0¼0on R0:
Rema k 1. The ace o u0on R0is well defined. Indeed, in iew o (1) and (2), we
ha e o all c>0
Zoc
j u0j2dxpb;
whence u0AðL2ðG0ÞÞ33:
Rema k 2. A simila esul can be p o ed in any dimension NX2:I is also clea
ha , o his heo em o hold, we only need he hypo heses o be sa isfied by a
sequence ðuenÞn;wi h en-0:On he o he hand, he esul s ill holds i we eplace (1)
by he weake assump ion
ZGe
j uejpdxpb;ð5Þ
wi h p>1:To see his, i su fices o adap he a gumen used in Sec ion 3.
Rema k 3. I Zpossesses an in a ian di ec ion, i.e., i (3) is no sa isfied, he
p e ious esul does no hold. Mo e p ecisely, he a gumen s used in Sec ion 3 show
ha , in ha case, one o he ollowing wo si ua ions is ound:
J. Casado-Dı´az e al. / J. Di e en ial Equa ions 189 (2003) 526–537 529
*Zis cons an ; hen he unique conclusion is ha
u0n¼0onR0:
Indeed, i such a field u0is p esc ibed, all assump ions a e sa isfied by he unc ions
ueðxÞ¼u0ðx1;x2;x3eZÞ:
*Zpossesses only one in a ian di ec ion xin ; hen one has
u0n¼0 and u0x>
in ¼0onR0:
This is he case o a wall co e ed wi h ible s: he fluid possibly slides in he di ec ion
xin o he ible s bu no in he o hogonal di ec ion.
The in a iance o Zin he di ec ion xin is equi alen o he ac ha Zonly
depends on a scala a iable which is y0x>
in ; ha is equi alen o he exis ence o a
unc ion *Zsuch ha Zðy0Þ¼*Zðy0x>
in Þ o all y0:
Rema k 4. The asse ions o Theo em 1 and Rema k 3 can be ga he ed oge he in a
single s a emen in which (3) is no equi ed: whene e he unc ions uesa is y (1), (2)
and (4), one has he ollowing o almos all xin R0:
u0ðxÞAðNðxÞÞ>;
whe e
NðxÞ¼Span nðxÞ @Z
@x1
ðy0Þ;@Z
@x2
ðy0Þ;0

:y0Að0;l1Þð0;l2Þ

¼Span nðxÞ;Mg
and
M¼Span @Z
@x1
ðy0Þ;@Z
@x2
ðy0Þ;0

:y0Að0;l1Þð0;l2Þ

:
In his s a emen , again (1) can be eplaced by (5). Assump ion (3) o Theo em 1 (i.e.
he ac ha Zpossesses no in a ian di ec ion) is equi alen o dim M¼2 and,
he e o e, o dim NðxÞ¼3 (since hen Mis he ho izon al plane and nðxÞis no
ho izon al).
The exis ence o exac ly one in a ian di ec ion examined in Rema k 3 (i.e., he
ac ha Zdepends only on one scala a iable) is equi alen o dim M¼1 and
he e o e o dim NðxÞ¼2:
The exis ence o many in a ian di ec ions (i.e., he ac ha Zis cons an ) is
equi alen o dim M¼0 and he e o e o dim NðxÞ¼1:
J. Casado-Dı´az e al. / J. Di e en ial Equa ions 189 (2003) 526–537530

3. P oo o Theo em 1
In he sequel, Cis a gene ic posi i e eal numbe ha can depend on S;a;b;Zand
0;bu no on e:
Fi s educ ion o he p oblem: The si ua ion is educed o he case 01 by means
o he change o a iable x/ˆ
x¼ðx0;1þðx3 0ðx0ÞÞ=aÞand es ic ion o he
subdomain whe e ˆ
x3>0:Consequen ly, we will assume om now on ha 01;
hen, R0¼ ðx0;1Þ:x0ASg:
Second educ ion o he p oblem: Fo each y0AR2;we se
lðy0Þ¼ @Z
@x1
ðy0Þ;@Z
@x2
ðy0Þ;1

:
Due o pe iodici y, Z eaches a maximum o e R2;say, a x1:Then lðx1Þ¼ð0;0;1Þ:
In iew o (3), he e exis wo poin s x2and x3such ha lðx1Þ;lðx2Þand lðx3Þa e
linea ly independen . Indeed, i his we e no he case, we would ha e lðxÞ¼
ðCa;Cb;1Þ o all x; o some fixed aand b; hus, we would also ha e he ollowing,
o all y1and y2;
d
d Zðy1þ b;y2 aÞ¼b@Z
@x1
ðy1þ b;y2 aÞa@Z
@x2
ðy1þ b;y2 aÞ
¼Cba þCab
¼0;
which is in con adic ion wi h (3). Acco dingly, i will be su ficien o p o e ha , o
all y0AR2and almos all x0AS;one has u0ðx0;1Þlðy0Þ¼0 o , equi alen ly,
u0ðx0;1Þnðy0Þ¼0;ð6Þ
whe e nðy0Þ¼lðy0Þ=jlðy0Þj:Le us deno e by S he ‘‘2D pe iod’’ o Z;i.e. he se
S¼ð0;c1Þð0;c2Þ;
and le Kbe an a bi a y nonemp y compac subse o S:Since u0AðH1ðG0ÞÞ3;see
Rema k 1, a con inuous unc ion 0is defined on ½0;1by
0ðx3Þ¼ZKZS
ju0ðx0;x3Þnðy0Þj2dy0dx0:ð7Þ
To ge (6), i will su fice o p o e ð1Þ¼0:Since 0is con inuous, i will be su ficien
o p o e ha
1
sZ1s
12s
0ðx3Þdx3-0ass-0:ð8Þ
J. Casado-Dı´az e al. / J. Di e en ial Equa ions 189 (2003) 526–537 531
P oo o (8). Le sbe gi en such ha 0oso1=2:Le us choose e>0 such ha
Kþey0CS o all y0AS;and such ha ejjZjjLNðR2Þos:On he o he hand, le ðun
eÞnbe
a sequence in ðC1ðGeÞÞ3con e ging s ongly in ðH1ðGeÞÞ3 o ue:Gi en x3Að1
2s;1sÞ;x0AKand y0AS;we in oduce a poin zARewhich is ‘‘close’’ o xby
pu ing
z0¼x0þey0;z3¼1þeZ z0
e

:
Then we ha e
un
eðx0;x3Þ¼un
eðz0;x3ÞeZ1
0
y0
x0un
eðx0þ ey0;x3Þd
¼un
eðz0;z3ÞZz3
x3
@un
e
@x3
ðz0;y3Þdy3eZ1
0
y0
x0un
eðx0þ ey0;x3Þd :
Taking scala p oduc s wi h neðzÞand using he inequali ies jneðzÞjp1;jz3
x3jpeZðz0=eÞ2spCðeþsÞand jy0jpC;we find he ollowing:
jun
eðx0;x3ÞneðzÞj2pCjun
eðz0;z3ÞneðzÞj2þðeþsÞZz3
0
@un
e
@x3
ðz0;y3Þ

2
dy3
þe2Z1
0
j x0un
eðx0þ ey0;x3Þj2d :
In eg a ing his inequali y wi h espec o x0in K;wi h espec o y0in Sand finally
wi h espec o x3in ð12s;1sÞ;we deduce ha
Z1s
12sZSZK
jun
eðx0;x3ÞneðzÞj2dx0dy0dx3
pCs ZSZK
jun
eðz0;z3ÞneðzÞj2dx0dy0
þCsðeþsÞZSZKZz3
0
@un
e
@x3
ðz0;y3Þ

2
dy3dx0dy0
þCe2Z1s
12sZSZKZ1
0
j x0un
eðx0þ ey0;x3Þj2d dx0dy0dx3
pCs ZSZK
jun
eðz0;z3ÞneðzÞj2dx0dy0þCðe2þs2ÞZGe
j un
eðxÞj2dx:
The las inequali y is implied by he ac ha KþeSCS:Now, aking limi s in his
inequali y as n-N;in iew o s a emen s (1) and (4) and Fubini’s Theo em, we find
Z1s
12sZKZS
jueðx0;x3ÞneðzÞj2dy0dx0dx3pCðe2þs2Þ:ð9Þ
J. Casado-Dı´az e al. / J. Di e en ial Equa ions 189 (2003) 526–537532
The no mal o Rea zis neðzÞ¼nðz0=eÞ;i.e. nðy0þx0=eÞ:Since nis a pe iodic unc ion
and since i s ‘‘2D pe iod’’ is S; his implies, o almos all ðx0;sÞin Kðs;2sÞ; he
iden i y
ZS
jueðx0;x3ÞneðzÞj2dy0¼ZS
jueðx0;x3Þnðy0Þj2dy0:
Then (9) can also be w i en in he o m
Z1s
12sZKZS
jueðx0;x3Þnðy0Þj2dy0dx0dx3pCðe2þs2Þ:
Taking limi s as e-0;we ob ain
1
sZ1s
12sZKZS
ju0ðx0;x3Þnðy0Þj2dy0dx0dx3pCs:
Consequen ly, we ha e p o ed (8). This ends he p oo o Theo em 1. &
4. A consequence: he asymp o ic beha io o a iscous fluid nea a wall wi h aspe i ies
Theo em 1 can be used o iden i y he limi o he solu ion o he s a iona y
Na ie –S okes sys em sa is ying Na ie ’s law on an oscilla ing bounda y. In o de o
fix ideas, le us in oduce he fluid domains Oeand O0;wi h
Oe¼ xAR3:0ox3o eðx0Þg
and
O0¼ xAR3:0ox3oc3g:
He e, eis gi en by
eðx0Þ¼c3þeZ x0
e

(c3is posi i e and cons an ) and ZAC1ðR2Þis pe iodic o pe iod ðc1;c2Þin he
a iable y0¼x0=e:We se
Ge¼ xAR3:x3¼ eðx0Þg
( he uppe bounda y o Oe), and
G0¼ xAR3:x3¼c3g;P¼ xAR3:x3¼0g:
J. Casado-Dı´az e al. / J. Di e en ial Equa ions 189 (2003) 526–537 533
Le us conside he s a iona y Na ie –S okes sys em in Oe
nDueþðue Þueþ pe¼0; ue¼0inOe;ð10Þ
comple ed wi h he slip and ic ion condi ions
uene¼0;ðseneÞ an þkue¼0onGeð11Þ
(neis he uni no mal ec o on Geand seis he s ess enso associa ed o ðue;peÞ),
uen¼0;ðsenÞ an þkðuegÞ¼0onPð12Þ
(gis a non-ze o ec o o he o m g¼ðg1;g2;0Þ) and he ollowing addi ional
condi ion:
ðue;peÞis x0-pe iodic;o pe iod ðc1;c2Þ:ð13Þ
Le Lbe gi en by
L¼maxðc1;c2;c3Þ
(a cha ac e is ic leng h o O0) and le us in oduce he associa ed Reynolds numbe
Re ¼Ljgj
n:
Fo simplici y, we assume ha Re is su ficien ly small. Then, sys em (10)–(13)
possesses exac ly one solu ion
ðue;peÞAðH1
locðOeÞÞ3L2
locðOeÞ:
sa is ying
ZOe- jx0joKg
j uej2dx þZOe- jx0joKg
juej2dxpbKð14Þ
o all K>0;whe e bKis independen o e( he p oo o his asse ion is essen ially
gi en in Re s. [1,2]). F om (14), i is no di ficul o deduce he exis ence o a unc ion
u0AðH1
locðO0ÞÞ3such ha , a leas o a subsequence, we ha e
ue-u0weakly in ðH1
locðocÞÞ3and s ongly in ðL2
locðocÞÞ3
o all c>0;whe e oc¼ xAR3:0ox3oc3cg:
Then, as a consequence o Theo em 1, we ob ain he ollowing:
Theo em 2. Assume ha Re is su icien ly small,Zsa is ies (3) and e-0:Then ue
con e ges o u0;i.e., oge he wi h some p0; he unique solu ion o he s a iona y
Na ie –S okes equa ions
nDu0þðu0 Þu0þ p0¼0; u0¼0in O0;
J. Casado-Dı´az e al. / J. Di e en ial Equa ions 189 (2003) 526–537534