This article contributes to a better understanding of traveling waves over periodically undulated inclines. Therefore we used various measurement techniques to combine multiple information: (a) linear stability measurements, (b) measurement of the evolution of traveling free surface waves, and (c) single particle tracking. Thereby, we revealed two distinct wave shapes for different substrates, namely, convex and concave. Furthermore, we investigated the influence of the excitation amplitude and frequency on the flow’s linear stability and the evolution of nonlinear traveling waves. By varying the substrate shape and the wavelength to amplitude ratio, the dependence on the underlying flow field and the geometric constraints is examined. We report (a) an energy transfer from the excitation frequency to its higher harmonics, (b) a strongly varying strength of beat frequencies of the traveling wave and the substrate wave for different substrates, (c) similarities of the traveling wave for different substrate shapes but the same wavelength to amplitude ratio, and (d) a strong interaction between the traveling waves and the steady-state flow with even an eddy breakup for some substrates.
REFERENCES
1.
P. L.
Kapitza
, “Wave flow of thin layers of a viscous fluid
,” Zh. Eksp. Teor. Fiz.
18
, 1
–28
(1948
).2.
P. L.
Kapitza
and S. P.
Kapitza
, “Wave flow of thin layers of a viscous fluid
,” Zh. Eksp. Teor. Fiz.
19
, 105
–120
(1949
).3.
T. B.
Benjamin
, “Wave formation in laminar flow down an inclined plane
,” J. Fluid Mech.
2
, 554
–574
(1957
). 4.
C. S.
Yih
, “Stability of liquid flow down an inclined plane
,” Phys. Fluids
6
, 321
–334
(1963
). 5.
S. V.
Alekseenko
, V. Y.
Nakoryakov
, and B. G.
Pokusaev
, “Wave formation on a vertical falling liquid film
,” AIChE J.
31
, 1446
–1460
(1985
). 6.
B.
Gjevik
, “Occurrence of finite-amplitude surface waves on falling liquid films
,” Phys. Fluids
13
, 1918
–1925
(1970
). 7.
J.
Liu
, J. D.
Paul
, and J. P.
Gollub
, “Measurements of the primary instabilities of film flows
,” J. Fluid Mech.
250
, 69
–101
(1993
). 8.
H. C.
Chang
, “Wave evolution on a falling film
,” Annu. Rev. Fluid Mech.
26
, 103
–136
(1994
). 9.
J.
Liu
and J. P.
Gollub
, “Solitary wave dynamics of film flows
,” Phys. Fluids
6
, 1702
–1712
(1994
). 10.
M.
Vlachogiannis
and V.
Bontozoglou
, “Observations of solitary wave dynamics of film flows
,” J. Fluid Mech.
435
, 191
–215
(2001
). 11.
A.
Oron
, S. H.
Davis
, and S. G.
Bankoff
, “Long-scale evolution of thin liquid films
,” Rev. Mod. Phys.
69
, 931
–980
(1997
). 12.
H. C.
Chang
and E. A.
Demekhin
, Complex Wave Dynamics on Thin Films
(Elsevier
, Amsterdam
, 2002
).13.
R. V.
Craster
and O. K.
Matar
, “Dynamics and stability of thin liquid films
,” Rev. Mod. Phys.
81
, 1131
–1198
(2009
). 14.
M.
Scholle
and N.
Aksel
, “An exact solution of visco-capillary flow in an inclined channel
,” Z. Angew. Math. Phys. ZAMP
52
, 749
–769
(2001
). 15.
M.
Scholle
and N.
Aksel
, “Thin film limit and film rupture of the visco–capillary gravity–driven channel flow
,” Z. Angew. Math. Phys. ZAMP
54
, 517
–531
(2003
). 16.
A.
Haas
, T.
Pollak
, and N.
Aksel
, “Side wall effects in thin gravity-driven film flow–steady and draining flow
,” Phys. Fluids
23
, 062107
(2011
). 17.
V.
Leontidis
, J.
Vatteville
, M.
Vlachogiannis
, N.
Andritsos
, and V.
Bontozoglou
, “Nominally two-dimensional waves in inclined film flow in channels of finite width
,” Phys. Fluids
22
, 112106
(2010
). 18.
M.
Vlachogiannis
, A.
Samandas
, V.
Leontidis
, and V.
Bontozoglou
, “Effect of channel width on the primary instability of inclined film flow
,” Phys. Fluids
22
, 012106
(2010
). 19.
A.
Georgantaki
, J.
Vatteville
, M.
Vlachogiannis
, and V.
Bontozoglou
, “Measurements of liquid film flow as a function of fluid properties and channel width: Evidence for surface-tension-induced long-range transverse coherence
,” Phys. Rev. E
84
, 026325
(2011
). 20.
T.
Pollak
, A.
Haas
, and N.
Aksel
, “Side wall effects on the instability of thin gravity-driven films—From long-wave to short-wave instability
,” Phys. Fluids
23
, 094110
(2011
). 21.
V.
Bontozoglou
, “Laminar film flow along a periodic wall
,” Comput. Modell. Eng. Sci.
1
, 133
–142
(2000
).22.
M. I.
Pak
and G. H.
Hu
, “Numerical investigations on vortical structures of viscous film flows along periodic rectangular corrugations
,” Int. J. Multiphase Flow
37
, 369
–379
(2011
).23.
C.
Pozrikidis
, “The flow of a liquid film along a periodic wall
,” J. Fluid Mech.
188
, 275
–300
(1988
). 24.
A.
Wierschem
, M.
Scholle
, and N.
Aksel
, “Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers
,” Phys. Fluids
15
, 426
–435
(2003
). 25.
M.
Scholle
, A.
Wierschem
, and N.
Aksel
, “Creeping films with vortices over strongly undulated bottoms
,” Acta Mech.
168
, 167
–193
(2004
). 26.
Y. Y.
Trifonov
, “Viscous liquid film flows over a periodic surface
,” Int. J. Multiphase Flow
24
, 1139
–1161
(1999
). 27.
A.
Wierschem
and N.
Aksel
, “Influence of inertia on eddies created in films creeping over strongly undulated substrates
,” Phys. Fluids
16
, 4566
–4574
(2004
). 28.
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
). 29.
A.
Wierschem
, T.
Pollak
, C.
Heining
, and N.
Aksel
, “Suppression of eddies in films over topography
,” Phys. Fluids
22
, 113603
(2010
). 30.
A.
Wierschem
, V.
Bontozoglou
, C.
Heining
, H.
Uecker
, and N.
Aksel
, “Linear resonance in viscous films on inclined wavy planes
,” Int. J. Multiphase Flow
34
, 580
–589
(2008
). 31.
C.
Heining
, V.
Bontozoglou
, N.
Aksel
, and A.
Wierschem
, “Nonlinear resonance in viscous films on inclined wavy planes
,” Int. J. Multiphase Flow
35
, 78
–90
(2009
).32.
A.
Wierschem
and N.
Aksel
, “Hydraulic jumps and standing waves in gravity-driven flows of viscous liquids in wavy open channels
,” Phys. Fluids
16
, 3868
–3877
(2004
). 33.
Y. Y.
Trifonov
, “Viscous liquid film flows over a vertical corrugated surface and the film free surface stability
,” Russ. J. Eng. Thermophys.
10
(2
), 129
–145
(2000
).34.
M.
Vlachogiannis
and V.
Bontozoglou
, “Experiments on laminar film flow along a periodic wall
,” J. Fluid Mech.
457
, 133
–156
(2002
). 35.
A.
Wierschem
and N.
Aksel
, “Instability of a liquid film flowing down an inclined wavy plane
,” Phys. D
186
, 221
–237
(2003
). 36.
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
–442
(2002
). 37.
A.
Wierschem
, C.
Lepski
, and N.
Aksel
, “Effect of long undulated bottoms on thin gravity-driven films
,” Acta Mech.
179
, 41
–66
(2005
). 38.
Y. Y.
Trifonov
, “Stability of a viscous liquid film flowing down a periodic surface
,” Int. J. Multiphase Flow
33
, 1186
–1204
(2007
). 39.
Y. Y.
Trifonov
, “Stability and nonlinear wavy regimes in downward film flows on a corrugated surface
,” J. Appl. Mech. Tech. Phys.
48
, 91
–100
(2007
).40.
L. A.
Dávalos-Orozco
, “Nonlinear instability of a thin film flowing down a smoothly deformed surface
,” Phys. Fluids
19
, 074103
(2007
). 41.
L. A.
Dávalos-Orozco
, “Instabilities of thin films flowing down flat and smoothly deformed walls
,” Microgravity Sci. Technol.
20
, 225
–229
(2008
). 42.
Y. Y.
Trifonov
, “Stability and bifurcations of the wavy film flow down a vertical plate: The results of integral approaches and full-scale computations
,” Fluid Dyn. Res.
44
, 031418
(2012
). 43.
C.
Heining
and N.
Aksel
, “Bottom reconstruction in thin-film flow over topography: Steady solution and linear stability
,” Phys. Fluids
21
, 083605
(2009
). 44.
S. J. D.
D’Alessio
, J. P.
Pascal
, and H. A.
Jasmine
, “Instability in gravity-driven flow over uneven surfaces
,” Phys. Fluids
21
, 062105
(2009
). 45.
C.
Heining
and N.
Aksel
, “Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline
,” Int. J. Multiphase Flow
36
, 847
–857
(2010
). 46.
D.
Tseluiko
, M. G.
Blyth
, and T. D.
Papageorgiou
, “Stability of film flow over inclined topography based on a long-wave nonlinear model
,” J. Fluid Mech.
729
, 638
–671
(2013
). 47.
T.
Pollak
and N.
Aksel
, “Crucial flow stabilization and multiple instability branches of gravity-driven films over topography
,” Phys. Fluids
25
, 024103
(2013
). 48.
Y. Y.
Trifonov
, “Stability of a film flowing down an inclined corrugated plate: The direct Navier-Stokes computations and Floquet theory
,” Phys. Fluids
26
, 114101
(2014
). 49.
M.
Schörner
, D.
Reck
, and N.
Aksel
, “Stability phenomena far beyond the Nusselt flow—Revealed by experimental asymptotics
,” Phys. Fluids
28
, 022102
(2016
).50.
M.
Schörner
, D.
Reck
, N.
Aksel
, and Y. Y.
Trifonov
, “Switching between different types of stability isles in films over topographies
,” Acta Mech. SI Ziegler
(to be published).51.
Z.
Cao
, M.
Vlachogiannis
, and V.
Bontozoglou
, “Experimental evidence for a short-wave global mode in film flow along periodic corrugations
,” J. Fluid Mech.
718
, 304
–320
(2013
). 52.
M.
Schörner
, D.
Reck
, and N.
Aksel
, “Does the topography’s specific shape matter in general for the stability of film flows?
,” Phys. Fluids
27
, 042103
(2015
). 53.
Y. Y.
Trifonov
, “Counter-current gas-liquid flow between vertical corrugated plates
,” Chem. Eng. Sci.
66
, 4851
–4866
(2011
). 54.
K.
Argyriadi
, M.
Vlachogiannis
, and V.
Bontozoglou
, “Experimental study of inclined film flow along periodic corrugations: The effect of wall steepness
,” Phys. Fluids
18
, 012102
(2006
). 55.
D.
Reck
and N.
Aksel
, “Experimental study on the evolution of traveling waves over an undulated incline
,” Phys. Fluids
25
, 102101
(2013
). 56.
Y. Y.
Trifonov
, “Nonlinear waves on a liquid film falling down an inclined corrugated surface
,” Phys. Fluids
29
, 054104
(2017
). 57.
58.
J. C.
Crocker
and D. G.
Grier
, “Methods of digital video microscopy for colloidal studies
,” J. Colloid Interface Sci.
179
, 298
–310
(1996
). 59.
D.
Allan
, T.
Caswell
, N.
Keim
, and C.
van der Wel
, “ Trackpy version 0.3.2,” Zenodo
, 2016
, .© 2017 Author(s).
2017
Author(s)
You do not currently have access to this content.