Effective diffusion coefficient in 2D periodic channels Pavol Kalinay Citation: The Journal of Chemical Physics 141, 144101 (2014); doi: 10.1063/1.4897250 View online: http://dx.doi.org/10.1063/1.4897250 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Erratum: “Effective diffusion coefficient in 2D periodic channels” [J. Chem. Phys.141, 144101 (2014)] J. Chem. Phys. 141, 169902 (2014); 10.1063/1.4900656 Effective diffusivity through arrays of obstacles under zero-mean periodic driving forces J. Chem. Phys. 137, 154109 (2012); 10.1063/1.4758703 Diffusion in one-dimensional channels with zero-mean time-periodic tilting forces J. Chem. Phys. 136, 114103 (2012); 10.1063/1.3693332 Diffusion in periodic two-dimensional channels formed by overlapping circles: Comparison of analytical and numerical results J. Chem. Phys. 135, 224101 (2011); 10.1063/1.3664179 Time-dependent diffusion in tubes with periodic partitions J. Chem. Phys. 131, 104705 (2009); 10.1063/1.3224954

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THE JOURNAL OF CHEMICAL PHYSICS 141, 144101 (2014)

Effective diffusion coefficient in 2D periodic channels Pavol Kalinaya) Institute of Physics, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia

(Received 29 July 2014; accepted 23 September 2014; published online 8 October 2014) Calculation of the effective diffusion coefficient D(x), depending on the longitudinal coordinate x in 2D channels with periodically corrugated walls, is revisited. Instead of scaling the transverse lengths and applying the standard homogenization techniques, we propose an algorithm based on formulation of the problem in the complex plane. A simple model is solved to explain the behavior of D(x) in the channels with short periods L, observed by Brownian simulations of Dagdug et al. [J. Chem. Phys. 133, 034707 (2010)]. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4897250] I. INTRODUCTION

An effective one-dimensional (1D) description of diffusion in non homogeneous channels appears to be a powerful tool for analysis of peculiar phenomena in quasi 1D systems, like rectification in the Brownian ratchets,1–6 stochastic resonance,7–9 transport over the entropic barriers,6, 10–13 etc. The simplest kind of the non homogeneity is corrugation of the walls, so the cross-section area, or width of the 2D channels A(x), depends on the longitudinal coordinate x. The 1D density p(x, t) of the particles diffusing inside is then governed by an effective equation, obtained by the dimensional reduction of the original 2D or 3D diffusion equation14 supplemented by the no-flux (Neumann) boundary conditions (BC). The simplest one is the Fick-Jacobs (FJ) equation,15 ∂t p(x, t) = ∂x D0 A(x)∂x

p(x, t) , A(x)

(1.1)

understood usually as the Smoluchowski equation for the entropic potential Ue (x), A(x) = exp [−Ue (x)/kB T], T denotes the temperature and D0 is the diffusion constant of particles under no confinement. The FJ equation (1.1) assumes (almost) infinitely fast equilibration of the 2D or 3D density ρ(x, y, t) in the transverse direction(s) y, then ρ(x, y, t) = p(x, t)/A(x). It is justified by smallness of the typical width of the channel if compared to the typical length in the longitudinal direction. Of course, in a periodic channel, the typical length may correspond to the period rather than the length of the whole channel. The typical width and length may be comparable and thus the FJ equation is not working correctly; it is usable only for |A (x)|  1. Zwanzig16 and later Reguera and Rubí17 suggested to replace D0 by a spatially dependent effective diffusion coefficient D(x), ∂t p(x, t) = ∂x D(x)A(x)∂x

p(x, t) , A(x)

(1.2)

reflecting deviations of the true 2D density ρ(x, y, t) from the averaged value p(x, t)/A(x) considered in Eq. (1.1). Aside from the phenomenological formulas,16, 17 like D(x) = D0 [1 + h2 (x)]−d , a) Electronic mail: [email protected]

0021-9606/2014/141(14)/144101/5/$30.00

(1.3)

where h(x) denotes half-width or radius of the symmetric 2D or 3D channels with d = 1/3 or 1/2, respectively, D(x) is derived mainly by the recurrence homogenization procedures.18–20 They introduce √ scaling of the transverse lengths by a small parameter , enabling us to find the deviations of the true ρ(x, y, t) from p(x, t)/A(x), and so D(x) as a series of corrections in , e.g.,  2 D(x)/D0 = 1 − h2 (x) + h (x)[9h3 (x) 3 45 + h(x)h (x)h (x) − h2 (x)h(3) (x)] + ... (1.4) for 2D channels. Summing the terms depending only on h (x) (i.e., neglecting h (x) and the higher derivatives) approves the formula (1.3) for 3D channels after removing the scaling,  = 1. In 2D channels, it gives D(x) = D0 arctan[h (x)]/ h (x), differing numerically only slightly from the 2D version of Eq. (1.3) for |h | ≤ 1. Applying the formula (1.3) in Eq. (1.2) considerably extends validity of the FJ equation, up to |h (x)| ∼ 1, as verified by the Brownian simulations.21–23 Still, the rapid changes of h(x), cusps,24 or jumps25, 26 require to fix D(x) in a specific way; the formula (1.3), depending only on h (x) may not work, too. A good example is diffusion in a “zigzag” channel, formed by a periodic sequence of enlarging and narrowing cones, h(x) = h0 + α|x| for x ∈ ( − L/2, L/2), h(x) = h(x + L), see Fig. 1. An automatic use of Eq. (1.3) returns a constant value of D(x) = Dα < D0 , depending on α. The Brownian simulations27 show consistency with this value only for large L, while in the limit L/h0 → 0, D(x) approaches D0 . In general, the failures of Eq. (1.3) are caused by neglecting h (x) and the higher derivatives in the exact expansion (1.4). Here, of course, the problem is to express the derivatives at the cusps. The homogenization techniques require h(x) to be analytic. In the opposite case, it is necessary to use other methods to find D(x). A considerable advantage of diffusion in 2D domains is possibility to formulate the problem in the complex plane.28 Then several models with non analytic h(x) become solvable using, e.g., the conformal transformation.26, 29 In the present paper, we demonstrate a method enabling us to calculate D(x) in 2D periodic channels without introducing the scaling and homogenization. In Sec. II, the method based on calculus in

141, 144101-1

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J. Chem. Phys. 141, 144101 (2014)

FIG. 1. A “zigzag” channel of period L, A(x) = h0 + |x| for x ∈ (−L/2, L/2).

the complex plane is explained. A simple example of h(x) with cusps is solved in Sec. III; the results are compared with the Brownian simulations in the zigzag channels.27 II. FORMULATION OF THE METHOD

We present here a method calculating D(x) for an arbitrary periodic 2D channel based on formulation of the problem in the complex plane. For simplicity, we deal here with only a channel bounded by the x axis and some periodic function y = h(x) = A(x) > 0, A(x) = A(x + L). First, let us recall28 that the coefficient D(x) is completely determined by the stationary flow. Eq. (1.2) represents the 1D mass conservation, ∂ t p(x, t) = −∂ x J(x, t), hence the net flux J (x, t) = −D(x)A(x)∂x [p(x, t)/A(x)].

(2.1)

In the stationary regime, the density and the flux are time independent, so J(x, t) = J is constant also in x (but nonzero, unlike the equilibrium). If the stationary 2D density ρ(x, y) is known, e.g., for some exactly solvable model, the corresponding 1D density p(x) and the flux J can be expressed from their definitions,  A(x) ρ(x, y, t), (2.2) p(x, t) = 0

 J (x, t) = −D0

A(x)

0

∂x ρ(x, y, t)dy,

and finally, D(x) is fixed from the stationary Eq. (2.1),   p(x) −1 D(x) = −J A(x)∂x . A(x)

(2.3)

for ρ(x, y), satisfying the Neumann (no-flux) BC (2.6)

in a very effective way. Any analytic function f(z) of the complex variable z = x + iy, f (z) = f (x + iy) = φ(x, y) + iρ(x, y),

0

(2.8)

In the method presented, an analytic complex function f(z) instead of only ρ(x, y) is sought. The analyticity warrants that ρ(x, y) solves Eq. (2.5), BC are held by φ(x, 0) = Re[f (x)] = 0,     φ x, A(x) = Re[f x + iA(x) ] = J /D0

(2.7)

solves the Laplace equation (2.5). Its real and complex components, φ(x, y) and ρ(x, y), satisfy Cauchy-Riemann’s rela-

(2.9)

for any x. The last quantity necessary for calculation of D(x) according to Eq.(2.4) is p(x). It is obtained from the primitive function F(z) = f(z)dz using Eq. (2.2),  A(x)  A(x) dy p(x) = ρ(x, y)dy = [f (x + iy) − f¯(x − iy)] 2i 0 0 1 = − [F (x + iy) + F¯ (x − iy)]A(x) 0 2 = −Re[F (x + iA(x)) − F (x)]

(2.10)

the bars at f, F denote conjunction. The constant J/D0 from Eq. (2.9) appears as a multiplication factor of ∂ x [p(x)/A(x)] and so D(x) ∼ D0 and independent of the flux J according to Eq. (2.4). We can take D0 = 1, J = 1 to simplify our next calculations. Now the method is applied to the periodic channels, A(x) = A(x + L). The flux lines exhibit the same symmetry, so φ(x, y) = φ(x + L, y). Its Fourier series,

(2.4)

The following calculus in the complex plane enables us to solve the 2D stationary diffusion equation:   (2.5) 0 = ∂t ρ(x, y) = D0 ∂x2 + ∂y2 ρ(x, y)

∂y ρ(x, y)|y=A(x) = A (x)∂x ρ(x, y)|y=A(x) ,

0

  = D0 φ x, A(x) .

φ(x, y) =

28

∂y ρ(x, y)|y=0 = 0,

tions, ∂ x φ(x, y) = ∂ y ρ(x, y), ∂ y φ(x, y) = −∂ x ρ(x, y). If combined, Eq. (2.5) either for φ(x, y), or ρ(x, y) is obtained. We can interpret ρ(x, y) as the 2D stationary density. Then φ(x, y) = φ describes the flux lines; its gradient is orthogonal to the flux density j (x, y) = −D0 (∂x ρ, ∂y ρ) = D0 (∂y φ, −∂x φ) according to Cauchy-Riemann. The noflux BC (2.6) require the flux lines at the boundaries to be parallel to them; we can put φ(x, 0) = 0 without loss of generality to satisfy the BC at y = 0. The value of φ at the upper boundary is given by the net flux (2.3),  A(x)  A(x) ∂x ρ(x, y)dy = D0 ∂y φ(x, y)dy J = −D0

∞ 

φn (y)einkx ,

(2.11)

n=−∞

k = 2π /L, can be generated only by an analytic function f(z) = f(x + iy) of the form f (z) = cz +

∞ 

cn einkz ,

(2.12)

n=−∞

where c = a + ib, cn = an + ibn are complex constants fixed by the BC, represented by Eq. (2.9). The first condition, Re[f(x)] = 0, gives a = 0, a−n = −an , and b−n = bn ; c0 is an irrelevant offset. The second one, J /D0 = −bA(x) +

∞  [2bn sinh[nkA(x)] sin(nkx) n=1

−2an cosh[nkA(x)] cos(nkx)],

(2.13)

after Fourier’s transform (FT) becomes a system of equations for the real coefficients an , bn , and b, depending on the Fourier

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J. Chem. Phys. 141, 144101 (2014)

y

coefficients of A(x). The corresponding function f (z) = ibz + 2i

∞ 

a ac

1.5

[an sin(nkz) + bn cos(nkz)]

(2.14)

n=1

(2.15)

The method presented reduces the calculation of the stationary 2D density ρ(x, y) and D(x) to solving the system of linear equations for b, an , bn , obtained by FT of Eq. (2.13). There is no limitation on analyticity of A(x); the cusps or jumps do not represent any problem. It is worth to analyze the first term of f(z), ibz. It describes the flow through a flat channel, A(x) = A0 constant. FT of Eq. (2.13) gives the only nonzero coefficient b = −1/A0 (for J = D0 = 1). Then f(z) = −i(x + iy)/A0 ; φ(x, y) = y/A0 describes the flux lines parallel to the x axes with φ growing from 0 at y = 0 up to 1 = J/D0 at the upper boundary. The imaginary part, ρ(x, y) = −x/A0 (plus an irrelevant constant), expresses descent of the 2D density along the channel due to the constant flow. In general, it is the only term in Eq. (2.14) violating periodicity of ρ(x, y) in x. If b = 0, then ρ(x + L, y) = ρ(x, y), i.e., there is no drop of the density over one period at a constant nonzero flux J. It is possible only if there is a singularity of ρ(x, y) at some x between x and x + L, generated by a closed bottleneck, A(x ) = 0; ρ(x, 0) → ±∞ depending on whether x → x from the left or right. In the regular flow, b is always nonzero. In the limit of small and frequent oscillations of A(x), as considered by Dagdug et al.,27 it represents the major term, determining ρ(x, y) close to the density in the flat channel and so D(x) → D0 . We demonstrate this phenomenon on a specific geometry with cusps in Sec. III. III. A CHANNEL WITH CUSPS

We consider here a specific class of channels with cusps and calculate D(x) by the method explained above. We show that D(x) really approaches D0 if the period and amplitude of corrugation go to zero, as observed in the Brownian simulations.27 We avoid the most technical part of the method, calculation of the coefficients an , bn , and b from a given A(x). Instead, we suppose f(z) of the form f (z) = −i(z + a sin kz),

0

1

(3.2)

for y = A(x); we put J = D0 = 1. The first term [b = −1 in Eq. (2.14)] corresponds to the flat channel with A(x) = 1,

2

3

4

6 x

5

FIG. 2. Channels defined by Eq. (3.2) for k = 1 and a = 0, 0.1, 0.2, and ac = 0.2759.

a controls amplitude of the corrugation, as shown in Fig. 2. The corrugation is not harmonic; there is an upper limit of a = ac , when the maxima of A(x) at kx = (2n + 1)π , n = ... − 1, 0, 1, ... become cusps. It is given by solution of Eq. (3.2), visualized in Fig. 3. For α(x) = −acos (kx) < 0, we have always one positive solution y < 1. For small positive α(x), there are two solutions, one has to take the smaller one, y > 1. α(x) achieves its maximum a at kx = π . For a = ac , the function y − 1 becomes the tangent of ac sinh (ky), so 1 = kac cosh(kyc )

(3.3)

at the only positive solution y = yc of Eq. (3.2), yc = 1 + ac sinh(kyc ) = 1 +

1 tanh(kyc ). k

(3.4)

This is the solution we are interested in, especially for small L, i.e., large k. In this limit, tanh (kyc ) → 1, hence yc 1 + 1/k and the corresponding kac 1/cosh (k + 1). For an arbitrary x, the solution y = A(x) of Eq. (3.2) depends on x. Derivating this equation in x, we find A (x) =

ka sin(kx) sinh[kA(x)] . 1 + ka cos(kx) cosh[kA(x)]

(3.5)

To obtain the limit Ac of A (x) for a = ac and kx = π + δ approaching π , the functions in Eq. (3.5) are expanded in δ. The numerator kac sin(π + δ) sinh[kyc + Ac δ] = − tanh(kyc )δ + · · ·, the denominator is −tanh(kyc )Ac δ + O(δ 2 ) in a  similar way. Then in the limit δ → 0, we get A2 c = 1. A (x) approaches ±1 depending on the sign of δ, i.e., the cusp at kx = π is approved. Notice that A2 c does not depend on k. rhs, lhs

ac

1

0

1

(3.1)

k = 2π /L, and find the corresponding function A(x) forming the upper boundary, satisfying Eq. (2.9), Re[f (x + iy)] = y + a cos(kx) sinh(ky) = 1

1 0.5

is easily integrable to get F(z), the 1D density p(x), Eq. (2.10), and finally D(x) according to Eq. (2.4), ∞  D0 sinh[nkA(x)] = −A(x)∂x bx + 2 D(x) nkA(x) n=1

×(an sin nkx + bn cos nkx) .

0.2

y 1

yc

ac

y

0

1 FIG. 3. Graphic solution of Eq. (3.2) for k = 1, y − 1 = αsinh (y) and α = −acos (x). LHS is described by dashed line, RHS for various α by solid lines. For ac = 0.2759, y − 1 is the tangent of RHS, the only solution yc = 1.9612.

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144101-4

Pavol Kalinay

J. Chem. Phys. 141, 144101 (2014)

Approximating cosh (ky) sinh (ky) exp (ky)/2 in Eq. (3.2), k(y − 1) = − cos(kx)ek(y−1)−1 ,

1.5

1

(3.6)

which is equivalent to the replacement tanh (kyc ) 1, enables us to find an approximative analytic formula for A(x), usable for large k (k > 2 in practice). If (kx)2 is expressed from Eq. (3.6) and expanded in k(y − 1) up to the 2nd order, we obtain the equation of ellipse, 2  π π3 2 , = (kx) + e(π − e) k(y − 1) − 2(π − e) 4(π − e) (3.7) hence, π e − π 3 e − 4e(π − e)(kx)2 . (3.8) A(x) 1 + 2ke(π − e) Notice that Eqs. (3.6) and (3.7) in the “scaled” coordinates kx and k(y − 1) remain unchanged for various k. In summary, the function (3.1) for a = ac describes the stationary flow in a channel of width ∼1, with the upper boundary corrugated by a periodic pattern, formed by a part of ellipse, see Fig. 4. The size of the pattern is controlled by k in a similar way as the sequence of zigzag channels27 by decreasing period L. Of course, the cusps here point only upward. The exact D(x) is calculated from Eqs. (2.4) and (2.10), using F(z) = f(z)dz = −i[z2 /2 − (a/k)cos (kz)], 

 1 tan kx 1 = A(x)∂x x − 1 − D(x) A(x) k A (x) tan kx 1 − A(x) sin2 kx − , (3.9) = cos2 kx kA(x) (A(π /2k) = 1, so there is no problem at kx = π /2). The approximative formula (3.8) is usable for k ≥ 2, otherwise A(x) has to be calculated from Eq. (3.2) numerically and then A (x) is obtained from Eq. (3.5). The plots of D(x) according to Eq. (3.9) are described in Fig. 5 by the solid lines. These data are compared to D(x) coming from the formula (1.3), the dashed lines. We observe a huge difference between the results near the cusps; mainly the exact D(x) approaches there unity (or D0 ). To understand this result, it is necessary to notice the shape of the lines of the same density, ρ(x, y) = ρ, in Fig. 4. They deviate from the normal to the x axis only in the vicinity of the upper boundary near the cusp. So the 2D density ρ(x, y) at some x π /k is almost constant across the channel, ρ(x, y) p(x)/A(x) like in the FJ approximation, and thus D(x) → D0 . Let us stress that the reason is not the fast transverse relaxation, but the specific geometry of the channel. For growing k, i.e., decreasing period L, or the size of the elliptic pattern, the lines ρ(x, y) = ρ are curved only in a layer of width ∼L at the corrugated boundary. So D(x) approaches D0 everywhere, as can be seen in Fig. 5, comparing the thin solid line (k = 4) to the thick solid line (k = 2). In other words, the particles diffusing in the “bulk” of the channel do not recognize tiny corrugations of the upper boundary. It is reflected correctly in the exact solution, Eq. (3.9), but not in the standard formulas of the type (1.3), taking only the

y 0.5

0

1

2

3

x FIG. 4. Stationary flow in the channel defined by Eq. (3.2) for k = 2 and the corresponding a = ac = 0.04991. Decreasing 2D density for growing x is depicted by different gray level. The dashed curve parallel to the boundary plots Eq. (3.8).

1 0.95 Dx

0.9 0.85 0.8 0

1

2

3

x FIG. 5. D(x) calculated for the channels with cusps, defined by Eq. (3.2) for k = 2 (thick lines, a = ac = 0.04991) and k = 4 (thin lines, a = ac = 0.003369). The solid lines describe the exact solutions, Eq. (3.9); the dashed lines plot the standard formula (1.3).

derivative h (x) into account. The span of the derivative A (x) in our example is (−1, 1) independently of k, leading incorrectly to the same course of D(x) (1.3) along the period of the channel, the dashed lines in Fig. 5. As a result, the effective diffusion coefficient Deff , calculated according to the Lifson-Jackson formula,30  L  L D0 dx 1 , (3.10) = 2 A(x)dx Deff L 0 0 A(x)D(x) modified for the x-dependent D(x), approaches D0 in the limit L → 0 for the exact Eq. (3.9), depicted as the solid line in Fig. 6, unlike Eq. (1.3), the dashed line. On the other hand, the coefficient D(x), calculated incorrectly according to Eq. (1.3) near the cusp, represents only a small contribution to Deff for large L. Both lines in Fig. 6 become close to one another here, similar to the simulations of diffusion in the zigzag channels.27 Of course, the boundary is far from the ellipse for L 1 and one has to solve Eq. (3.2) numerically to find A(x), instead of taking Eq. (3.8). These results of Deff , calculated for the exact and the approximate D(x), Eqs. (3.9), (1.3), are depicted by disks and squares, respectively, in Fig. 6. IV. CONCLUSION

The first aim of this paper was to present a method calculating the effective diffusion coefficient D(x) in periodic channels without use of the transverse scaling and homogenization.14, 18, 19 Our method is based on formulation

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J. Chem. Phys. 141, 144101 (2014)

1

0.9 Deff D0 0.8

0.7

0.1

1 L

10

FIG. 6. Effective diffusion coefficient Deff over one period (3.10) with D(x) calculated according to the exact formula (3.9), blue solid line, and Eq. (1.3), violet dashed line. The disks and squares represent continuation of the solid and dashed lines, respectively, in the region L > 1, with A(x) obtained by numerical solution of Eq. (3.2) instead of using the approximation (3.8).

of the problem in the complex plane and Fourier’s analysis. We took advantage of properties of the complex analytic functions, which automatically, satisfying Cauchy-Riemann’s relations, satisfy the stationary diffusion equation (2.5), too. To obtain the stationary 2D density ρ(x, y), determining also D(x), one needs only to comply with the BC (2.6), i.e., to find the coefficients an , bn , and b of Eq. (2.14), describing the stationary flow in the channel. The advantage of the method is its usability in the channels, where the formulas derived by the homogenization methods are not working properly; especially those with the boundaries defined by non analytic functions h(x), exhibiting cusps or jumps. Unfortunately, the present formulation in the complex plane is applicable only for 2D channels. Still, D(x) for periodic 3D channels with cylindrical symmetry should be also expressed by Fourier’s analysis, but using a different technique. Section III demonstrates the method on a class of the channels with cusps. Our aim is also to interpret the simulations of diffusion in the “zigzag” channels, where Dagdug et al.27 found another restrictions of usability of the formula (1.3). Despite of |h (x)| ≤ 1, the effective diffusion constant approaches a different value than predicted by Eq. (1.3) in the limit of small period L of the zigzag corrugation. We present an analytic solution for the channels defined by function (3.2), exhibiting cusps for specific values of a = ac . Although our channels differ from the zigzag geometry, both sets of channels exhibit a common property; the pattern of corrugation of the upper boundary (a part of ellipse in our case, Fig. 4) remains approximately the same, but its size is controlled by L. We show that our exact solution (3.9) corresponds to the behavior observed on the zigzag channels; in the limit L → 0, the effective diffusion coefficient approaches D0 , valid in a flat channel, unlike the prediction of Eq. (1.3). The explanation is based on analysis of the lines of equal stationary density ρ(x, y) = ρ. The deviation of D(x) from D0 is given by deviation (curving) of these lines from the normal to the x axis. No deviation means that the profile of the density is flat, ρ(x, y) = p(x)/A(x), like in the Fick-Jacobs equation, giving D(x) = D0 . The flat profile of ρ(x, y) can be a result of a specific geometry (symmetry) of the channel, although h (x)

= 0 and the transverse equilibration is not fast. An example in Fig. 4 is the cusp at x = π /2, where ρ(x, y) does not depend on y, although diffusion across the channel and along one period takes comparable times. If the amplitude of the corrugation is small, the lines of equal density are curved only in a thin layer near the corrugated boundary, so D(x) approaches D0 , reflecting mainly diffusion in the “bulk” of the channel. The same situation occurs near the cusps, compare Figs. 4 and 5. The formulas taking only the first derivatives h (x) into account, e.g., Eq. (1.3), reflect mainly the properties of the stationary flow near the curved boundary and thus they give wrong results. ACKNOWLEDGMENTS

Support from VEGA Grant No. 2/0049/14 is gratefully acknowledged. This work was inspired by discussions with L. Dagdug and S. Martens during the International Workshop on Brownian Motion in Confined Geometries in Dresden, March 2014. 1 P.

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Effective diffusion coefficient in 2D periodic channels.

Calculation of the effective diffusion coefficient D(x), depending on the longitudinal coordinate x in 2D channels with periodically corrugated walls,...
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