Why viscous fluids adhere to rugose walls: A mathematical explanation

J. Differential Equations 189 (2003) 526–537 Why viscous fluids adhere to rugose walls: A mathematical explanation Juan Casado-D! ıaz, a Enrique Ferna ´ndez-Cara, a, * and Jacques Simon b a Dpto. E.D.A.N., Univ. de Sevilla, Aptdo. 1160, 41080 Sevilla, Espan˜a, Spain b CNRS, Laboratoire de Mathe ´matiques Applique ´es, Univ. Blaise Pascal (Clermont-Ferrand 2), 63177 Aubie `re cedex, France Received October 12 2001; revised April 29 2002 Abstract The main purpose of this paper is to justify rigorously the following assertion: A viscous fluid cannot slip on a wall covered by microscopic asperities because, due to the viscous dissipation, the surface irregularities bring to rest the fluid particles in contact with the wall. In mathematical terms, this corresponds to an asymptotic property established in this paper for any family of fields that slip on oscillating boundaries and remain uniformly bounded in the H1-norm. r2002 Elsevier Science (USA). All rights reserved. 1. Introduction This paper is devoted to justify rigorously the fact that, asymptotically, a fluid cannot slip on a wall covered by microscopic asperities: the slip condition, i.e. the requirement un¼0 on the wall; where uis the velocity and n¼nðxÞis a normal vector at a boundary point x;which expresses the fact that the wall is not permeable to the fluid particles, provides sufficient information to ensure that, as the size of asperities goes to 0, the fluid *Corresponding author. Fax: +34-954-55-28-98. E-mail addresses: [email protected] (J. Casado-D! ıaz), [email protected] (E. Fern! andez-Cara), [email protected] (J. Simon). 0022-0396/03/$ - see front matter r2002 Elsevier Science (USA). All rights reserved. PII: S 0 0 2 2 - 0396(02)00115-8 satisfies the no-slip condition, i.e. u¼0 on the wall: This was noticed and justified for a 2D periodic Stokes flow in [11] and was mathematically proved for a 3D periodic Navier–Stokes flow in [1]. However, the periodicity of the flow at the microscopic scale assumed in these papers is very restrictive. Indeed, it prevents any vortex or any other structure larger than asperities to occur and it implies that the mean velocity over a period is a Couette flow (this enables a satisfactory analysis in this case with a particular proof based on scaling arguments, see [1]). In the present paper, we will give a mathematical proof of the previous assertion for any 3D flow whatever the governing equation (in fact, no equation is prescribed). This can be viewed as a property of the limit u0of a family of vector fields uethat slip on a boundary covered by asperities of size e;with an enstrophy Rjruej2dx that remains bounded as e-0 (Theorem 1). Roughly speaking, this is due to the fact that sliping with a non-zero velocity dissipates energy on asperities because the direction of velocity suddenly varies as the slope does. For instance, in a 2D domain with a serrated boundary whose slope is alternately þ1 and 1;if the horizontal velocity is v;then the vertical velocity is alternately þvand v:When the size eof asperities goes to 0, the energy dissipated by each asperity goes to 0 but not fast enough to compensate the fact that there are many of them. Therefore, the total dissipation grows to infinity and the unique possibility for enstrophy to be uniformly bounded is that the limit velocity vanishes on the wall. A rigorous formulation of this assertion will be given in (8). We will also prove that our general result applies to a flow governed by the Navier–Stokes equations together with Navier’s law un¼0;ðsnÞtan þku¼0; where sdenotes the stress tensor and the subscript tan denotes the tangential component, i.e. ftan ¼fðfnÞnfor any vector field f:Of course, the second previous equality means that the friction forces on the wall are proportional to the tangential velocity. Indeed, in this situation the enstrophy remains bounded as e-0 and, therefore, the limit velocity u0vanishes on the limit boundary whatever the friction coefficient k(see Theorem 2). This generalizes, to non-periodic flows, the above-mentioned results of [1,11]. It is worth mentioning that this result is in contradiction with a statement in [8], but the argument used in that reference is false, as we will explain in Remark 5, at the end of Section 4. Our argument relies on the internal viscous dissipation in the fluid and the geometry of the domain only. It does not require any dissipation of energy due to the friction (or molecular interaction) of the fluid particles in contact with the solid walls. J. Casado-Dı´az et al. / J. Differential Equations 189 (2003) 526–537 527 The effective relative importance of surface roughness and fluid/solid molecular interactions is discussed in [14]. There, the authors show that roughness dominates except for very smooth walls. The reader is referred to [5,7] for an analysis of molecular interaction by molecular dynamics simulation and to [3] for a similar analysis in the case of a two-component fluid. The flow at the surface of a porous medium is extensively discussed in [6] and references therein. In this case our argument does not apply, since the slip condition un¼0 is not imposed. In particular, we do not find in the limit the no-slip condition when a rugose interface is modeled by Fourier’s law snþku¼0 on the wall (see [2], where a homogeneized friction coefficient k0is obtained in the limit). Let us finally mention that many physical and numerical experiments have shown that, when a fluid flows between two plates, the occurrence of asperities on the walls is not irrelevant. In particular, it is known that small riblets (tiny asperities parallel to the flow) can be used to reduce considerably the drag experienced by the fluid; see [4,12] and references therein. This paper is organized as follows. The main result (Theorem 1) is stated and commented in Section 2. It is proved in Section 3. Finally, Section 4 is concerned with the application of Theorem 1 to a viscous fluid near a wall with asperities. 2. Main result Let us now present our main result with precision. Let SCR2be a bounded open set and assume that, for each ewith 0oepe0;the function reis given by reðx0Þ¼r0ðx0ÞþeZ x0 e  ; where r0AC1ð% SÞ;r0ðx0ÞXa>0 and ZAC1ðR2Þis a periodic function of period ðc1;c2Þ in the variable y0¼x0=e:Let Gebe the open set Ge¼fxAR3:x0AS;0ox3oreðx0Þg and let us put Re¼fxAR3:x0AS;x3¼reðx0Þg (the oscillating piece of boundary). We also set G0¼fxAR3:x0AS;0ox3or0ðx0Þg (the limit domain) and R0¼fxAR3:x0AS;x3¼r0ðx0Þg: J. Casado-Dı´az et al. / J. Differential Equations 189 (2003) 526–537528 Assume that for each ewe have ueAðH1ðGeÞÞ3;with ZGe jruej2dxpb;ð1Þ where bis independent of e:Also, assume that u0is a distribution on G0such that, as e-0;one has for all c>0 ue-u0in ðL2ðocÞÞ3;ð2Þ where oc¼fxAR3:x0AS;0ox3or0ðx0Þcg:Finally, assume that Zvaries in any direction y0;at least at one point z0;that is 8y0AR2;y0a0;there exists z0AR2and cARsuch that Zðz0þcy0ÞaZðz0Þ:ð3Þ Then the following holds: Theorem 1. If,for every e>0;we have uene¼0on Re;ð4Þ then u0¼0on R0: Remark 1. The trace of u0on R0is well defined. Indeed, in view of (1) and (2), we have for all c>0 Zoc jru0j2dxpb; whence ru0AðL2ðG0ÞÞ33: Remark 2. A similar result can be proved in any dimension NX2:It is also clear that, for this theorem to hold, we only need the hypotheses to be satisfied by a sequence ðuenÞn;with en-0:On the other hand, the result still holds if we replace (1) by the weaker assumption ZGe jruejpdxpb;ð5Þ with p>1:To see this, it suffices to adapt the argument used in Section 3. Remark 3. If Zpossesses an invariant direction, i.e., if (3) is not satisfied, the previous result does not hold. More precisely, the arguments used in Section 3 show that, in that case, one of the following two situations is found: J. Casado-Dı´az et al. / J. Differential Equations 189 (2003) 526–537 529 *Zis constant; then the unique conclusion is that u0n¼0onR0: Indeed, if such a field u0is prescribed, all assumptions are satisfied by the functions ueðxÞ¼u0ðx1;x2;x3eZÞ: *Zpossesses only one invariant direction xinv;then one has u0n¼0 and u0x> inv ¼0onR0: This is the case of a wall covered with riblets: the fluid possibly slides in the direction xinv of the riblets but not in the orthogonal direction. The invariance of Zin the direction xinv is equivalent to the fact that Zonly depends on a scalar variable which is y0x> inv;that is equivalent to the existence of a function *Zsuch that Zðy0Þ¼*Zðy0x> invÞfor all y0: Remark 4. The assertions of Theorem 1 and Remark 3 can be gathered together in a single statement in which (3) is not required: whenever the functions uesatisfy (1), (2) and (4), one has the following for almost all xin R0: u0ðxÞAðNðxÞÞ>; where NðxÞ¼Span nðxÞ @Z @x1 ðy0Þ;@Z @x2 ðy0Þ;0  :y0Að0;l1Þð0;l2Þ  ¼SpanfnðxÞ;Mg and M¼Span @Z @x1 ðy0Þ;@Z @x2 ðy0Þ;0  :y0Að0;l1Þð0;l2Þ  : In this statement, again (1) can be replaced by (5). Assumption (3) of Theorem 1 (i.e. the fact that Zpossesses no invariant direction) is equivalent to dim M¼2 and, therefore, to dim NðxÞ¼3 (since then Mis the horizontal plane and nðxÞis not horizontal). The existence of exactly one invariant direction examined in Remark 3 (i.e., the fact that Zdepends only on one scalar variable) is equivalent to dim M¼1 and therefore to dim NðxÞ¼2: The existence of many invariant directions (i.e., the fact that Zis constant) is equivalent to dim M¼0 and therefore to dim NðxÞ¼1: J. Casado-Dı´az et al. / J. Differential Equations 189 (2003) 526–537530 3. Proof of Theorem 1 In the sequel, Cis a generic positive real number that can depend on S;a;b;Zand r0;but not on e: First reduction of the problem: The situation is reduced to the case r01 by means of the change of variable x/ˆ x¼ðx0;1þðx3r0ðx0ÞÞ=aÞand restriction to the subdomain where ˆ x3>0:Consequently, we will assume from now on that r01; then, R0¼fðx0;1Þ:x0ASg: Second reduction of the problem: For each y0AR2;we set lðy0Þ¼ @Z @x1 ðy0Þ;@Z @x2 ðy0Þ;1  : Due to periodicity, Zreaches a maximum over R2;say, at x1:Then lðx1Þ¼ð0;0;1Þ: In view of (3), there exist two points x2and x3such that lðx1Þ;lðx2Þand lðx3Þare linearly independent. Indeed, if this were not the case, we would have lðxÞ¼ ðCa;Cb;1Þfor all x;for some fixed aand b;thus, we would also have the following, for all y1and y2; d dt Zðy1þtb;y2taÞ¼b@Z @x1 ðy1þtb;y2taÞa@Z @x2 ðy1þtb;y2taÞ ¼Cba þCab ¼0; which is in contradiction with (3). Accordingly, it will be sufficient to prove that, for all y0AR2and almost all x0AS;one has u0ðx0;1Þlðy0Þ¼0 or, equivalently, u0ðx0;1Þnðy0Þ¼0;ð6Þ where nðy0Þ¼lðy0Þ=jlðy0Þj:Let us denote by Sthe ‘‘2D period’’ of Z;i.e. the set S¼ð0;c1Þð0;c2Þ; and let Kbe an arbitrary nonempty compact subset of S:Since u0AðH1ðG0ÞÞ3;see Remark 1, a continuous function f0is defined on ½0;1by f0ðx3Þ¼ZKZS ju0ðx0;x3Þnðy0Þj2dy0dx0:ð7Þ To get (6), it will suffice to prove fð1Þ¼0:Since f0is continuous, it will be sufficient to prove that 1 sZ1s 12s f0ðx3Þdx3-0ass-0:ð8Þ J. Casado-Dı´az et al. / J. Differential Equations 189 (2003) 526–537 531 Proof of (8). Let sbe given such that 0oso1=2:Let us choose e>0 such that Kþey0CSfor all y0AS;and such that ejjZjjLNðR2Þos:On the other hand, let ðun eÞnbe a sequence in ðC1ðGeÞÞ3converging strongly in ðH1ðGeÞÞ3to ue:Given x3Að1 2s;1sÞ;x0AKand y0AS;we introduce a point zARewhich is ‘‘close’’ to xby putting z0¼x0þey0;z3¼1þeZ z0 e  : Then we have un eðx0;x3Þ¼un eðz0;x3ÞeZ1 0 y0r x0un eðx0þtey0;x3Þdt ¼un eðz0;z3ÞZz3 x3 @un e @x3 ðz0;y3Þdy3eZ1 0 y0r x0un eðx0þtey0;x3Þdt: Taking scalar products with neðzÞand using the inequalities jneðzÞjp1;jz3 x3jpeZðz0=eÞ2spCðeþsÞand jy0jpC;we find the following: jun eðx0;x3ÞneðzÞj2pCjun eðz0;z3ÞneðzÞj2þðeþsÞZz3 0 @un e @x3 ðz0;y3Þ  2 dy3 þe2Z1 0 jrx0un eðx0þtey0;x3Þj2dt: Integrating this inequality with respect to x0in K;with respect to y0in Sand finally with respect to x3in ð12s;1sÞ;we deduce that 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 jrx0un eðx0þtey0;x3Þj2dt dx0dy0dx3 pCs ZSZK jun eðz0;z3ÞneðzÞj2dx0dy0þCðe2þs2ÞZGe jrun eðxÞj2dx: The last inequality is implied by the fact that KþeSCS:Now, taking limits in this inequality as n-N;in view of statements (1) and (4) and Fubini’s Theorem, we find Z1s 12sZKZS jueðx0;x3ÞneðzÞj2dy0dx0dx3pCðe2þs2Þ:ð9Þ J. Casado-Dı´az et al. / J. Differential Equations 189 (2003) 526–537532 The normal to Reat zis neðzÞ¼nðz0=eÞ;i.e. nðy0þx0=eÞ:Since nis a periodic function and since its ‘‘2D period’’ is S;this implies, for almost all ðx0;sÞin Kðs;2sÞ;the identity ZS jueðx0;x3ÞneðzÞj2dy0¼ZS jueðx0;x3Þnðy0Þj2dy0: Then (9) can also be written in the form Z1s 12sZKZS jueðx0;x3Þnðy0Þj2dy0dx0dx3pCðe2þs2Þ: Taking limits as e-0;we obtain 1 sZ1s 12sZKZS ju0ðx0;x3Þnðy0Þj2dy0dx0dx3pCs: Consequently, we have proved (8). This ends the proof of Theorem 1. & 4. A consequence: the asymptotic behavior of a viscous fluid near a wall with asperities Theorem 1 can be used to identify the limit of the solution of the stationary Navier–Stokes system satisfying Navier’s law on an oscillating boundary. In order to fix ideas, let us introduce the fluid domains Oeand O0;with Oe¼fxAR3:0ox3oreðx0Þg and O0¼fxAR3:0ox3oc3g: Here, reis given by reðx0Þ¼c3þeZ x0 e  (c3is positive and constant) and ZAC1ðR2Þis periodic of period ðc1;c2Þin the variable y0¼x0=e:We set Ge¼fxAR3:x3¼reðx0Þg (the upper boundary of Oe), and G0¼fxAR3:x3¼c3g;P¼fxAR3:x3¼0g: J. Casado-Dı´az et al. / J. Differential Equations 189 (2003) 526–537 533 Let us consider the stationary Navier–Stokes system in Oe nDueþðuerÞueþrpe¼0;rue¼0inOe;ð10Þ completed with the slip and friction conditions uene¼0;ðseneÞtan þkue¼0onGeð11Þ (neis the unit normal vector on Geand seis the stress tensor associated to ðue;peÞ), uen¼0;ðsenÞtan þkðuegÞ¼0onPð12Þ (gis a non-zero vector of the form g¼ðg1;g2;0Þ) and the following additional condition: ðue;peÞis x0-periodic;of period ðc1;c2Þ:ð13Þ Let Lbe given by L¼maxðc1;c2;c3Þ (a characteristic length of O0) and let us introduce the associated Reynolds number Re ¼Ljgj n: For simplicity, we assume that Re is sufficiently small. Then, system (10)–(13) possesses exactly one solution ðue;peÞAðH1 locðOeÞÞ3L2 locðOeÞ: satisfying ZOe-fjx0joKg jruej2dx þZOe-fjx0joKg juej2dxpbKð14Þ for all K>0;where bKis independent of e(the proof of this assertion is essentially given in Refs. [1,2]). From (14), it is not difficult to deduce the existence of a function u0AðH1 locðO0ÞÞ3such that, at least for a subsequence, we have ue-u0weakly in ðH1 locðocÞÞ3and strongly in ðL2 locðocÞÞ3 for all c>0;where oc¼fxAR3:0ox3oc3cg: Then, as a consequence of Theorem 1, we obtain the following: Theorem 2. Assume that Re is sufficiently small,Zsatisfies (3) and e-0:Then ue converges to u0;i.e.,together with some p0;the unique solution to the stationary Navier–Stokes equations nDu0þðu0rÞu0þrp0¼0;ru0¼0in O0; J. Casado-Dı´az et al. / J. Differential Equations 189 (2003) 526–537534