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Journal of Fluid Mechanics
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Effective slip boundary conditions for arbitrary one dimensional surfaces
Evgeny S. Asmolov and Olga I. Vinogradova
Journal of Fluid Mechanics / Volume 706 / September 2012, pp 108 117 DOI: 10.1017/jfm.2012.228, Published online:

Link to this article: http://journals.cambridge.org/abstract_S0022112012002285 How to cite this article: Evgeny S. Asmolov and Olga I. Vinogradova (2012). Effective slip boundary conditions for arbitrary onedimensional surfaces. Journal of Fluid Mechanics,706, pp 108117 doi:10.1017/jfm.2012.228 Request Permissions : Click here

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J. Fluid Mech. (2012) , vol. 706, pp. 108­117. doi:10.1017/jfm.2012.228

c Cambridge University Press 2012

108

Effective slip boundary conditions for arbitrary one -dimensional surfaces
Evgeny S. Asmolov
1 ,2 ,3

and Olga I. Vinogradova1

,4 ,5

1 A. N. Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences,

31 Leninsky Prospect, 119991 Moscow, Russia
2 Central Aero-Hydrodynamic Institute, 140180 Zhukovsky, Moscow region, Russia 3 Institute of Mechanics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia 4 Department of Physics, M. V. Lomonosov Moscow State University, 119991 Moscow, Russia 5 DWI, RWTH Aachen, Forckenbeckstr. 50, 52056 Aachen, Germany

(Received 3 March 2012; revised 21 April 2012; accepted 14 May 2012; first published online 7 June 2012)

In many applications it is advantageous to construct effective slip boundary conditions, which could fully characterize flow over patterned surfaces. Here we focus on laminar shear flows over smooth anisotropic surfaces with arbitrary scalar slip b(y), varying in only one direction. We derive general expressions for eigenvalues of the effective slip-length tensor, and show that the transverse component is equal to half of the longitudinal one, with a two times larger local slip, 2b(y). A remarkable corollary of this relation is that the flow along any direction of the one-dimensional surface can be easily determined, once the longitudinal component of the effective slip tensor is found from the known spatially non-uniform scalar slip. Key words: low-Reynolds-number flows, microfluidics, effective slip

1. Introduction With recent advances in microfluidics (Stone, Stroock & Ajdari 2004), renewed interest has emerged in quantifying the effects on fluid motion of surface chemical heterogeneities with a different local scalar slip. In this situation it is advantageous to construct the effective slip boundary condition, which is applied at the imaginary smooth homogeneous, but generally anisotropic, surface and mimics the actual one along the true heterogeneous slip surface (Kamrin, Bazant & Stone 2010; Vinogradova & Belyaev 2011). Such an effective condition fully characterizes the flow at the real surface (on a scale larger than the pattern characteristic length) and can be used to solve complex hydrodynamic problems without tedious calculations. For an anisotropic texture, the effective boundary condition generally depends on the eff direction of the flow and is a tensor, beff {bij } represented by a symmetric, positive definite 2 в 2 matrix (Bazant & Vinogradova 2008)
beff = S

beff 0

0 b eff

S- ,

(1.1)

Email address for correspondence: AES50@yandex.ru


Effective slip boundary conditions for arbitrary one-dimensional surfaces diagonalized by a rotation
S =

109

cos - sin

sin cos

.

(1.2)

Therefore, (1.1) allows us to calculate an effective slip in any direction given by an angle . In other words, the general problem reduces to computing the two eigenvalues, beff ( = 0) and b ( = /2), which attain the maximal and eff minimal directional slip lengths, respectively. This tensorial slip approach, based on a consideration of a `macroscale' fluid motion instead of solving hydrodynamic equations at the scale of the individual pattern, has been supported by statistical diffusion arguments (Bazant & Vinogradova 2008), and was recently justified for the case of Stokes flow over a broad class of periodic surfaces (Kamrin et al. 2010). The concept of an effective tensorial slip has recently been proved by simulations (Priezjev 2011; Schmieschek et al. 2012), and has already been used to obtain simple solutions of several complex problems. It may be useful in many situations, such as drainage of thin films (Belyaev & Vinogradova 2010b; Asmolov, Belyaev & Vinogradova 2011), mixing in superhydrophobic channels (Vinogradova & Belyaev 2011), and electrokinetics of patterned surfaces (Bahga, Vinogradova & Bazant 2010; Belyaev & Vinogradova 2011). However, to the best of our knowledge all these analytical solutions were obtained for a flow on alternating slip and no-slip stripes, and the quantitative understanding of effective slippage past other types of anisotropic surfaces is still a challenge. In this paper, we study the Stokes flow past flat surfaces, where the local (scalar) slip length b varies only in one direction. Our focus is on the limit of a thick (compared to texture period) channel or a single interface, so that effective slip is a characteristic of a heterogeneous interface solely and does not depend on the channel thickness. We derive a simple universal relationship between eigenvalues of the sliplength tensor. This allows one to avoid tedious calculations of flows in a transverse configuration, by reducing the problem to an analysis of longitudinal flows, which is much easier to evaluate. Our results open the possibility of solving a broad class of hydrodynamic problems for one-dimensional textured surfaces.
2. Theory 2.1. General consideration

We consider a creeping flow along a plane anisotropic wall, and a Cartesian coordinate system (x, y, z) (figure 1). The origin of coordinates is placed at the flat interface, characterized by a slip length b(y), spatially varying in one direction, and the texture varies over a period L. Our analysis is based on the limit of a thick channel or a single interface, so that the velocity profile sufficiently far above the surface may be considered as a linear shear flow. Note that our results do not apply to a thin or an arbitrary channel situation, where the effective slip scales with the channel width (Feuillebois, Bazant & Vinogradova 2009; Schmieschek et al. 2012). Dimensionless variables are defined by using L as a reference length scale, the shear rate sufficiently far above the surface, G, and the fluid kinematic viscosity, . We seek the solution for the velocity profile in the form v = U + u1 , (2.1)


110

E. S. Asmolov and O. I. Vinogradova

F I G U R E 1 . Schematic representation of periodic textures with scalar slip boundary conditions, varying in direction y. The patterns of slip boundary conditions are depicted as alternating stripes with piecewise-constant slip lengths, but our discussion is more general and applies to any one-dimensional distribution of a local slip (e.g. sinusoidal, trapezoidal, etc.).

where U = zel , l = x, y, is the undisturbed linear shear flow, and el are the unit vectors. The perturbation of the flow, u1 = (u, v , w), which is caused by the presence of the texture and decays far from the surface at small Reynolds number Re = GL2 / satisfies dimensionless Stokes equations, · u1 = 0, p - u1 = 0, (2.2a) (2.2b)

where p is pressure. The boundary conditions at the wall and at infinity are defined in the usual way u1 = (y) el , (2.3) z w = 0, (2.4) u1 = 0, (2.5) z: z where u1 = (u, v , 0) is the velocity along the wall and = b/L is the normalized slip length. A local slip length can be expanded in a Fourier series z=0: u1 - (y) (y) =
n=-

b (n) exp (ikn y) ,

(2.6) (2.7)

Similarly, the solution to (2.2) for u1 and p has the form u1 =
n=-

kn = 2n.

u (n, z) exp(ikn y),

p=

n=-

p (n, z) exp(ikn y),

(2.8)

and the Stokes equations can then be rewritten as


· u = 0, p - u = 0,



(2.9) (2.10)


Effective slip boundary conditions for arbitrary one-dimensional surfaces d d , =2 dz dz We have therefore reduced the problem to a system of (ODE), which can be now solved analytically. A similar to address different hydrodynamic problems (Asmolov Kamrin et al. 2010). The zero-mode solution represents a constant, u eigenvalues of the effective slip-length tensor can be of u (0, z) :


111 (2.11)

= 0, ikn ,

2

2 - kn .

ordinary differential equations strategy has been used before 2008; Davis & Lauga 2010; (0, z) = (cx (0), cy (0), 0). The obtained as the components (2.12)

b

eff

Below we analyse in more detail two configurations of the shear flow, where the slip regions are distributed parallel ( = 0) and transverse ( = /2) to the shear flow direction.
2.2. Longitudinal configuration For longitudinal patterns, U = zex , so that the perturbation of the velocity has only one component, u1 = (u, 0, 0), and the system (2.9)­(2.10) reduces to a single ODE:


= Lcx (0) ,

b

eff

= Lcy (0) .

Its solution that decays at infinity for non-zero modes has the form The boundary condition at the wall, (2.3), then determines constants cx (n). Indeed, from (2.8) and (2.14) we get z=0: u=
n=-

u = 0.

(2.13) (2.14)

u = cx (n) exp(-|kn |z).

cx (n) exp(ikn y),

u =- z

n=-

|kn |cx (n) exp(ikn y),

(2.15)

so that (2.3) takes the form cx (n) +
m=-

|km |cx (m)b (n - m) = b (n),

(2.16)

where b (n) is the Fourier coefficient of the slip length. Thus we reduce the longitudinal problem to an infinite linear system for cx (n), which can be found by truncating the system and by using standard routines for linear systems.
2.3. Transverse configuration For transverse patterns, U = zey and u1 = (0, v , w), the solution is usually constructed using the vorticity = в u1 = (0, 0, z ), which significantly complicates the analysis compared to a longitudinal case (Cottin-Bizonne et al. 2004; Priezjev, Darhuber & Troian 2005; Belyaev & Vinogradova 2010a). However, simple analytical solutions can be obtained directly for the Fourier coefficients of velocities (Asmolov 2008; Kamrin et al. 2010). Indeed, (2.9)­(2.10) for transverse stripes can be written as

dw = 0, dz ikn p - v = 0, dp - w = 0. dz ikn v +

(2.17a) (2.17b) (2.17c)


112


E. S. Asmolov and O. I. Vinogradova w = 0, z + : w = 0.
2

By excluding p and v , one can transform them to a single ODE for the z-component: z = 0; (2.18) (2.19)

A general solution of the fourth-order ODE (2.18) decaying at infinity can be written as Condition (2.19) then determines the coefficient, cz = 0. The velocity Fourier coefficient in the y-direction can also be obtained from (2.17a): with cy = idz /kn , so that v = cy exp(-|kn |z)(1 - |kn |z), w = -ikn cy z exp(-|kn |z). (2.21) (2.22) w = (cz + dz z) exp(-|kn |z). (2.20)

Thus the solution for a given Fourier mode involves only one unknown constant cy (n), which can be found by applying boundary conditions (2.3). Using (2.8) and (2.21), we then get z=0: v=
n=-

cy (n) exp(ikn y),

v = -2 z

n=-

|kn |cy (n) exp(ikn y),

(2.23)

so that the slip boundary condition (2.3) can be rewritten as cy (n) + 2
m=-

|km |cy (m)b (n - m) = b (n).

(2.24)

The system (2.24) is very similar differs only by the prefactor of 2. expressed in terms of coefficients cx local slip, b2 (y) = 2b(y). Since b (n) 2 c x 2 ( n) + 2
m=-

to that for the longitudinal patterns, (2.16), and This means that the solution for cy (n) can be 2 (n) for the longitudinal flow with two times the = 2b (n), we obtain from (2.16) (2.25)

|km | cx2 (m) b (n - m) = 2b (n) .

The left-hand sides of the linear systems (2.24) and (2.25) are identical, but the right-hand sides differ by the factor of 2. In what follows, cy (n) = Whence by using (2.12) we get beff [2b(y)/L] . (2.27) 2 Thus, the longitudinal and transverse effective slip lengths are affine, being related by a simple formula. b [b(y)/L] = eff
3. Discussion In this section we compare (2.27) with the results obtained earlier for some particular periodic textures spatially varying in one direction y along the surface.

cx2 (n) . 2

(2.26)


Effective slip boundary conditions for arbitrary one-dimensional surfaces

113

We also make use of (2.26) to prove a similarity of velocity profiles in eigendirections, again by giving some supporting examples from prior work. Finally, we show that our results are valid for arbitrary, not necessarily periodic, one-dimensional textures (varying on a characteristic scale L).
3.1. Effective slip length

During the last few decades several theoretical papers have been concerned with the flow past alternating (parallel or transverse) stripes characterized by piecewise-constant slip lengths, b+ and b- , with area fractions + and - = 1 - + , correspondingly. Without loss of generality we consider below that 0 b- b+ < . A large fraction of these papers deals with an ideal case of stripes with b+ L (perfect slip) and b- = 0 (no-slip). A formula describing the effective slip in the longitudinal direction for such a texture was proposed by Philip (1972): bideal = L ln sec 2
+

.

(3.1)

Later Lauga & Stone (2003) derived an expression for the transverse configuration: b
ideal

=

L ln sec 2

2

+

,

(3.2)

which suggested that transverse and longitudinal components of the slip-length tensor are related as b
ideal

=

b

ideal

2

.

(3.3)

This relationship is consistent with predictions of (2.27). Note that (3.3) should be valid not only for regular textures, such as shown in figure 1, but also for periodic textures including several stripes of differing widths, e.g. for hierarchical fractal surfaces (Cottin-Bizonne, Barentin & Bocquet 2012). A few authors have discussed the situation of b- = 0 and finite b+ . Numerical data obtained by Cottin-Bizonne et al. (2004) seem to satisfy (2.27), and their beff , b eff asymptotically tend to the limiting values, bideal and b , when b+ /L becomes large. ideal Belyaev & Vinogradova (2010a) derived analytical expressions for this case: L ln sec 1+ L ln sec b+ + 2 + + tan 2 + 2 + + tan 2

b

|| eff

2

+

,

(3.4a)

b

eff

L 2

ln sec L 1+ ln sec 2b+

2

+

,

(3.4b)

which are again in agreement with predictions of (2.27).


114 b
+

E. S. Asmolov and O. I. Vinogradova

Ng & Wang (2009) addressed the problem of effective slip lengths for stripes with L and partial b- , and obtained b
eff

b

ideal

+

b- , -

b

eff

b

ideal

+

b- . -

(3.5)

It can be seen that this result also satisfies (2.27). For a weak-slip anisotropic texture, b(x, y) L, the area-averaged isotropic slip length has been predicted (Belyaev & Vinogradova 2010a). This means that the slip-length tensor becomes isotropic and for all in-plane directions, the flow aligns with the applied shear stress. A similar conclusion has been made by Kamrin et al. (2010). They proposed asymptotic solutions for beff for a weak-slip interface, = max |b (n)| 1. The two-term expansions of the effective slip lengths in can be obtained for an arbitrary local slip: b|| /L = b (0) - eff
n=-

|kn | |b (n)|2 ,

b /L = b (0) - 2 eff

n=-

|kn | |b (n)|2 .

(3.6)

Since b (n) = 2b (n), (3.6) is also consistent with the relation (2.27). 2 These examples fully support our predictions, but of course (2.27) is universal and should hold for any one-dimensional surface. Moreover, (2.27) still holds when the flow is unsteady. Ng & Wang (2011) have considered pressure-driven oscillatory flow in a channel with walls patterned by stripes of b+ L and b- = 0. For a thick channel they found that both real and imaginary parts of the effective slip lengths roughly satisfy (3.3). The reason is that (2.14), (2.16) 2 and (2.21)­(2.24) remain valid in this case if one replaces |kn | by kn + i.
3.2. Flow field We remark and stress that (2.26) allows one to express the entire flow field for the transverse configuration of patterns in terms of the longitudinal flow field, u2 (y, z) = u [y, z, 2b (y)] . Indeed, by applying the inverse Fourier transform of (2.21) and (2.22), one can easily derive by using (2.14), (2.17) and (2.26):

v=

1 2

u2 + z

u2 z

, u2 . y

w=-

z u2 , 2 y

(3.7) (3.8)

p=- Whence we conclude that at the wall z=0:

1 v = u2 , 2

v u2 = . z z

(3.9)

Several important conclusions follow from (3.7), (3.8) for alternating perfect-slip ( b+ L) and no-slip stripes (b- = 0). In this case u2 = u, and hence, at the wall the velocity along this one-dimensional texture is always twice that perpendicular to it, u = 2v . The same relation between velocities in eigendirections has been found by Teo & Khoo (2009) and Ng, Chu & Wang (2010) for the pressure-driven flow in a wide grooved superhydrophobic channel. Analytical solutions have been obtained for the longitudinal velocity u and its gradient u/ y at the wall for the shear flow (Sbragaglia & Prosperetti 2007). Equation (3.8) enables us to obtain the pressure for the transverse


Effective slip boundary conditions for arbitrary one-dimensional surfaces
5

115

p0

­5 ­ 0.5

0

0.5

y

F I G U R E 2 . Pressure distribution in the transverse flow over alternating perfect-slip, |y| + /2, and no-slip, + /2 |y| 1/2, stripes. Solid and dashed lines are (3.10) and (3.11), respectively.

flow over the perfect-slip region, |y| z=0: p=-

+ /2: sin(y) cos (y) - cos2 ( + /2)
2

u = y

.

(3.10)

The pressure over the no-slip regions, + /2 |y| 1/2, where u/ y = 0, is zero. The distribution (3.10) calculated for + = 0.7 is shown in figure 2. It grows infinitely near the jump in b(y) (from b+ L to b- = 0) at y = ± + /2: p Note that a solution for a velocity in an arbitrary direction of the undisturbed flow, U = z(ex cos + ey sin ), represents a superposition of solutions in eigendirections: Therefore, knowledge about the longitudinal velocity at the wall allows us to calculate the flow field in any direction given by an angle .
3.3. Surfaces with an arbitrary one-dimensional texture Up to this point, our focus has been on periodic one-dimensional surfaces. Now, we can remove all periodicity requirements that we used to derive (2.26) by applying a Fourier analysis. Indeed, if u2 is the solution of the longitudinal problem with a non-periodic slip length 2L (y), satisfying

± sin( + /2) [ sin( + )s]

-1/2

as s = ( + /2 - |y|) +0.

(3.11)

u1 = uex cos + (v ey + wez ) sin .

(3.12)

u2 = 0, (3.13) u2 z = 0 : u2 - 2 (y) = 2 (y), (3.14) z then one can verify directly that transverse solutions, (3.7), (3.8), taking into account (3.9), satisfy the Stokes equation (2.2) and boundary conditions (2.3), (2.4). Thus, our


116

E. S. Asmolov and O. I. Vinogradova

results are valid for any arbitrary patterned, not necessarily periodic, one-dimensional surfaces.
Acknowledgement This research was supported by the Russian Academy of Sciences through its Priority Programme `Assembly and Investigation of Macromolecular Structures of New Generations'.
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