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J. Phys.: Condens. Matter 23 (2011) 184104 (15pp)

doi:10.1088/0953-8984/23/18/184104

Wetting, roughness and flow boundary conditions
Olga I Vinogradova1,2 and Aleksey V Belyaev1
1

,3

A N Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences, 31 Leninsky Prospect, 119991 Moscow, Russia 2 ITMC and DWI, RWTH Aachen, Pauwelsstraúe 8, 52056 Aachen, Germany 3 Physics Department, M V Lomonosov Moscow State University, 119991 Moscow, Russia E-mail: oivinograd@yahoo.com

Received 31 May 2010, in final form 18 October 2010 Published 20 April 2011 Online at stacks.iop.org/JPhysCM/23/184104 Abstract We discuss how the wettability and roughness of a solid impacts its hydrodynamic properties. We see in particular that hydrophobic slippage can be dramatically affected by the presence of roughness. Owing to the development of refined methods for setting very well controlled micro- or nanotextures on a solid, these effects are being exploited to induce novel hydrodynamic properties, such as giant interfacial slip, superfluidity, mixing and low hydrodynamic drag, that could not be achieved without roughness. (Some figures in this article are in colour only in the electronic version)

1. Introduction
Fluid mechanics is one of the oldest and most useful of the `exact' sciences. For hundreds of years it has relied upon the no-slip boundary condition at a solid­liquid interface that was applied successfully to model many macroscopic experiments [1]. However, the problem is not that simple and has been revisited in recent years. One reason for such a strong interest in this `old' problem is purely fundamental. The noslip boundary condition is an assumption that cannot be derived from first principles even for a molecularly smooth hydrophilic (the contact angle, fixed by the chemical nature of a solid, lies between 0 and 90 ) surface. Therefore, the success of the no-slip postulate may not always reflect its accuracy but in fact rather the insensitivity of the experiment. Another reason for current interest in flow boundary conditions lies in the potential applications in many areas of engineering and applied science that deal with small size systems, including micro- and nanofluidics [2], flow in porous media, friction and lubrication, and biological fluids. The driving and mixing of liquids when the channel size decreases represent very difficult problems [3]. There is therefore a great hope that changes in hydrodynamic behavior can be modified by interfacial phenomena on the flow. For example, even ideal solids, which are both flat and chemically homogeneous, can have a contact angle that exceeds 90 (the hydrophobic case). This can modify the hydrodynamic boundary conditions, as has already been shown
0953-8984/11/184104+15$33.00

in early work [4]. Furthermore, solids are not ideal but rough. This can further change, and quite dramatically, the boundary conditions. It is of course interesting and useful to show how the defects or pores of the solids modify the conditions. But today, the question has slightly shifted. Thanks to techniques coming from microelectronics, we are able to elaborate substrates whose surfaces are patterned (often at the micro- and nanometer scale) in a very well controlled way, thus providing properties (e.g. optical or electrical) that the solid did not have when flat or slightly disordered. A texture affects the wettability and boundary conditions on a substrate, and can induce unique properties that the material could not have without these micro- and nanostructures. In particular, in the case of super-hydrophobic solids, which are generated by a combination of surface chemistry and patterns, roughness can dramatically lower the ability of drops to stick, leading to the remarkable mobility of liquids. At the macroscopic scale this renders them `self-cleaning' and causes droplets to roll (rather than slide) under gravity and rebound (rather than spread) upon impact instead of spreading [5]. At the smaller scale, reduced wall friction and a superlubricating potential are almost certainly associated with the breakdown of the no-slip hypothesis. In this paper we concentrate on the understanding and expectations for the fluid­solid boundary conditions in different situations where hydrophobicity and roughness impact the flow properties. After introducing the terminology,
1
© 2011 IOP Publishing Ltd Printed in the UK & the USA


J. Phys.: Condens. Matter 23 (2011) 184104

O I Vinogradova and A V Belyaev

(a)

z

Liquid u b
s

Solid

with slip

no-slip plane

Figure 1. Schematic representation of the definition of intrinsic (a), apparent (b) and effective (c) slip lengths.

and describing new developments and instruments that give the possibility of investigating fluid behavior at the micro- and nanoscale, in the following section we present results obtained for smooth surfaces, by highlighting the role of wettability. Then follows the results for rough hydrophilic and, especially, hydrophobic surfaces. In the latter case we show, and this is perhaps the main message of our paper, how roughness can enhance hydrodynamic slip and thus the efficiency of transport phenomena.

2. Terminology
We will refer to slip as being any situation where the value of the tangential component of velocity appears to be different from that of the solid surface. The simplest possible relation assumes that the tangential force per unit area exerted on the solid surface is proportional to the slip velocity. Combining this with the constitutive equation for the bulk Newtonian fluid one gets the so-called (scalar) Navier boundary condition

us = b

u , z

(1)

positive slip velocity. It can, however, be negative, although in this case it would not have a long-range effect on the flow [7]. Obviously the control of slip lengths is of major importance for flow at an interface and in a confined geometry. It would be useful to distinguish between three different situations for a boundary slip since the dynamics of fluids at the interface introduce various length scales. Molecular (or intrinsic) slip, which allows liquid molecules to slip directly over a solid surface (figure 1(a)). Such a situation is not of main concern here since molecular slip cannot lead to a large b [4, 8, 9] and its calculations require a molecular consideration of the interface region. In particular, recent molecular dynamics simulations predicted a molecular b below 10 nm for realistic contact angles [10, 11]. Therefore, it is impossible to benefit from such a slip in a larger scale applications. The intrinsic boundary condition may be rather different from what is probed in flow experiments at larger length scale. It has been proposed [6] to describe the interfacial region as a lubricating `gas film' of thickness e of viscosity g different from its bulk value . A straightforward calculations give apparent slip (figure 1(b))

where u s is the (tangential) slip velocity at the wall, u / z the local shear rate and b the slip length. This slip length represents a distance inside the solid to which the velocity has to be extrapolated to reach zero. The standard no-slip boundary condition corresponds to b = 0, and the shear-free boundary condition corresponds to b [6]. In the most common situation b is finite (a partial slip) and associated with the 2

b=e

-1 g

e

. g

(2)

This represents the so-called `gas cushion model' of hydrophobic slippage [6], which has a clear microscopic foundation in terms of a prewetting transition [12]. Being a schematic representation of a depletion close to a wall [13],


J. Phys.: Condens. Matter 23 (2011) 184104

O I Vinogradova and A V Belyaev

this model provides a useful insight into the sensitivity of the interfacial transport to the structure of an interface. Similarly, electrokinetic flow displays apparent slip. Another situation is that of effective slip, beff , which refers to a situation where slippage at a complex heterogeneous surface is evaluated by averaging of a flow over the length scale of the experimental configuration (e.g. a channel etc) [3, 14­16]. In other words, rather than trying to solve equations of motion at the scale of the individual corrugation or pattern, it is appropriate to consider the `macroscale' fluid motion (on the scale larger than the pattern characteristic length or the thickness of the channel) by using effective boundary conditions that can be applied at the imaginary smooth surface. Such an effective condition mimics the actual one along the true heterogeneous surface. It fully characterizes the flow at the real surface and can be used to solve complex hydrodynamic problems without tedious calculations. Such an approach is supported by a statistical diffusion arguments (being treated as an example of commonly used Onsager­ Casimir relations for non-equilibrium linear response) [14] theory of heterogeneous porous materials [15], and has been justified for the case of Stokes flow over a broad class of surfaces [16]. For anisotropic textures beff depends on the flow direction and is generally a tensor [14]. Effective slip also depends on the interplay between typical length scales of the system as we will see below. Well-known examples of such a heterogeneous system include composite super-hydrophobic (Cassie) surfaces, where a gas layer is stabilized with a rough wall texture (figure 1(c)). For these surfaces effective slip lengths are often very large compared with the value on flat solids, similar to what has been observed for wetting, where the contact angle can be dramatically enhanced when the surface is rough and heterogeneous [17].

Figure 2. Schematic of the dynamic AFM force experiment.

3. Experimental methods
The experimental challenge has generated considerable progress in the experimental tools for investigating flow boundary conditions, using the most recent developments in optics and scanning probe techniques. A great variability still exists in the results of slip experiments so it is important first to consider the different experimental methods used to measure slip. Two broad classes of experimental approaches have been used so far: indirect and direct (local) methods. High-speed force measurements can be performed with the SFA (surface forces apparatus) [18­20] or AFM (atomic force microscope) [21]. In particular, in the drainage method [18, 21] the end of the spring away from the attached sphere is driven toward the (fixed) plane with a constant driving speed (as shown in figure 2). The sphere itself, however, does not move at a constant speed, so that the spring is deflected as a result of both the surface force (which should be measured separately) and the hydrodynamic forces. The solution of the (differential) equation of motion allows one to deduce a drag force, with the subsequent comparison with a theory of a film drainage [6, 22]. This approach, being extremely accurate at the nanoscale, does not provide visualization of the flow profile, so these measurements are identified as indirect. 3

Direct approaches to flow profiling, or velocimetry, take advantage of various optics to monitor tracer particles. These methods include TIR-FRAP (total internal reflection fluorescence recovery after photo-bleaching) [23], -PIV [24, 25] (particle image velocimetry), TIRV (total internal reflection velocimetry) [26], EW -PIV (evanescent wave micro-particle image velocimetry) [27], and multilayer nano-particle image velocimetry (nPIV) [28]. Their accuracy is normally much lower than that of force methods due to relatively low optical resolution, system noise due to polydispersity of tracers and difficulties in decoupling flow from diffusion (the tracer distribution in the flow field is affected by Taylor dispersion [29]). As a consequence, it has been always expected that a slippage of the order of a few tens of nanometers cannot be detected by a velocimetry technique. However, recently, direct high-precision measurements at the nanoscale have been performed with a new optical technique, based on a DF-FCS (double-focus spatial fluorescence crosscorrelation) [30, 29] (as is schematically shown in figure 3). As the fluorescence tracers flow along the channel they cross the two foci in turn, producing two time-resolved fluorescence intensities, I1 (t ) and I2 (t ), recorded independently. The time cross-correlation function can be calculated and typically exhibits a local maximum. The position of this maximum M is characteristic of the local velocity of the tracers. Another example of high resolution applications of FCS consists in the determination of the average transverse diffusion coefficient to probe slippage [31]. Since FCS methods allow consideration of N 106 particles, this gives a satisfactory signal to noise ratio N of the order 103 , providing extremely good resolution as compared with other direct velocimetry methods. Coupling this with TIRF [32], which allows the measurements of the distance of tracers from the wall through the exponential decay of an evanescent wave, should further improve the accuracy of the approach. Many experiments have been performed on the subject, with somewhat contradictory results. Experimental work focused mostly on bare (smooth) surfaces, more recent investigations have turned towards rough and structured surfaces, in particular super-hydrophobic surfaces [33]. We refer the reader to comprehensive review articles [8, 34] for detailed accounts of the early experimental work. Below we


J. Phys.: Condens. Matter 23 (2011) 184104

O I Vinogradova and A V Belyaev

Figure 3. Schematics of the double-focus spatial fluorescence cross-correlation method. Two laser foci are located along the x axis, separated by a distance of a few micrometers. They independently record the time-resolved fluorescence intensities I1 (t ) and I2 (t ). The forward cross-correlation of these two signals yields G (t ). Two foci are scanned simultaneously along the z axis to probe the velocity profile u (z ).

mention only what we believe is the most relevant recent contribution to the subject of flow past `simple' smooth surfaces, which clarified the situation, highlighted the reasons for the existing controversies and resolved apparent paradoxes. We focus, however, more on the implication of micro- and nanostructuring on fluidic transport, which is still in its infancy and remains to be explored.

microfluidic applications. This is why in the discussion of super-hydrophobic slippage below we often ignore a slip past hydrophobic solids. However, a hydrophobic slippage is likely to be of major importance in nanochannels (highly confined hydrophobic pores, biochannels, etc), where ordinary Poiseuille flow is fully suppressed.

5. Rough surfaces 4. Smooth surfaces: slippage versus wetting
From the theoretical [9, 35] and simulation [11, 36] point of view, slippage should not appear on a hydrophilic surface, except probably at a very high shear rate [37]. A slip length of the order of a hundred nanometers or smaller is, however, expected for a hydrophobic surface [6, 9, 12, 38]. On the experimental side, no consensus was achieved until recently. While some experimental data were consistent with the theoretical expectations both for hydrophilic and hydrophobic surfaces [19, 21], some other reports completely escaped from this picture with both qualitative (slippage over hydrophilic surfaces, shear rate dependent slippage, rate thresholds for slip, etc) and quantitative (slip length of several micrometers) discrepancies (for a review see [8]). More recent experiments, performed with various new experimental methods, finally concluded that water does not slip on smooth hydrophilic surfaces, and develops a slip only on hydrophobic surfaces [29, 31, 39­42]. One can therefore conclude that a concept of hydrophobic slippage is now widely accepted. An important issue is the amplitude of hydrophobic slip. The observed slip length reached the range 20­100 nm, which is above predictions of the models of molecular slip [10, 43]. This suggests the apparent slip, such as the `gas cushion model', equation (2). Water glides on air, owing to the large viscosity ratio between water and air (typically a factor of 50). Experimental values of b suggest that the thickness of this `layer' is below 2 nm. A modification of this scenario would be a nanobubble coated surface [44­47]. Another important conclusion is that it is impossible to benefit from such a nanometric slip at separations O(m) and larger, i.e. in 4 Only a very few solids are molecularly smooth. Most of them are naturally rough, often at a micro- and nanoscale, due to their structure, methods of preparation, various coatings, etc. These surfaces are very often in the Wenzel (impaled) state, where the solid­liquid interface has the same area as the solid surface (figure 4(a)). However, even for rough hydrophilic Wenzel surfaces the situation is not very clear, and opposing experimental conclusions have been made: one is that roughness generates extremely large slip [48], whereas another is that it decreases the degree of slippage [49, 50]. More recent experimental data suggests that the description of flow near rough surfaces has to be corrected, but for a separation, not slip [39]. The theoretical description of such a flow represents a difficult, nearly insurmountable, problem. It has been solved only approximately, and only for a case of the periodic roughness and far-field flow with a conclusion that it may be possible to approximate the actual surface by a smooth one with the slip boundary condition [51­53]. This issue was recently resolved in a LB (lattice Boltzmann) simulation study [36], where the hydrodynamic interaction between a smooth sphere of radius R and a randomly rough plane was studied (as shown in figure 5). Beside its significance as a geometry of SFA/AFM dynamic force experiments, this allows one to explore both far- and near-field flows in a single `experiment'. The `measured' hydrodynamic force was smaller than predicted for two smooth surfaces (with the separation defined at the top of the asperities) when the standard no-slip boundary conditions were used in the calculation. Moreover, at small separations the force was even weaker and showed different asymptotics


J. Phys.: Condens. Matter 23 (2011) 184104

O I Vinogradova and A V Belyaev

Figure 4. Schematic representation of the (a) Wenzel and (b) Cassie pictures with the local flow profiles at the gas and solid areas.

Figure 5. Hydrodynamic force acting on a hydrophilic sphere of radius R approaching a smooth hydrophilic (diamonds), smooth hydrophobic (circles) and randomly rough hydrophilic (triangles) wall with 2 = 4% (adapted from [36]). Here FSt = 6 RU is the Stokes drag. The separation h is defined on top of the surface roughness, as shown in the inset. Simulation results (symbols) compared with theoretical curves: F / FSt = 1 + 9 R /(8h ) (solid), F / FSt = 1 + 9 Rf /(8h ) with f = f (b/ h ) taken from [7] (dash-dotted) and F / FSt = 1 + 9 R /(8[h + s ]) (dashed). Values of b and s were determined by fitting the simulation data.

than those expected if one invokes slippage at the smooth fluid­solid interfaces. This can only be explained by the model of a no-slip wall, located at an intermediate position (controlled by the density of roughness elements) between the top and bottom of the asperities (illustrated by the dashed line figure 4(a)). Calculations based on this model provided an excellent description of the simulation data (figure 5).

6. Super-hydrophobicity and effective hydrodynamic slippage
On hydrophobic solids, the situation is different from that on hydrophilic solids. If the solid is rough enough, we do not expect that the liquid will conform to the solid surface, as assumed in the Wenzel or impaled state. Rather, air pockets should form below the liquid, provided that the energetic cost associated with all the corresponding liquid­vapor interfaces is smaller than the energy gained not to follow the solid [5]. This is the so-called Cassie or fakir state. Hydrophobic Cassie materials generate large contact angles and small hysteresis, ideal conditions for making water drops very mobile. It is natural to expect a large effective slip in a Cassie situation. 5

Indeed, taking into account that the variation of the texture height, e, is in the typical interval 0.1­10 m, according to equation (2) we get b = 5­500 m at the gas area. The composite nature of the texture requires regions of very low slip (or no slip) in direct contact with the liquid, so the effective slip length of the surface, beff , is smaller than b . Still, one can expect that a rational design of such a texture could lead to a large values of beff . Below we make these arguments more quantitative. We will examine an idealized super-hydrophobic surface in the Cassie state, sketched in figure 4(b), where a liquid slab lies on top of the surface roughness. The liquid­gas interface is assumed to be flat with no meniscus curvature, so that the modeled super-hydrophobic surface appears as perfectly smooth with a pattern of boundary conditions. In the simplified description the latter are taken as no-slip (b1 = 0) over solid­ liquid areas and as partial slip (b2 = b ) over gas­liquid regions (as we have shown above, b1 is of the orders of tens of nanometers, so that we could neglect it since b2 is of the order of tens of micrometers). We denote as the typical length scale of gas­liquid areas. The fraction of solid­liquid areas will be denoted 1 = ( L - )/ L , and of gas­liquid area 2 = 1 - 1 = / L . Overall, the description of a super-hydrophobic surface we use is similar to those considered in [15, 54­57]. In this idealization, some assumptions may have a possible influence on the friction properties and, therefore, a hydrodynamic force. First, by assuming a flat interface, we have neglected an additional mechanism for a dissipation connected with the meniscus curvature [58­60]. Second, we ignore a possible transition towards the impaled (Wenzel) state that can be provoked by additional pressure in the liquid phase [61, 62]. Finally, for the sake of brevity we focus below only on the canonical microfluidic geometry where the fluid is confined between flat plates, and only on the asymmetric case, where one (upper) surface is smooth hydrophilic and the other (lower) represents a super-hydrophobic wall in the Cassie state. Such a configuration is relevant for various setups, where the alignment of opposite textures is inconvenient or difficult. We also restrict the discussion to a pressure-driven flow governed by the Stokes equations

2 u = p ,

· u = 0,

(3)

where u is the velocity vector, and p is pressure. Extensions of our analysis to study other configuration geometries and types of flow would be straightforward.


J. Phys.: Condens. Matter 23 (2011) 184104

O I Vinogradova and A V Belyaev

gradient p = (-, 0, 0). Essentially, since along these orthogonal directions there are no transverse hydrodynamic couplings [14], the pressure gradient p coincides with the direction of slip for longitudinal and transverse stripes. We seek the solution for a velocity u by perturbation analysis: u = u0 + u1 , (6)

where u0 is the velocity of the Poiseuille flow, and the effective slip length beff at the super-hydrophobic surface is defined as

b
Figure 6. Sketch of a flat channel of thickness H , with notation for directions along the plates. One wall represents an anisotropic super-hydrophobic texture.

eff

=

(

u z=0 u z )z =0

,

(7)

6.1. Anisotropic surfaces Many natural and synthetic textures are isotropic. However, it can be interesting to design directional structures, such as arrays of parallel grooves or microwrinkles, that consequently generate anisotropic effective slip in the Cassie regime. The hydrodynamic slippage is quite different along and perpendicular to the grooves. Axial motion is preferred, and such designs are appropriate when liquid must be guided. There are examples of such patterns in nature, such as the wings of butterflies or the legs of water striders. The flow past such surfaces becomes misaligned with the pressure gradient and has been analyzed in a number of studies [63, 64]. Such phenomena have motivated a tensorial version of (1), as discussed in [3, 14]:

where u denotes x -component of the velocity and ··· means the average value in plane xO y . The effective slip length can now be calculated by using the dual series technique suggested in recent work [60, 55]. By employing a family of Fourier series solutions to equations (3), together with boundary conditions u(x , y , 0) = b (x , y ) ·

u (x , y , 0), z

z · u(x , y , 0) = 0, ^ (8) (9)

u(x , y , H ) = 0,

z · u(x , y , H ) = 0, ^

we obtain the dual series problem (in dimensionless form) for the longitudinal and transverse configurations

0 1 +
n =1

h

+

n =1

n [1 + nV (nh )] cos(n ) = ,
(10)

0<

2 ,

0 +

n cos(n ) = 0,

2 <

,

(11)

ui |

A

=
j ,k

ef bij f n

k

u x

j kA

,

(4)

where {, h , } = (2/ L ) ·{z , H , b } and the function V (x ) is defined as V (x ) = coth(x ) (12) for a longitudinal flow and sinh(2x ) - 2x cosh(2x ) - 2x 2 - 1

where u| A is the effective slip velocity, averaged over the surface pattern and n is a unit vector normal to the surface ef A. The second-rank effective slip tensor beff {bij f } characterizes the surface anisotropy and is represented by symmetric, positive definite 2 â 2 matrix diagonalized by a rotation: beff = S

V (x ) = 2

(13)

for a transverse flow. The effective slip length is then

b

eff

=

0 L . 2 1 - 0 / h

(14)

beff 0

(5) As proven in [14], for all anisotropic surfaces the eigenvalues beff and beff of the slip-length tensor correspond to the fastest (greatest forward slip) and slowest (least forward slip) directions, which are always orthogonal (see figure 6). To illustrate the calculation of the slip-length tensor, below we consider the geometry where the liquid is confined between two plates separated by a distance H , and one of them represents a super-hydrophobic striped wall (figure 7(a)). 6.1.1. General solution. We calculate the effective slip lengths in the eigendirections (which in this case are obviously parallel and orthogonal to the stripes), by solving the Stokes equations (3). The x -axis is directed along the pressure 6

0 beff

S- ,

S =

cos - sin

sin cos

.

Following [60] we can now use the orthogonality of trigonometric sine and cosine functions to obtain a system of linear algebraic equations


A
n =0

nm n

= Bm ,

(15)

that can be solved in respect to n . Figure 8(a) shows the typical calculation results (the numerical example corresponds to b / L = 20 and 2 = 0.75) and demonstrates that the effective slip lengths increase with H and saturate for a thick gap. This points to the fact that an effective boundary condition is not a characteristic of the liquid­solid interface alone, but depends on the flow configuration and interplay between the typical length scales, L , H and b , of the problem. Next we discuss the asymptotic limits (of small and large gaps) of our semi-analytical solution.


J. Phys.: Condens. Matter 23 (2011) 184104

O I Vinogradova and A V Belyaev

Figure 7. Special textures arising in the theory: (a) stripes, which attain the Wiener bounds of maximal and minimal effective slip, if oriented parallel or perpendicular to the pressure gradient, respectively; (b) the Hashin­Shtrikman fractal pattern of nested circles, which attains the maximal/minimal slip among all isotropic textures (patching should fill up the whole space, but their number is limited here for clarity); and (c) the Schulgasser and (d) chessboard textures, whose effective slip follows from the phase interchange theorem.

Figure 8. Eigenvalues beff (solid curve) and beff (dashed curve) of the slip-length tensor for stick­slip stripes of period L with local slip length at the liquid­gas interface b/ L = 20 and slipping area fraction 2 = 0.75, as a function of the thickness of the channel, H .

6.1.2. Thin channel. vicinity of x = 0

The Taylor expansion of V (x ) in the

coth x | 2

x 0

=x

-1

+ O (x ), = 4x
-1

(16) (17)

sinh(2x ) - 2x cosh(2x ) - 2x 2 - 1

x 0

+ O (x ),

allows us to find analytical expressions for 0 and beff in the limit of H L . By substituting them into (10) and (11), we get bH 2 bH 2 beff , beff . (18) H + b 1 H + 4b 1 These expressions are independent of L , but depend on H ,and suggest a way to distinguish between two separate cases. If b H we obtain

b

eff

b

eff

b 2 ,

(19)

so that despite the surface anisotropy we predict a simple surface-averaged effective slip. Although this limit is less important for pressure-driven microfluidics, it may have relevance for amplifying transport phenomena [65]. When H b we derive

b

eff

H

1 , 2

b

eff

1 b 4 eff

.

(20)

The above formula implies the effective slip length is generally four times as large for parallel versus perpendicular pressuredriven flow. Both asymptotic results, equations (19) and (20), 7

are surprising taking into account that for anisotropic Stokes flow in a thick channel a factor of two is often expected, as reminiscent of results for striped pipes [66], sinusoidal grooves [16] and the classical result that a rod sediments twice as fast in creeping flow if aligned vertically rather than horizontally [67]. A very important conclusion from our analysis is that this standard scenario can significantly differ in a thin super-hydrophobic channel, giving a whole spectrum of possibilities, from isotropic to highly anisotropic flow, depending on the ratio b / H . Note that in case of a thin channel the flow can be described by an expression of Darcy's law, which relates the depth-averaged fluid velocity to an average pressure gradient along the plates through the effective permeability of the channel. The permeability, Keff , is in turn expressed through the effective slip length beff , and permeability and slip-length tensors are coaxial. Such an approach allows one to use the theory of transport in heterogeneous media [68], which provides exact results for an effective permeability over length scales much larger than the heterogeneity. This theory allows one to derive rigorous bounds on an effective slip length for arbitrary textures, given only the area fraction and local (any) slip lengths of the low-slip (b1 ) and high-slip (b2 ) regions [15, 69]. These bounds constrain the attainable effective slip and provide theoretical guidance for texture optimization, since they are attained only by certain special textures in the theory. In some regimes, the bounds are close enough to obviate the need for tedious calcu