^{1,a)}, Henry C. W. Chu

^{1}and C. Y. Wang

^{2}

### Abstract

Comparisons between slip lengths predicted by a liquid-gas coupled model and that by an idealized zero-gas-shear model are presented in this paper. The problem under consideration is pressure-driven flow of a liquid through a plane channel bounded by two superhydrophobic walls which are patterned with longitudinal or transverse gas-filled grooves. Effective slip arises from lubrication on the liquid-gas interface and intrinsic slippage on the solid phase of the wall. In the mathematical models, the velocities are analytically expressed in terms of eigenfunction series expansions, where the unknown coefficients are determined by the matching of velocities and shear stresses on the liquid-gas interface. Results are generated to show the effects due to small but finite gas viscosity on the effective slip lengths as functions of the channel height, the depth of grooves, the gas area fraction of the wall, and intrinsic slippage of the solid phase. Conditions under which even a gas/liquid viscosity ratio as small as 0.01 may have appreciable effects on the slip lengths are discussed.

The work was supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, through Project No. HKU 715609E, and also by the University of Hong Kong through the Small Project Funding Scheme under Project Code No. 200807176081, and the Seed Funding Programme for Basic Research under Project Code No. 200911159024.

I. INTRODUCTION

II. COUPLED MODELS AND SOLUTIONS

A. Longitudinal flow

B. Transverse flow

III. RESULTS

A. Longitudinal slip length

B. Transverse slip length

C. Phenomenological relations

IV. CONCLUDING REMARKS

### Key Topics

- Viscosity
- 53.0
- Gas liquid interfaces
- 24.0
- Gas liquid flows
- 18.0
- Poiseuille flow
- 11.0
- Interfacial properties
- 10.0

## Figures

Flow through a plane channel with grooved walls; longitudinal flow is normal to the plane, while transverse flow is along the -axis. The coordinates and length dimensions are normalized with respect to half the period of the wall pattern. The liquid-gas interface is a flat surface in alignment with the top of the ribs.

Flow through a plane channel with grooved walls; longitudinal flow is normal to the plane, while transverse flow is along the -axis. The coordinates and length dimensions are normalized with respect to half the period of the wall pattern. The liquid-gas interface is a flat surface in alignment with the top of the ribs.

Streamwise velocity profiles of the liquid phase on the liquid-gas interface for (a) longitudinal flow, (b) transverse flow, where , , , and (solid), 0.01 (dashed), 0.02 (dashed-dotted). The channel height is , except in one case, , as specified in (b). The symbols are the results adopted from previous studies: Teo and Khoo (Ref. 6) (crosses), Maynes *et al.* (Ref. 11) (squares and circles), and Davies *et al.* (Ref. 10) (triangles and inverted triangles).

Streamwise velocity profiles of the liquid phase on the liquid-gas interface for (a) longitudinal flow, (b) transverse flow, where , , , and (solid), 0.01 (dashed), 0.02 (dashed-dotted). The channel height is , except in one case, , as specified in (b). The symbols are the results adopted from previous studies: Teo and Khoo (Ref. 6) (crosses), Maynes *et al.* (Ref. 11) (squares and circles), and Davies *et al.* (Ref. 10) (triangles and inverted triangles).

Longitudinal slip length as a function of the channel height and gas area fraction of the wall , where , , and (a) , (b) . The dashes are for ideal gas .

Longitudinal slip length as a function of the channel height and gas area fraction of the wall , where , , and (a) , (b) . The dashes are for ideal gas .

Longitudinal slip length as a function of the channel height and gas area fraction of the wall , where , , and (a) , (b) . The dashes are for ideal gas .

Longitudinal slip length as a function of the viscosity ratio and gas area fraction of the wall , where , and (a) , , (b) , . The dotted lines are the limits for ideal gas .

Longitudinal slip length as a function of the viscosity ratio and gas area fraction of the wall , where , and (a) , , (b) , . The dotted lines are the limits for ideal gas .

Longitudinal slip length as a function of the groove depth , for , where , , and .

Longitudinal slip length as a function of the groove depth , for , where , , and .

Transverse slip length as a function of the channel height and gas area fraction of the wall , where , , and (a) , (b) . The dashes are for ideal gas .

Transverse slip length as a function of the channel height and gas area fraction of the wall , where , , and (a) , (b) . The dashes are for ideal gas .

Transverse slip length as a function of the channel height and gas area fraction of the wall , where , , and (a) , (b) . The dashes are for ideal gas .

Transverse slip length as a function of the viscosity ratio and gas area fraction of the wall , where , , and (a) , (b) . The dotted lines are the limits for ideal gas .

Transverse slip length as a function of the viscosity ratio and gas area fraction of the wall , where , , and (a) , (b) . The dotted lines are the limits for ideal gas .

For , , and , (a) transverse slip length as a function of the intrinsic slip length , for , 0.005, 0.01, and 0.02, where the dotted lines are the values computed by the approximation formula (36); (b) reciprocal of the gas slip length as a function of the intrinsic slip length , for , 0.005, 0.01, and 0.02.

For , , and , (a) transverse slip length as a function of the intrinsic slip length , for , 0.005, 0.01, and 0.02, where the dotted lines are the values computed by the approximation formula (36); (b) reciprocal of the gas slip length as a function of the intrinsic slip length , for , 0.005, 0.01, and 0.02.

Comparison between the modeling values of effective slip length, , and the predictions (a) using the formula (38), (b) using the formula (39). The symbols denote the following groups: (i) squares for and , (ii) diamonds for and , (iii) circles for and , (iv) triangles for and , and (v) inverted triangles for and . Each group contains the following cases: , , . The data points lying near the lower left corner of the graphs are those cases with , while the others are those with .

Comparison between the modeling values of effective slip length, , and the predictions (a) using the formula (38), (b) using the formula (39). The symbols denote the following groups: (i) squares for and , (ii) diamonds for and , (iii) circles for and , (iv) triangles for and , and (v) inverted triangles for and . Each group contains the following cases: , , . The data points lying near the lower left corner of the graphs are those cases with , while the others are those with .

Ratio of the effective transverse to longitudinal slip lengths, , as a function of the gas area fraction of the wall, , for , , , and . The dashes are for ideal gas .

Ratio of the effective transverse to longitudinal slip lengths, , as a function of the gas area fraction of the wall, , for , , , and . The dashes are for ideal gas .

## Tables

Approximate values representing the deviation of the effective longitudinal and transverse slip lengths based on gas viscosity from the corresponding values based on ideal inviscid gas for and . The upper and lower limits are for and , respectively. The values can be extended to other values of by linear interpolation as long as . Recall that all length dimensions are normalized by half the pitch of the wall micropattern.

Approximate values representing the deviation of the effective longitudinal and transverse slip lengths based on gas viscosity from the corresponding values based on ideal inviscid gas for and . The upper and lower limits are for and , respectively. The values can be extended to other values of by linear interpolation as long as . Recall that all length dimensions are normalized by half the pitch of the wall micropattern.

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