Eddy formation and presence in a plane laminar shear flow configuration consisting of two infinitely long plates orientated parallel to each other is investigated theoretically. The upper plate, which is planar, drives the flow; the lower one has a sinusoidal profile and is fixed. The governing equations are solved via a full finite element formulation for the general case and semianalytically at the Stokes flow limit. The effects of varying geometry (involving changes in the mean plate separation or the amplitude and wavelength of the lower plate) and inertia are explored separately. For Stokes flow and varying geometry, excellent agreement between the two methods of solution is found. Of particular interest with regard to the flow structure is the importance of the clearance that exists between the upper plate and the tops of the corrugations forming the lower one. When the clearance is large, an eddy is only present at sufficiently large amplitudes or small wavelengths. However, as the plate clearance is reduced, a critical value is found, which triggers the formation of an eddy in an otherwise fully attached flow for any finite amplitude and arbitrarily large wavelength. This is a precursor to the primary eddy to be expected in the lid-driven cavity flow, which is formed in the limit of zero clearance between the plates. The influence of the flow driving mechanism is assessed by comparison with corresponding solutions for the case of gravity-driven fluid films flowing over an undulating substrate. When inertia is present, the flow generally becomes asymmetrical. However, it is found that for large mean plate separations the flow local to the lower plate becomes effectively decoupled from the inertia dominated overlying flow if the wavelength of the lower plate is sufficiently small. In such cases the local flow retains its symmetry. A local Reynolds number based on the wavelength is shown to be useful in characterizing these large-gap flows. As the mean plate separation is reduced, the form of the asymmetry caused by inertia changes and becomes strongly dependent on the plate separation. For lower plate wavelengths which do not exhibit a kinematically induced secondary eddy, an inertially induced secondary eddy can be created if the mean plate separation is sufficiently small and the global Reynolds number is sufficiently large.

1.
D. J.
Tritton
,
Physical Fluid Dynamics
, 2nd ed. (
Oxford Science, Clarendon
,
Oxford
,
1988
).
2.
J. H.
Spurk
and
N.
Aksel
,
Fluid Mechanics
, 2nd ed. (
Springer
,
New York
,
2008
).
3.
B. J.
Hamrock
,
Fundamentals of Fluid Film Lubrication
(
McGraw-Hill
,
New York
,
1994
).
4.
R.
Oliveira
, “
Understanding adhesion: A means for preventing fouling
,”
Exp. Therm. Fluid Sci.
14
,
316
(
1997
).
5.
M.
Scholle
, “
Hydrodynamical modelling of lubricant friction between rough surfaces
,”
Tribol. Int.
40
,
1004
(
2007
).
6.
I.
Etsion
, “
State of the art in laser surface texturing
,”
J. Rheol.
127
,
248
(
2005
).
7.
J. J. L.
Higdon
, “
Stokes flow in arbitrary two-dimensional domains: Shear flow over ridges and cavities
,”
J. Fluid Mech.
159
,
195
(
1985
).
8.
M.
Arghir
,
N.
Roucou
,
M.
Helene
, and
J.
Frene
, “
Theoretical analysis of the incompressible laminar flow in a macro-roughness cell
,”
Trans. ASME, J. Tribol.
125
,
309
(
2003
).
9.
F.
Sahlin
,
S. B.
Glavatskih
,
T.
Almqvist
, and
R.
Larson
, “
Two-dimensional CFD-analysis of micro-patterned surface in hydrodynamic lubrication
,”
J. Tribol.
127
,
96
(
2005
).
10.
P. H.
Gaskell
,
H. M.
Thompson
, and
M. D.
Savage
, “
Stagnation-saddle points and flow patterns in Stokes flow between contra-rotating cylinders
,”
J. Fluid Mech.
370
,
221
(
1998
).
11.
J. L.
Summers
,
H. M.
Thompson
, and
P. H.
Gaskell
, “
Flow structure and transfer jets in a contra-rotating rigid roll coating system
,”
Theor. Comput. Fluid Dyn.
17
,
189
(
2004
).
12.
M. C. T.
Wilson
,
J. L.
Summers
,
N.
Kapur
, and
P. H.
Gaskell
, “
Stirring and transport enhancement in a continuously modulated free-surface flow
,”
J. Fluid Mech.
565
,
319
(
2006
).
13.
P. H.
Gaskell
,
M. D.
Savage
, and
M.
Wilson
, “
Stokes flow in a half-filled annulus between rotating coaxial cylinders
,”
J. Fluid Mech.
337
,
263
(
1997
).
14.
M. C. T.
Wilson
,
P. H.
Gaskell
, and
M. D.
Savage
, “
Nested separatrices in simple shear flow: The effect of localized disturbances on stagnation lines
,”
Phys. Fluids
17
,
093601
(
2005
).
15.
A. E.
Perry
and
M. S.
Chong
, “
A description of eddying motions and flow patterns using critical-point concepts
,”
Annu. Rev. Fluid Mech.
19
,
125
(
1987
).
16.
D. J.
Jeffrey
and
J. D.
Sherwood
, “
Streamline patterns and eddies in low Reynolds-number flow
,”
J. Fluid Mech.
96
,
315
(
1980
).
17.
C.
Pozrikidis
,
Little Book of Streamlines
(
Academic
,
San Diego
,
1999
).
18.
M.
Scholle
,
A.
Rund
, and
N.
Aksel
, “
Drag reduction and improvement of material transport in creeping films
,”
Arch. Appl. Mech.
75
,
93
(
2006
).
19.
M.
Scholle
,
A.
Haas
,
N.
Aksel
,
H. M.
Thompson
,
R. W.
Hewson
, and
P. H.
Gaskell
, “
The effect of locally induced flow structure on global heat transfer for plane laminar shear flow
,”
Int. J. Heat Fluid Flow
30
,
175
(
2009
).
20.
C.
Pozrikidis
, “
Creeping flow in two-dimensional channels
,”
J. Fluid Mech.
180
,
495
(
1987
).
21.
M.
Scholle
, “
Creeping Couette flow over an undulated plate
,”
Arch. Appl. Mech.
73
,
823
(
2004
).
22.
A. E.
Malevich
,
V. V.
Mityushev
, and
P. M.
Adler
, “
Stokes flow through a channel with wavy walls
,”
Acta Mech.
182
,
151
(
2006
).
23.
A. E.
Malevich
,
V. V.
Mityushev
, and
P. M.
Adler
, “
Couette flow in channels with wavy walls
,”
Acta Mech.
197
,
247
(
2008
).
24.
M.
Scholle
,
A.
Haas
,
N.
Aksel
,
M. C. T.
Wilson
,
H. M.
Thompson
, and
P. H.
Gaskell
, “
Competing geometric and inertial effects on local flow structure in thick gravity-driven fluid films
,”
Phys. Fluids
20
,
123101
(
2008
).
25.
P. H.
Gaskell
,
P. K.
Jimack
,
M.
Sellier
,
H. M.
Thompson
, and
M. C. T.
Wilson
, “
Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography
,”
J. Fluid Mech.
509
,
253
(
2004
).
26.
C. B.
Millikan
, “
On the steady motion of viscous, incompressible fluids; with particular reference to a variation principle
,”
Philos. Mag.
7
,
641
(
1929
).
27.
K. -H.
Anthony
, “
Hamilton’s action principle and thermodynamics of irreversible processes—a unifying procedure for reversible and irreversible processes
,”
J. Non-Newtonian Fluid Mech.
96
,
291
(
2001
).
28.
M.
Scholle
,
A.
Wierschem
, and
N.
Aksel
, “
Creeping films with vortices over strongly undulated bottoms
,”
Acta Mech.
168
,
167
(
2004
).
29.
A.
Wierschem
,
M.
Scholle
, and
N.
Aksel
, “
Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers
,”
Phys. Fluids
15
,
426
(
2003
).
30.
A.
Wierschem
,
M.
Scholle
, and
N.
Aksel
, “
Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel
,”
Exp. Fluids
33
,
429
(
2002
).
31.
H. K.
Moffatt
, “
Viscous and resistive eddies near a sharp corner
,”
J. Fluid Mech.
18
,
1
(
1964
).
32.
P. H.
Gaskell
,
H. M.
Thompson
, and
M. D.
Savage
, “
A finite element analysis of steady viscous flow in triangular cavities
,”
Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci.
213
,
263
(
1999
).
33.
A.
Wierschem
and
N.
Aksel
, “
Influence of inertia on eddies created in film creeping over strongly undulated substrates
,”
Phys. Fluids
16
,
4566
(
2004
).
34.
N. G.
Wright
and
P. H.
Gaskell
, “
An efficient multigrid approach to solving highly recirculating flows
,”
Comput. Fluids
24
,
63
(
1995
).
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