Soft Matter

PAPER

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aA.N. Frumkin Institute of Physical Chemistr

of Sciences, 31 Leninsky Prospect, 119071 MbCentral Aero-Hydrodynamic Institute, 1401cInstitute of Mechanics, M.V. Lomonosov M

RussiadFaculty of Physics, M.V. Lomonosov Mos

Russia. E-mail: oivinograd@yahoo.comeDWI, RWTH Aachen, Forckenbeckstr. 50, 52

Cite this: DOI: 10.1039/c3sm51850g

Received 7th July 2013Accepted 11th October 2013

DOI: 10.1039/c3sm51850g

www.rsc.org/softmatter

This journal is ª The Royal Society of

Flow in channels with superhydrophobic trapezoidaltextures

Tatiana V. Nizkaya,a Evgeny S. Asmolovabc and Olga I. Vinogradova*ade

Superhydrophobic one-dimensional surfaces reduce drag and generate transverse hydrodynamic

phenomena by combining hydrophobicity and roughness to trap gas bubbles in microscopic textures.

Recent studies in this area have focused on specific cases of superhydrophobic stripes. Here we provide

some theoretical results to guide the optimization of the forward flow and transverse hydrodynamic

phenomena in a parallel-plate channel with a superhydrophobic trapezoidal texture, varying on scales

larger than the channel thickness. The permeability of such a thin channel is shown to be equivalent to

that of a striped one with greater average slip. The maximization of a transverse flow normally requires

largest possible slip at the gas areas, similar to a striped channel. However, in the case of trapezoidal

textures with a very small fraction of the solid phase this maximum occurs at a finite slip at the gas

areas. Exact numerical calculations show that our analysis, based on a lubrication theory, can also be

applied for a larger gap. However, in this case it overestimates a permeability of the channel, and

underestimates an anisotropy of the flow. Our results provide a framework for the rational design of

superhydrophobic surfaces for microfluidic applications.

1 Introduction

Superhydrophobic (SH) textured materials have been extensivelyinvestigated over the last decade, since texture can bring excep-tional wetting properties.1,2 Beside that, SH materials in theCassie state, where the texture is lled with gas, are important incontext of uid dynamics and their superlubricating proper-ties.3–5 The drag reduction can be quantied by an effective sliplength of the heterogeneous SH surface, which can be of theorder of several mm or even more,6–9 and for anisotropic surfacesvaries with the orientation of the wall texture relative to ow.9,10

Despite signicant advances in the eld, previous investigationsof anisotropic textures were limited mostly by SH stripes. Theproblem of ow past striped SH surfaces has previously beenstudied in the context of a drag reduction of pressure-driven orshear forward ow in thick,11–14 thin15 and intermediate16,17

channels, and it is directly relevant for mixing,18,19 and for ageneration of a tensorial electro-osmotic ow.19–21

In this paper we focus on the trapezoidal surface texture asillustrated in Fig. 1. This one-dimensional patterned SH surface,where the local (scalar) slip length b(y) varies only in one direction,

y and Electrochemistry, Russian Academy

oscow, Russia

80 Zhukovsky, Moscow region, Russia

oscow State University, 119991 Moscow,

cow State University, 119991 Moscow,

056 Aachen, Germany

Chemistry 2013

is more general since it provides more geometrical textureparameters, namely, trapezoid base width, angle, height, andspacing. Furthermore, the trapezoidal texture can be extended toits extreme, saw-tooth case, where the solid area vanishes. It canalso be transformed into a striped texture. These grooved surfaceswith a trapezoidal relief can be made easily using modern solithography techniques, and were used in wetting studies.22 Note

Fig. 1 (a) Sketch of the channel with a hydrophilic (top) and a SH (bottom) wallwith one-dimensional trapezoidal texture. (b) Illustration of a tensorial hydrody-namic response, with the direction of a flow misaligned from the pressure gradient(top view). (c) The cross-section on the channel flow with the trapezoidal texture.

Soft Matter

Soft Matter Paper

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that such trapezoidal textures can be naturally formed from diluterectangular grooves as a result of elasto-capillarity.23–25 Theyprovide a stable Cassie state26 and can sustain very small areas ofliquid/solid contact without losing mechanical stability. This typeof surface has already been intensively used in the slippageexperiment.27 However, how the trapezoidal relief of the textureimpacts hydrodynamic phenomena remains largely unknown.Here we investigate a pressure-driven ow in such a channel, byfocusing mostly on the so-called thin-channel limit, where a nitethickness,H, is small compared to a texture period, L. (Note that aow past an isolated superhydrophobic trapezoidal texture hasbeen considered in ref. 28).

Our paper is arranged as follows. In Section 2 we dene theproblem. The general expansions for the eigenvalues of thepermeability tensor of a thin channel, which fully describe itsbehavior at the macrolevel, are constructed in Section 3. Then,in Section 4 we present specic results for a channel with atrapezoidal textured wall. Here beside analytical resultsobtained in a thin-channel limit, we present exact numericaldata obtained at arbitrary channel gaps. We conclude in Section5. In Appendix A we give some simple arguments suggestingthat in the case of shallow grooves the local slip prole followsthe relief of the texture. Appendix B describes the details of ournumerical methods.

2 Model

Here we present the basic assumptions of our theoreticalmodel, and dene local slip boundary conditions.

We consider a pressure-driven ow in a channel consistingof two parallel walls located at z¼ 0 and z ¼ h and unboundedin the x and y directions as sketched in Fig. 1. The upper platerepresents a no-slip hydrophilic surface, and the lower plateis a SH surface. The origin of coordinates is placed at thecenter of the solid sector. The x-axis is directed along thetexture, z-axis is orthogonal to the channel walls andthe y-axis is normal both to x and z. This (SH vs. hydrophilic)geometry is relevant for various setups, where the alignmentof opposite textures is inconvenient or difficult. Besides, theadvantage of such a geometry is that it allows to avoid the gasbridging and long-range attractive capillary forces,28 whichappear when we deal with interactions of two hydrophobicsolids.29,30

As in most previous publications,11,13,31–34 we model the SHplate as a at interface. In such a description, we neglect anadditional mechanism for a dissipation connected with themeniscus curvature.35–37 We emphasize however, that such anideal situation is not unrealistic, and has been achieved inmanyrecent experiments.38–40

At the hydrodynamic level, the composite surface is modeledas spatially dependent local slip boundary conditions:

us � bðyÞ vusvz

¼ 0;

uz ¼ 0;

8<: (1)

where b is the slip length of the surface at point (x, y), and us ¼(ux, uy) is the tangential velocity of the uid. For smooth

Soft Matter

hydrophobic coating slip lengths are of the orders of tens of nmonly,41–44 so that for simplicity the boundary conditions at thesolid areas are set as no-slip (b ¼ 0).

Note that prior work oen assumed shear-free boundaryconditions over the gas sectors,11,13,31 so that the viscous dissi-pation in the underlying gas phase has been neglected. Here weuse the partial slip boundary conditions. These are the conse-quences of the “gas cushion model”, which takes into accountthat the dissipation at the gas/liquid interface is dominated bythe shearing of a continuous gas layer45,46

bðyÞx m

mg

eðyÞ: (2)

Here m and mg are the liquid and the gas dynamic viscosi-ties, respectively, and e(y) the local thickness of the gas layer.Eqn (2) represents an upper bound for a local slip at the gasarea, which is attained in the limit of a small fraction of thesolid phase. At a relatively large solid fraction, eqn (2) couldoverestimate the local slip.37 Beside that, net gas ux in the“pockets” of SH surfaces could be zero in case they areconsidered to have end walls,33 which is also expected toincrease the surface friction and to decrease the slip. However,even in this extreme situation the local slip length prole, b(y),necessarily follows the relief of the texture provided groovesare shallow enough (see Appendix A).

A striped SH texture is fully determined by the fraction of theno-slip phase, f1 ¼ a/L, and the local slip at the gas areas,

which is proportional to the depth of the groove bmaxxm

mge.

The trapezoidal texture provides an additional geometricparameter, the basement of the trapezoid a ¼ c/L, which is nolonger equal to its top side, f1 # a# 1 (see Fig. 1). Note that thestriped SH surfaces represent a limiting case of trapezoidaltextures with a ¼ f1. The other limiting cases correspond toa ¼ 1 (a vanishing distance between trapezoids) and f1 ¼0 (sawtooth-like proles with a single solid–liquid contactpoint).

Using eqn (2), we express the trapezoidal slip length prolein the form:

bðtÞ ¼

0; jtj#f1=2;

ð2jtj � f1ÞHb

a� f1

;f1

2# jtj#a

2;

Hb;a

2# jtj# 1

2:

8>>>>>>><>>>>>>>:

(3)

Here t ¼ yL

h i� 12is a dimensionless variable dened on the

periodic cell (brackets denote the fractional part of the number)

and b ¼ bmax

His the maximal slip length scaled by the channel

thickness.

3 Permeability of a thin superhydrophobicchannel

Here we provide some general theoretical results to guide theoptimization of hydrodynamic phenomena in channels with

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one SH wall. Such a SH channel is characterized by threedifferent length scales: the channel thickness H, the textureperiod L, and somemacroscopic length LN [ L. Our analysis isbased on the lubrication (or thin-channel) limit, where thetexture varies over a scale L [ H. In this limit the pressure isuniform across the channel, and the ow prole u(x, y, z) islocally parabolic in z:

uðx; y; zÞ ¼ �VPðx; yÞ2m

½zþ c0ðyÞ�ðH � zÞ; (4)

where the coefficient c0(y) is found from eqn (1):

c0ðyÞ ¼ bðyÞHH þ bðyÞ : (5)

The depth-averaged velocity U(x,y) is then proportional to thelocal pressure gradient:

Uðx; yÞ ¼ �kðyÞVPm

;

where k(y) is the local permeability of the channel, related to theslip length, b(y), as

kðyÞ ¼ H2

12

H þ 4bðyÞH þ bðyÞ : (6)

At the macroscopic scale, LN [ L, local details of the owbecome indiscernible, and the channel appears to be uniformand anisotropic. As usual for the lubrication approach, therelationship between the macroscale (averaged over the periodL) velocity hUi and the macroscale pressure gradient hVPi maybe written as a Darcy law

�U� ¼ � keff

m

�VP�:

Here

keff ¼ H2

12

k*k 0

0 k*t

!;

is the effective permeability tensor of the SH channel,15,47 k*k andk*t are the corrections to permeability of the hydrophilicchannel of the same thickness, H2/12, in the principal, i.e.longitudinal (x) and transverse (y) directions (without trans-verse ow). These corrections can be found by solving twoindependent problems on a microlevel, for the pressuregradient directed along x- and y-axes, and by averaging theresulting elds:

�U� ¼ 1

L

ðL0

UðyÞdy; hVPi ¼ 1

L

ðL0

VpðyÞdy; (7)

where U(y), Vp(y) are solutions of Stokes equations at themicrolevel, which take into account all the details of the SHtexture.

For the ow in the longitudinal direction the pressuregradient is uniform, Vxp ¼ hVx pi. In the transverse congura-tion the total ow and thus the depth-averaged velocity isuniform, Uy ¼ hUyi. This yields the following equations for thecorresponding effective permeabilities48

This journal is ª The Royal Society of Chemistry 2013

kk ¼ 1

L

ðL0

kðyÞdy; kt ¼ 1

L

� ðL0

dy

kðyÞ��1

: (8)

For a thin channel with a striped SH wall the resultingformulae correspond to familiar limits of resistors in parallel orin series.15 For the longitudinal conguration we get

kU ¼ f1k1 + f2k2, (9)

and for the transverse conguration

1/kL ¼ f1/k1 + f2/k2, (10)

where f2 ¼ 1 � f1 is the fraction of liquid surface in contactwith gas and k1,2 are local permeabilities of the correspondingparts of the channel, calculated using eqn (6). Values of kU andkL provide the upper and lower bounds for permeability for two-component texture with a piecewise constant slip b and xedfractions f1, f2.15 Taking into account eqn (6) and applyingboundary conditions one can obtain known formulae for thecorrections to permeability in the case of a SH channel with astriped wall:

k*Uðb;f1Þ ¼ f1 þ f2

1þ 4b

1þ b;

k*Lðb;f1Þ ¼

f1 þ f2

1þ b

1þ 4b

!�1

;

(11)

where kU,L ¼ k*U,LH2/12. Note that the bounds are fairly close

when b is very large, so in this situation the theory15 provides agood sense of the possible effective slip of any texture with apiecewise constant slip based only on the area fractions of gasand solid sectors, and on the local slip length.

4 Flow properties of a channel with atrapezoidal texture

In this section, we consider a channel with one SH wall with thelocal slip length given by eqn (3).

In a lubrication limit, the permeability of such a channel canbe calculated by using eqn (6) and eqn (8):

k*k ¼ k*

U � sðbÞða� f1Þ\k*U ;

k*t ¼ 1

k*L

� sð4bÞ4

ða� f1Þ� ��1

. k*L;

(12)

s ¼ 3

�lnð1þ bÞ

b� 1

1þ b

�. 0:

Note that the difference between the permeabilities for thetrapezoidal and striped channels with the same f1 decays slowlywith the increase in b since s(b)¼ O(ln b/b)/ 0 as b/N. Thismeans that for a very large b there is almost no difference inpermeability between striped and trapezoidal channels,provided f1 is the same. However, the permeability of channelswith striped and trapezoidal walls will be different at nite b.Below we explore this in more detail.

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Fig. 2 shows the dependence k*k(b), which characterizes themaximal possible drag reduction in the channel, at a xed valueof the base width (a ¼ 0.5) for different texture geometries,which obviously requires different f1. (The results for a trans-verse conguration are qualitatively similar, with only smallervalues of k*t, this is why we do not show them here.) We vary f1

in the interval from 0 to 0.5, so that we change the local slipprole from the triangular (saw-tooth texture) to the stripes.Regular trapezoidal SH textures (with a nite solid area) thencorrespond to intermediate values of f1. It can be seen thatqualitatively results for all channels are similar to a striped one.Namely, the correction to longitudinal permeability increaseswith b and saturates at b / N. However, it turns out that thetexture geometry (i.e. a decrease in f1) dramatically enhances k*k.Its maximal value can be evaluated with eqn (12) and reads

k*k(b / N, f1) ¼ 4 � 3f1. (13)

A similarity of the curves plotted in Fig. 2 suggests that wecan establish a certain equivalence between trapezoidal andstriped channels at a given b, since the correction to perme-ability for any trapezoidal texture will be the same as for astriped pattern, but with some different, “apparent” fraction ofliquid/solid contact f*

1. By using eqn (12) we can express this“apparent” fraction as:

f*1 ¼ f1 + r(b)(a � f1), (14)

r ¼ ð1þ bÞlnð1þ bÞb2

� 1

b: (15)

In particular, a channel with a partially slipping saw-toothtexture with f1 ¼ 0 and base a would be equivalent, from thepoint of view of a longitudinal permeability, to a striped channelwith f*

1 ¼ r(b)a.We emphasize that the analysis of function r(b) allows one to

interpret better the above results. Indeed, one can easilydemonstrate that r(b) < 1/2 for any b > 0. Therefore, the

Fig. 2 Corrections to permeability in the direction parallel to the texture for atrapezoidal wall with a fixed base width a ¼ 0.5 and (from top to bottom) f1 ¼ 0,0.1, 0.25 and 0.5.

Soft Matter

“apparent” fraction of no-slip surface, f*1, satises the following

inequality (see Fig. 3):

f*1\

f1 þ a

2¼ aþ c

2L:

This means that for equal average slip lengths hbi a trape-zoidal pattern provides a larger k*k. Beside that, r(b) vanishes atlarge b, and f*

1x f1, which claries the absence of a discernibledifference in permeability between striped and trapezoidalchannels at large local slip.

We remind that a pressure gradient in the eigen directionscannot produce any transverse ow. In other words, ow isaligned with forcing. In all other situations the direction of thevelocity hUi is at some angle with respect to VP (see Fig. 1). Themaximum angle q between hUi and VP provides the largesttransverse ow. The task of the optimization of its magnitudecan be transformed into the maximization of a so-calledanisotropy parameter

l ¼ffiffiffiffiffiffik*k

k*t

s$ 1; (16)

and then q ¼ arctan (l).18 The anisotropy parameter l forchannels with a trapezoidal textured wall can be calculated byusing eqn (12) and (16).

We rst investigate the effect of a on the anisotropyparameter l for the same textures as in Fig. 2. The results areshown in Fig. 4(a). We see that for all textures with large localslip l approaches its limit, which can be evaluated as

lðb/NÞ ¼ 1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4þ 9f1 � 9f1

2

q: (17)

We have shown that the permeability of a thin SH channelwith a trapezoidal wall is maximized by reducing the areafraction of solid in contact with the liquid (see Fig. 2). Incontrast, Fig. 4(a) and eqn (17) suggest that transverse ow in

Fig. 3 The function r versus b. The inset illustrates the striped (red) and thetrapezoidal (black) local height profiles, leading to the same longitudinalpermeability of the channel at a given b.

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Fig. 4 Anisotropy parameter for (a) the same patterns as in Fig. 2 (b) saw-toothtextures with f1 ¼ 0 and (from top to bottom) a ¼ 0.25, 0.5, 1.

Fig. 5 The boundary between two types of dependence of anisotropy param-eter l(b).

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such channels enhances for textures with a larger solid fraction.Thus, the transverse ow in thin channels is maximized bystripes with a rather large solid fraction, f1 ¼ 0.5, and very largeb. However, l becomes very small and even vanishes at large b

for a channel with a saw-tooth texture, f1 ¼ 0. For this texture lreaches its maximum value at some b ¼ O(1). In contrast, forregular trapezoids (with nite f1) l increases monotonouslywith b, which is qualitatively similar to a striped channel.

Next we examine the effect of varying a at xed f1 ¼ 0.Fig. 4(b) shows l as a function of b. The three data sets corre-spond to different a and again show the maximum at some b ¼O(1) and a decay to zero at large b.

The boundary between the two types of behavior of l can befound by using the conditions:

vlðb;f1;aÞvb

¼ 0;v2lðb;f1;aÞ

v2b¼ 0: (18)

Eqn (18) can be solved numerically. The results are shown inFig. 5 and clearly demonstrate that a transition between twotypes of behavior of l happens at small f1 and depends on aparameter a. The upper region in Fig. 5 corresponds to

This journal is ª The Royal Society of Chemistry 2013

trapezoidal SH channels, where l increases with the maximallocal slip, similar to a classical striped channel. The lowerregion corresponds to channels which showmaximal transverseow at some optimal nite b, similar to a channel with the saw-tooth slip at the wall.

The integral formulae for permeability (eqn (6)) used abovehave been obtained in the limit of a thin channel. To estimatethe range of validity of this approximation we solve the Stokesequations in the channel numerically and compare the calcu-lated longitudinal permeability and anisotropy parameter withthose obtained by using eqn (12). Our numerical approach isdescribed in Appendix B.

To test the analytical predictions we have chosen two “extreme”textures, which are equivalent in terms of k*k in a thin channellimit: a sawtooth texture with f1¼ 0, a¼ 1, b¼ 50 and stripes withan equivalent f*1 x 0.06, calculated using eqn (14). Due to avanishing and low f1 these patterns do not fully satisfy theassumptions of our thin channel theory, so that in this case thelargest possible deviations from the analytical results are expected.In our numerical calculationswe keep b constant and vary h¼H/L.This corresponds to the varying texture period L at xed values ofthe groove depth e and channel width H. Note that we can nowrelax the assumption of a thin channel to see if our analyticaltheory could be used outside the area of its formal applicability.

Fig. 6(a) includes curves for the maximum permeabilitycalculated for channels with the saw-tooth and equivalent stripedtextured wall. We remark that although the saw-tooth texturegives slightly greater permeability than the striped one, thedeviations between them are small. This conrms our conclusionon the hydrodynamic equivalence of these two channels. Bothcurves decay with h, which suggests that our analysis based onthe lubrication theory overestimates k*k. However, at small hresults approach the predictions obtained within the lubricationtheory, so that it can be safely used even at the “extreme” situa-tion of a vanishing solid area. At h¼ O(1) the deviations from thetheoretical predictions are signicant, so that the thin-channelanalysis cannot be applied. We have performed similar calcula-tion for different values of b, and have found similar behavior ofk*k(h). This is why we did not show them here.

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Fig. 6 Correction to a longitudinal permeability (a) and the anisotropy param-eter (b) as a function of h ¼ H/L calculated at fixed b ¼ 50. Solid curve shows theresults for a channel with a saw-tooth texture (f1 ¼ 0, a ¼ 1). Dashed curvepresents results for an equivalent striped channel (f*

1 x 0.06).

Fig. 7 Schematic representation of a flow in a closed gas cavity with slowlyvarying depth.

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Fig. 6(b) shows the anisotropy parameter for the same twochannels. We see that the difference between channels with thesaw-tooth and equivalent striped textured wall is again verysmall, so that both channels are expected to generate acomparable transverse ow. It can be seen that a thin channelapproach underestimates l, but could serve as its rough esti-mate when h is relatively small even for channels with f1 ¼ 0.Finally, we note that at h ¼ O(1) the transverse ow is maximal,

Soft Matter

which suggests that channels of such a thickness may bepromising for such microuidic applications as separation ormixing. This subject however is outside of the scope of ourpaper and will be reported elsewhere.

5 Conclusions

We have studied hydrodynamic properties of a thin channel withone wall decorated by a SH trapezoidal texture. General explicitexpressions for the permeability tensor of a thin channel with anarbitrary texture have been formulated. These expressions, beingapplied for a SH trapezoidal channel, allowed us to calculateseveral important hydrodynamic properties. We have demon-strated that the trapezoidal textured wall of the channel caninduce a rich and complex hydrodynamic behavior compared tothat expected for a classical striped channel, and can be advan-tageous in many situations. Furthermore, our theory givesanalytical guidance as to how we can choose the parameters ofthe trapezoidal texture, in order to optimize the forward andtransverse ows. Our theoretical predictions have been comparedwith the results of numerical calculations, which have been alsoperformed for thicker channels. Our results are directly relevantfor drag reduction and passive mixing in thin channels, as well asother microuidic applications.

AppendixA Local slip conditions on the gas–liquid interface forshallow gas cavity

Here we formulate the local slip conditions at the gas areas of aSH surface. We assume that the depth of the gas cavity is small,max[e(y)] � L, i.e. the lubrication (thin-channel) limit can beapplied to the gas ow as well. Then the velocity prole ug(x, y, z)is again locally parabolic in z (cf. eqn (4)):

ugðx; y; zÞ ¼ �VPgðx; yÞ2mg

½zþ c1ðyÞ�½zþ c2ðyÞ�: (19)

The velocity satises the no-slip condition at the bottom wallof the cavity, z ¼ �e, so we should require c1(y) ¼ e(y). The gascavity is assumed to be closed in all horizontal directions. Thismeans that the pressure gradient in the gas VPg is such that thenet ux of the gas ow is zero at any cross-section (see Fig. 7):

ð 0�e

½zþ c1ðyÞ�½zþ c2ðyÞ�dz ¼ 0: (20)

One readily has from eqn (20)

c2(y) ¼ e(y)/3.

The conditions at the gas–liquid interface are the continuityof the velocity and the shear stress:

z ¼ 0: u ¼ ug ¼ �VPge2

6mg

;

mvu

vz¼ mg

vug

vz¼ � 2VPge

3:

(21)

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Excluding VPg in (21) we obtain

z ¼ 0: u ¼ me

4mg

vu

vz;

which is equivalent to the slip condition for the liquid ow withthe local slip length

bðyÞ ¼ meðyÞ4mg

:

Thus the prole of local slip length follows the relief ofshallow texture, but with the factor four times less than the “gascushion model”, eqn (2).

B Numerical method for calculating permeabilities in achannel of arbitrary gap

The numerical method is based on the expansion of the solu-tion into Fourier series and solving a linear system to satisfyboundary conditions eqn (3) on a spatial grid over y ˛ [0, 1/2]. Asimilar method has been used in ref. 49 to nd the effective sliplength for a wall-bound linear shear ow. The only differencehere is that we keep both the growing and the decaying terms inz and require them to annihilate at z ¼ h. Also, we have to solvethe problem for the transverse ow explicitly, because therelationship between longitudinal and transverse effective sliplengths derived in ref. 50 is not valid for a channel of a nitethickness.

We seek a correction to the Poiseuille ow driven by a xedpressure gradient VP:

u ¼ �H2jVPj2m

�vp þ v

�; (22)

where np ¼ z(h � z)/h2ep is the undisturbed ow in non-dimensional variables (z ¼ Z/L and ep is the direction of thepressure gradient) and

v ¼ (vx, vy, vz),

is the perturbation of the ow, caused by the presence of thetexture.

The perturbation is zero at the upper wall (no-slip condi-tions) and satises the dimensionless Stokes equations,

V$v ¼ 0;Vp� Dv ¼ 0;

(23)

where p is pressure perturbation. Our trapezoidal texture isperiodic and symmetric in y: b(y) ¼ b(y + 1), b(1 � y) ¼ b(y).Therefore, the solution to (23) for v and p can be represented inthe form of a Fourier series. To nd the permeability tensor it issufficient to consider two special cases: the pressure gradientdirected parallel (x-direction) and perpendicular (y-direction) tothe texture.

B.1 Longitudinal conguration. When the pressuregradient is directed along the texture, the perturbation of thevelocity has the only one component, v ¼ (vx, 0, 0), and due tothe symmetry of b(y) can be sought in terms of a cosine series:

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vx ¼XNn¼0

v*xðz; nÞcosðknyÞ;

where kn ¼ 2pn.The Stokes equations eqn (23) are then reduced to a single

ordinary differential equation (ODE):

D*v*x ¼ 0, (24)

where D* ¼ d2

dz2� kn

2. The zero-mode solution is

v*x(z, 0) ¼ c0 + c1z.

The solution for non-zero modes has the form

v*x(z, n) ¼ c+(n)exp(knz) + c�(n)exp(�knz). (25)

The no-slip boundary condition at the upper wall (v*x ¼ 0)provides a relationship between c�(n) and c+(n):

c1 ¼ �c0/h, c+(n) ¼ �anc�(n),

where an ¼ exp(�2knh) is the coefficient describing the inu-ence of the upper wall: for innitely thick channel an ¼ 0.

Now we can express the solution and its derivatives in termsof c0 and c�(n) only. To obtain a linear system for these coeffi-cients we truncate the series to N terms and satisfy the partialslip boundary conditions (eqn (3)) in N points distributed overhalf of the period, yj ˛ [0;1/2], j ¼ 1.N (half of the period istaken since the slip length prole is symmetric, and the solu-tion is sought in cosines only). The resulting system of Nequations takes the following form:

c0

�1þ b

�yj�

h

�þXN�1

n¼1

c�ðnÞfn � b

�yj�gncos�knyj

� ¼ b�yj�

h;

j ¼ 1.N;

(26)

where fn ¼ (1 � an), gn ¼ �(1 + an)kn.The correction to channel permeability is obtained by aver-

aging eqn (22) over the channel gap and the texture period. Onlythe zero-mode term survives:

k*|| ¼ 1 + 3c0.

The coefficient c0 is found numerically by solving the linearsystem (26).

B.2 Transverse conguration. When the pressure gradientis directed across the texture, the velocity has two components,v ¼ (0, vy, vz), and the solution can be sought in terms of theFourier coefficients as follows:

vz ¼ v*zðz; 0Þ þXNn¼1

v*zðz; nÞsinðknyÞ;

vy ¼ v*yðz; 0Þ þXNn¼1

v*yðz; nÞcosðknyÞ;

p ¼ p*ðz; 0Þ þXNn¼1

p*ðz; nÞsinðknyÞ:

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The Stokes equations for transverse stripes can be written as:

�knv*y þ

dv*zdz

¼ 0;

�knp*� D*v*y ¼ 0;

dp*

dz� D*v*z ¼ 0:

(27)

for n ¼ 1.N. The solution for the zero-mode term is

v*zðz; 0Þ ¼ 0; v*yðz; 0Þ ¼ d0ð1� z=hÞ; p*ðz; 0Þ ¼ const:

By excluding p* and v*y, one can transform the equation fornon-zero modes into a single ODE for the z-component:

D*2v*z ¼ 0. (28)

A general solution of the fourth-order ODE (28) can bewritten in the following form:

v*z ¼ (c� + d�z)exp(�knz) + (c+ + d+z)exp(knz),

which involves four unknown constants per mode. The Fouriercoefficients for the y-component can be obtained from eqn (27):

v*y ¼1

kn

dv*zdz

:

The no-slip and impermeability conditions read

v*z(0, n) ¼ 0, v*z(h, n) ¼ 0,v*y(h, n) ¼ 0.

They provide three conditions per mode and enable us toexpress all the unknown coefficients in terms of d�(n). To satisfythe partial slip boundary conditions we have the followinglinear system:

d0

�1þ b

�yj�

h

�þXN�1

n¼1

d�ðnÞfn � b

�yj�gncos�knyj

� ¼ b�yj�

h;

j ¼ 1.N;

(29)

where

fn ¼ 2an � an2 � 1þ gn

2an

knðan þ gnan � 1Þ ;

gn ¼ � 2�an

2 � 1þ 2gnan

�ðaþ gnan � 1Þ ;

gn ¼ 2knh.

At large h we have pn ¼ 1 and qn ¼ �2kn, in full agreementwith the result of the innite channel.50 The linear system (29) issolved numerically, and the correction to transverse perme-ability is calculated as

k*t ¼ 1 + 3d0.

Soft Matter

Acknowledgements

This research was supported by the RAS through its priorityprogram “Assembly and Investigation of MacromolecularStructures of New Generations” and by the DFG throughSFB 985.

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