^{1}, Aleksey V. Belyaev

^{2,3}, Friederike Schmid

^{1}and Olga I. Vinogradova

^{2,3,4}

### Abstract

We report results of dissipative particle dynamics simulations and develop a semi-analytical theory of an anisotropicflow in a parallel-plate channel with two superhydrophobic striped walls. Our approach is valid for any local slip at the gas sectors and an arbitrary distance between the plates, ranging from a thick to a thin channel. It allows us to optimize area fractions, slip lengths, channel thickness, and texture orientation to maximize a transverse flow. Our results may be useful for extracting effective slip tensors from global measurements, such as the permeability of a channel, in experiments or simulations, and may also find applications in passive microfluidic mixing.

We are grateful to V. Lobaskin for discussions and advice. This research was supported by the Russian Academy of Sciences (RAS) through its priority program “Assembly and Investigation of Macromolecular Structures of New Generations,” and by the Deutsche Forschungsgemeinschaft (DFG) through SFB-TR6. The simulations were carried out using computational resources at the John von Neumann Institute for Computing (NIC Jülich), the High Performance Computing Center Stuttgart (HLRS), and Mainz University.

I. INTRODUCTION

II. MODEL AND GENERAL CONSIDERATION

III. THEORY FOR STRIPED PATTERNS

A. Longitudinal stripes

B. Transverse stripes

C. Transverse flow

IV. SIMULATION METHOD

V. RESULTS AND DISCUSSION

VI. CONCLUSION

### Key Topics

- Poiseuille flow
- 19.0
- Tensor methods
- 13.0
- Hydrodynamics
- 12.0
- Rough surfaces
- 12.0
- Anisotropy
- 11.0

## Figures

Sketch of the symmetric striped channel (a) Θ = π/2 corresponds to transverse stripes, Θ = 0 to longitudinal stripes; (b) situation in (a) is approximated by a periodic cell of size *L*, with equivalent flow boundary conditions on the gas-liquid and solid-liquid interfaces.

Sketch of the symmetric striped channel (a) Θ = π/2 corresponds to transverse stripes, Θ = 0 to longitudinal stripes; (b) situation in (a) is approximated by a periodic cell of size *L*, with equivalent flow boundary conditions on the gas-liquid and solid-liquid interfaces.

The relation between the slip length *b* and the wall friction parameter γ_{ L }, for a wall interaction cutoff 2.0σ. The inset shows an enlarged portion of the region where the slip length is zero. The no-slip boundary condition can be implemented by using . Dashed curves show analytical predictions.^{37}

The relation between the slip length *b* and the wall friction parameter γ_{ L }, for a wall interaction cutoff 2.0σ. The inset shows an enlarged portion of the region where the slip length is zero. The no-slip boundary condition can be implemented by using . Dashed curves show analytical predictions.^{37}

Typical density (a) and velocity (b) profiles simulated for a longitudinal flow and a texture with *L* = *H* = *b* = 50σ, ϕ_{2} = 0.5.

Typical density (a) and velocity (b) profiles simulated for a longitudinal flow and a texture with *L* = *H* = *b* = 50σ, ϕ_{2} = 0.5.

The effective downstream slip length, , as a function of tilt angle Θ for a pattern with *L* = *b* = 50σ and ϕ_{2} = 0.5. Symbols are simulation data. Solid lines are theoretical values calculated using Eq. (54) with eigenvalues obtained by a numerical solution of Eqs. (19), (20), (30), and (31), (a) *H* = 50σ. The thick channel limit (dashed line) is calculated with Eqs. (22) and (38). (b) *H* = 10σ. The thin channel limit (dashed line), is calculated with Eqs. (21) and (35).

The effective downstream slip length, , as a function of tilt angle Θ for a pattern with *L* = *b* = 50σ and ϕ_{2} = 0.5. Symbols are simulation data. Solid lines are theoretical values calculated using Eq. (54) with eigenvalues obtained by a numerical solution of Eqs. (19), (20), (30), and (31), (a) *H* = 50σ. The thick channel limit (dashed line) is calculated with Eqs. (22) and (38). (b) *H* = 10σ. The thin channel limit (dashed line), is calculated with Eqs. (21) and (35).

The ratio ⟨*Q*⟩_{ z }/⟨*Q*⟩_{ x } as a function of the tilt angle Θ obtained with Eq. (39) for the data sets from Fig. 4 for (a) *H* = 50σ and (b) *H* = 10σ. Symbols are simulation data, solid curves represent theoretical values, and dashed curves show asymptotic predictions in the limit of thick channels (a) and thin channels (b).

The ratio ⟨*Q*⟩_{ z }/⟨*Q*⟩_{ x } as a function of the tilt angle Θ obtained with Eq. (39) for the data sets from Fig. 4 for (a) *H* = 50σ and (b) *H* = 10σ. Symbols are simulation data, solid curves represent theoretical values, and dashed curves show asymptotic predictions in the limit of thick channels (a) and thin channels (b).

The eigenvalues of the effective slip length tensor (symbols) as a function of ϕ_{2}. Solid curves are theoretical values obtained by a numerical solution of Eqs. (19), (20), (30), and (31). Calculations were made for a pattern with *L* = *b* = 50σ, (a) *H* = 50σ. Dashed curves are calculated with Eqs. (22) and (38), (b) *H* = 10σ. Dashed curves are computed with Eqs. (21) and (35).

The eigenvalues of the effective slip length tensor (symbols) as a function of ϕ_{2}. Solid curves are theoretical values obtained by a numerical solution of Eqs. (19), (20), (30), and (31). Calculations were made for a pattern with *L* = *b* = 50σ, (a) *H* = 50σ. Dashed curves are calculated with Eqs. (22) and (38), (b) *H* = 10σ. Dashed curves are computed with Eqs. (21) and (35).

The ratio ⟨*Q*⟩_{ z }/⟨*Q*⟩_{ x } as a function of ϕ_{2} obtained with Eq. (39) for Θ = π/4 by using the data sets from Fig. 6 (a) *H* = 50σ and (b) *H* = 10σ. Symbols are simulation data, solid curves represent theoretical values, and dashed curves show asymptotic predictions in the limit of thick channels (a) and thin channels (b).

The ratio ⟨*Q*⟩_{ z }/⟨*Q*⟩_{ x } as a function of ϕ_{2} obtained with Eq. (39) for Θ = π/4 by using the data sets from Fig. 6 (a) *H* = 50σ and (b) *H* = 10σ. Symbols are simulation data, solid curves represent theoretical values, and dashed curves show asymptotic predictions in the limit of thick channels (a) and thin channels (b).

The longitudinal (a) and transverse (b) effective slip lengths as a function of the channel height *H* for a texture with *L* = *b* = 50σ and ϕ_{2} = 0.5. The theoretical curves are obtained by a numerical solution of Eqs. (19), (20), (30), and (31). Dashed lines show expected asymptotics in the limit of thin and thick channels.

The longitudinal (a) and transverse (b) effective slip lengths as a function of the channel height *H* for a texture with *L* = *b* = 50σ and ϕ_{2} = 0.5. The theoretical curves are obtained by a numerical solution of Eqs. (19), (20), (30), and (31). Dashed lines show expected asymptotics in the limit of thin and thick channels.

The ratio between the transverse and longitudinal flow rates ⟨*Q*⟩_{ z }/⟨*Q*⟩_{ x } as a function of channel thickness *H* for a pattern with *L* = *b* = 50σ, ϕ_{2} = 0.5, and Θ = π/4. Symbols are simulation data and the lines represent theoretical values obtained using Eq. (39). Dashed lines show expected asymptotics in the limit of thin and thick channels.

The ratio between the transverse and longitudinal flow rates ⟨*Q*⟩_{ z }/⟨*Q*⟩_{ x } as a function of channel thickness *H* for a pattern with *L* = *b* = 50σ, ϕ_{2} = 0.5, and Θ = π/4. Symbols are simulation data and the lines represent theoretical values obtained using Eq. (39). Dashed lines show expected asymptotics in the limit of thin and thick channels.

The velocity profile across the channel at Θ = π/4, (a) *H* = 10σ, *y* from −5σ to 5σ, (b) *H* = 50σ, *y* from −25σ to 25σ, (c) *H* = 50σ, *y* from 15σ to 25σ, enlarged part near the striped-pattern, and (d) *H* = 50σ, *y* from −5σ to 5σ, enlarged part near the channel center. The *z* components in (c) and (d) have been increased five times for better demonstration.

The velocity profile across the channel at Θ = π/4, (a) *H* = 10σ, *y* from −5σ to 5σ, (b) *H* = 50σ, *y* from −25σ to 25σ, (c) *H* = 50σ, *y* from 15σ to 25σ, enlarged part near the striped-pattern, and (d) *H* = 50σ, *y* from −5σ to 5σ, enlarged part near the channel center. The *z* components in (c) and (d) have been increased five times for better demonstration.

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