Why viscous fluids adhere to rugose walls: A mathematical explanation

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