In the present study on the linear stability of gravity-driven Newtonian films flowing over inclined topographies, we consider a fundamental question: Is there a universal principle, being valid to describe the parametric evolution of the flow’s stability chart for variations of different system parameters? For this sake, we first screened all experimental and numerical stability charts available in the literature. In a second step, we performed experiments to fill the gaps which remained. Variations of the fluid’s viscosity and the topography’s specific shape, amplitude, wavelength, tip width, and inclination were considered. That way, we identified a set of six characteristic patterns of stability charts to be sufficient to describe and unify all results on the linear stability of Newtonian films flowing over undulated inclines. We unveiled a universal pathway—the stability cycle—along which the linear stability charts of all considered Newtonian films flowing down periodically corrugated inclines evolved when the system parameters were changed.
REFERENCES
1.
I.
Luca
, K.
Hutter
, Y. C.
Tai
, and C. Y.
Kuo
, “A hierarchy of avalanche models on arbitrary topography
,” Acta Mech.
205
, 121
–149
(2009
).2.
R.
Greve
and H.
Blatter
, Dynamics of Ice Sheets and Glaciers
(Springer
, Berlin
, 2009
).3.
K.
Hutter
, B.
Svendsen
, and D.
Rickenmann
, “Debris flow modeling: A review
,” Continuum Mech. Thermodyn.
8
, 1
–35
(1996
).4.
A.
Kumar
, D.
Karig
, R.
Acharya
, S.
Neethirajan
, P. P.
Mukherjee
, S.
Retterer
, and M. J.
Doktycz
, “Microscale confinement features can affect biofilm formation
,” Microfluid. Nanofluid.
14
, 895
–902
(2013
).5.
J. L.
Bull
and J. B.
Grotberg
, “Surfactant spreading on thin viscous films: Film thickness evolution and periodic wall stretch
,” Exp. Fluids
34
, 1
–15
(2003
).6.
A.
Kraus
, A.
Bar-Cohen
, and A. A.
Wative
, Cooling Electronic Equipment
, 3rd ed. (Wiley Online Library
, 2006
).7.
S. F.
Kistler
and P. M.
Schweizer
, Liquid Film Coating
(Springer
, Netherlands
, 1997
).8.
S. J.
Weinstein
and K. J.
Ruschak
, “Coating flows
,” Annu. Rev. Fluid Mech.
36
, 29
–53
(2004
).9.
G.
Gugler
, R.
Beer
, and M.
Mauron
, “Operative limits of curtain coating due to edges
,” Chem. Eng. Process.: Process Intensif.
50
, 462
–465
(2011
).10.
J. R.
Fair
and J. R.
Bravo
, “Distillation columns containing structure packing
,” Chem. Eng. Prog.
86
, 19
–29
(1990
).11.
J. M.
de Santos
, T. R.
Melli
, and L. E.
Scriven
, “Mechanics of gas-liquid flow in packed-bed contactors
,” Annu. Rev. Fluid Mech.
23
, 233
–260
(1991
).12.
13.
P.
Vlasogiannis
, G.
Karagiannis
, P.
Argyropoulos
, and V.
Bontozoglou
, “Air-water two-phase flow and heat transfer in a plate heat exchanger
,” Int. J. Multiphase Flow
28
, 757
–772
(2002
).14.
P.
Valluri
, O. K.
Matar
, G. F.
Hewitt
, and M. A.
Mendes
, “Thin film flow over structured packings at moderate Reynolds numbers
,” Chem. Eng. Sci.
60
, 1965
–1975
(2005
).15.
K.
Helbig
, R.
Nasarek
, T.
Gambaryan-Roisman
, and P.
Stephan
, “Effect of longitudinal minigrooves on flow stability and wave characteristics of falling liquid films
,” J. Heat Transfer
131
, 011601
(2009
).16.
P. L.
Kapitza
, “Wavy flow of thin layers of a viscous fluid
,” Zh. Eksp. Teor. Fiz.
18
, 3
–28
(1948
).17.
P. L.
Kapitza
and S. P.
Kapitza
, “Wavy flow of thin layers of a viscous fluid
,” Zh. Eksp. Teor. Fiz.
19
, 105
–120
(1949
).18.
W.
Nusselt
, “Die Oberflächenkondensation des Wasserdampfes
,” VDI Z.
60
, 541
–546
(1916
).19.
T. B.
Benjamin
, “Wave formation in laminar flow down an inclined plane
,” J. Fluid Mech.
2
, 554
–574
(1957
).20.
C. S.
Yih
, “Stability of liquid flow down an inclined plane
,” Phys. Fluids
6
, 321
–334
(1963
).21.
S. V.
Alekseenko
, V. Y.
Nakoryakov
, and B. G.
Pokusaev
, “Wave formation on a vertical falling liquid film
,” AIChE J.
31
, 1446
–1460
(1985
).22.
B.
Gjevik
, “Occurrence of finite-amplitude surface waves on falling liquid films
,” Phys. Fluids
13
, 1918
–1925
(1970
).23.
J.
Liu
, J. D.
Paul
, and J. P.
Gollub
, “Measurements of the primary instabilities of film flows
,” J. Fluid Mech.
250
, 69
–101
(1993
).24.
H. C.
Chang
, “Wave evolution on a falling film
,” Annu. Rev. Fluid Mech.
26
, 103
–136
(1994
).25.
J.
Liu
and J. P.
Gollub
, “Solitary wave dynamics of film flows
,” Phys. Fluids
6
, 1702
–1712
(1994
).26.
M.
Vlachogiannis
and V.
Bontozoglou
, “Observations of solitary wave dynamics of film flows
,” J. Fluid Mech.
435
, 191
–215
(2001
).27.
D.
Reck
and N.
Aksel
, “Recirculation areas underneath solitary waves on gravity-driven film flows
,” Phys. Fluids
27
, 112107
(2015
).28.
A.
Oron
, S. H.
Davis
, and S. G.
Bankoff
, “Long-scale evolution of thin liquid films
,” Rev. Mod. Phys.
69
, 931
–980
(1997
).29.
H. C.
Chang
and E. A.
Demekhin
, Complex Wave Dynamics on Thin Films
(Elsevier
, Amsterdam
, 2002
).30.
R. V.
Craster
and O. K.
Matar
, “Dynamics and stability of thin liquid films
,” Rev. Mod. Phys.
81
, 1131
–1198
(2009
).31.
S.
Kalliadasis
, C.
Ruyer-Quil
, B.
Scheid
, and M. G.
Velarde
, Falling Liquid Films
(Springer
, 2012
).32.
V.
Bontozoglou
, “Laminar film flow along a periodic wall
,” Comput. Model. Eng. Sci.
1
, 133
–142
(2000
).33.
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
).34.
C.
Pozrikidis
, “The flow of a liquid film along a periodic wall
,” J. Fluid Mech.
188
, 275
–300
(1988
).35.
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
).36.
M.
Scholle
, A.
Wierschem
, and N.
Aksel
, “Creeping films with vortices over strongly undulated bottoms
,” Acta Mech.
168
, 167
–193
(2004
).37.
Y. Y.
Trifonov
, “Viscous liquid film flows over a periodic surface
,” Int. J. Multiphase Flow
24
, 1139
–1161
(1998
).38.
A.
Wierschem
and N.
Aksel
, “Influence of inertia on eddies created in films creeping over strongly undulated substrates
,” Phys. Fluids
16
, 4566
–4574
(2004
).39.
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
).40.
A.
Wierschem
, T.
Pollak
, C.
Heining
, and N.
Aksel
, “Suppression of eddies in films over topography
,” Phys. Fluids
22
, 113603
(2010
).41.
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
).42.
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
).43.
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
).44.
N. A.
Malamataris
and V.
Bontozoglou
, “Computer aided analysis of viscous film flow along an inclined wavy wall
,” J. Comput. Phys.
154
, 372
–392
(1999
).45.
V.
Bontozoglou
and G.
Papapolymerou
, “Laminar film flow down a wavy incline
,” Int. J. Multiphase Flow
23
, 69
–79
(1997
).46.
M.
Vlachogiannis
and V.
Bontozoglou
, “Experiments on laminar film flow along a periodic wall
,” J. Fluid Mech.
457
, 133
–156
(2002
).47.
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
).48.
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
).49.
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
).50.
A.
Wierschem
, C.
Lepski
, and N.
Aksel
, “Effect of long undulated bottoms on thin gravity-driven films
,” Acta Mech.
179
, 41
–66
(2005
).51.
A.
Wierschem
and N.
Aksel
, “Instability of a liquid film flowing down an inclined wavy plane
,” Phys. D
186
, 221
–237
(2003
).52.
L. A.
Dávalos-Orozco
, “Nonlinear instability of a thin film flowing down a smoothly deformed surface
,” Phys. Fluids
19
, 074103
(2007
).53.
L. A.
Dávalos-Orozco
, “Instabilities of thin films flowing down flat and smoothly deformed walls
,” Microgravity Sci. Technol.
20
, 225
–229
(2008
).54.
Y. Y.
Trifonov
, “Stability of a viscous liquid film flowing down a periodic surface
,” Int. J. Multiphase Flow
33
, 1186
–1204
(2007
).55.
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
).56.
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
).57.
D.
Tseluiko
, M. G.
Blyth
, and D. T.
Papageorgiou
, “Stability of film flow over inclined topography based on a long-wave nonlinear model
,” J. Fluid Mech.
729
, 638
–671
(2013
).58.
T.
Pollak
and N.
Aksel
, “Crucial flow stabilization and multiple instability branches of gravity-driven films over topography
,” Phys. Fluids
25
, 024103
(2013
).59.
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
).60.
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
).61.
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
).62.
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
).63.
M.
Sellier
, “Inverse problems in free surface flows: A review
,” Acta Mech.
227
, 913
–935
(2016
).64.
M.
Schörner
, D.
Reck
, and N.
Aksel
, “Stability phenomena far beyond the Nusselt flow—Revealed by experimental asymptotics
,” Phys. Fluids
28
, 022102
(2016
).65.
M.
Schörner
, D.
Reck
, N.
Aksel
, and Y. Y.
Trifonov
, “Switching between different types of stability isles in films over topographies
,” Acta Mech.
(published online).66.
M.
Dauth
, M.
Schörner
, and N.
Aksel
, “What makes the free surface waves over topographies convex or concave? A study with Fourier analysis and particle tracking
,” Phys. Fluids
29
, 092108
(2017
).67.
Y. Y.
Trifonov
, “Nonlinear waves on a liquid film falling down an inclined corrugated surface
,” Phys. Fluids
29
, 054104
(2017
).68.
69.
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
).70.
D.
Reck
and N.
Aksel
, “Experimental study on the evolution of traveling waves over an undulated incline
,” Phys. Fluids
25
, 102101
(2013
).71.
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
).© 2018 Author(s).
2018
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