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.

1.
C.
Neto
,
D. R.
Evans
,
E.
Bonaccurso
,
H. J.
Butt
, and
V. S. J.
Craig
, “
Boundary slip in Newtonian liquids: A review of experimental studies
,”
Rep. Prog. Phys.
68
,
2859
(
2005
).
2.
E.
Lauga
,
M. P.
Brenner
, and
H. A.
Stone
, in
Handbook of Experimental Fluid Dynamics
(
Springer
,
New York
,
2007
), Chap. 19, pp.
1219
1240
.
3.
X.
Zhang
,
F.
Shi
,
J.
Niu
,
Y.
Jiang
, and
Z.
Wang
, “
Superhydrophobic surfaces: From structural control to functional application
,”
J. Mater. Chem.
18
,
621
(
2008
).
4.
J. P.
Rothstein
, “
Slip on superhydrophobic surfaces
,”
Annu. Rev. Fluid Mech.
42
,
89
(
2010
).
5.
C. L. M. H.
Navier
, “
Memoire sur les lois du movement des fluids
,”
Memoires de l’Academie Royale des Sciences de l’Institut de France
6
,
389
(
1823
).
6.
C. J.
Teo
and
B. C.
Khoo
, “
Analysis of Stokes flow in microchannels with superhydrophobic surfaces containing a periodic array of micro-grooves
,”
Microfluid. Nanofluid.
7
,
353
(
2009
).
7.
C.
Ybert
,
C.
Barentin
, and
C.
Cottin-Bizonne
, “
Achieving large slip with superhydrophobic surfaces: Scaling laws for generic geometries
,”
Phys. Fluids
19
,
123601
(
2007
).
8.
Y. P.
Cheng
,
C. J.
Teo
, and
B. C.
Khoo
, “
Microchannel flows with superhydrophobic surfaces: Effects of Reynolds number and pattern width to channel height ratio
,”
Phys. Fluids
21
,
122004
(
2009
).
9.
C. J.
Teo
and
B. C.
Khoo
, “
Flow past superhydrophobic surfaces containing longitudinal grooves: Effects of interface curvature
,”
Microfluid. Nanofluid.
9
,
499
(
2010
).
10.
J.
Davies
,
D.
Maynes
,
B. W.
Webb
, and
B.
Woolford
, “
Laminar flow in a microchannel with superhydrophobic walls exhibiting transverse ribs
,”
Phys. Fluids
18
,
087110
(
2006
).
11.
D.
Maynes
,
K.
Jeffs
,
B.
Woolford
, and
B. W.
Webb
, “
Laminar flow in a microchannel with hydrophobic surface patterned microribs oriented parallel to the flow direction
,”
Phys. Fluids
19
,
093603
(
2007
).
12.
C. Y.
Wang
, “
Flow over a surface with parallel grooves
,”
Phys. Fluids
15
,
1114
(
2003
).
13.
C. O.
Ng
and
C. Y.
Wang
, “
Stokes shear flow over a grating: Implications for superhydrophobic slip
,”
Phys. Fluids
21
,
013602
(
2009
).
14.
J. R.
Philip
, “
Flows satisfying mixed no-slip and no-shear conditions
,”
Z. Angew. Math. Phys.
23
,
353
(
1972
).
15.
C.
Cottin-Bizonne
,
C.
Barentin
,
E.
Charlaix
,
L.
Bocquet
, and
J. -L.
Barrat
, “
Dynamics of simple liquids at heterogeneous surface: Molecular-dynamics simulations and hydrodynamic description
,”
Eur. Phys. J. E
15
,
427
(
2004
).
16.
S. C.
Hendy
and
N. J.
Lund
, “
Effective slip boundary conditions for flows over nanoscale chemical heterogeneities
,”
Phys. Rev. E
76
,
066313
(
2007
).
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