Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T15:52:14.121Z Has data issue: false hasContentIssue false
  • English
  • Français

Electrified viscous thin film flow over topography

Published online by Cambridge University Press:  01 February 2008

D. TSELUIKO ,
M. G. BLYTH ,
D. T. PAPAGEORGIOU  and
J.-M. VANDEN-BROECK
D. TSELUIKO
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
M. G. BLYTH
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
D. T. PAPAGEORGIOU
Affiliation:
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102, USA
J.-M. VANDEN-BROECK
Affiliation:
School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. The limiting behaviour of the film profile as the steepness of the wall geometry is increased is discussed. Using substantial numerical evidence, it is established that as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. The effect of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates is also considered.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 597 , 25 February 2008 , pp. 449 - 475
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Abdelouhab, A., Bona, J. L., Felland, M. & Saut, J.-C. 1989 Nonlocal models for nonlinear dispersive waves. Physica D 40, 360–392.Google Scholar
Alekseenko, S. V., Nakoryakov, V. E. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460.Google Scholar
Argyriadi, K., Vlachogiannis, M. & Bontozoglou, V. 2006 Experimental study of inclined film flow along periodic corrugations: The effect of wall steepness. Phys. Fluids 18, 012102.CrossRefGoogle Scholar
Bankoff, S. G., Griffing, E. M. & Schluter, R. A. 2002 Use of an electric field in an electrostatic liquid film radiator. Ann. N.Y. Acad. Sci. 974, 1–9.Google Scholar
Bankoff, S. G., Miksis, M. J. & Gwinner, H., Kim, R. 1994 Design considerations for the rotating electrostatic liquid-film radiator. Nucl. Engng Des. 149, 441–447.CrossRefGoogle Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefGoogle Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150–155.Google Scholar
Bertozzi, A. L. & Brenner, M. P. 1997 Linear stability and transient growth in driven contact lines. Phys. Fluids 9, 530–539.Google Scholar
Bielarz, C. & Kalliadasis, S. 2003 Time-dependent free-surface thin film flows over topography. Phys. Fluids 15 (9), 2512–2524.CrossRefGoogle Scholar
Davis, J. M. & Troian, S. M. 2005 Generalized linear stability of noninertial coating flows over topographical features. Phys. Fluids 17, 072103.CrossRefGoogle Scholar
Decré, M. M. J. & Baret, J. C. 2003 Gravity-driven flows of low viscosity liquids over two-dimensional topographies. J. Fluid Mech. 487, 147–166.CrossRefGoogle Scholar
Diez, J. A. & Kondic, L. 2001 Contact line instabilities of thin liquid films. Phys. Rev. Lett. 86, 632–635.Google Scholar
Fernandez-Parent, C., Lammers, J. H. & Decré, M. M. J. 1998 Flow of a gravity driven thin liquid film over one-dimensional topographies. Phillips Research Unclassified Report No. UR 823/28.Google Scholar
Gaskell, P. H., Jimack, P. K., Sellier, M. & Thompson, H. M. 2006 Flow of evaporating, gravity-driven thin liquid films over topography. Phys. Fluids 18, 013601.Google Scholar
Gaskell, P. H., Jimack, P. K., Sellier, M., Thompson, H. M. & Wilson, M. C. T. 2004 Gravity-driven flow of continuous thin liquid films on non-porous substrates with topography. J. Fluid Mech. 509, 253–280.Google Scholar
Gonzalez, A. & Castellanos, A. 1996 Nonlinear electrohydrodynamic waves on films falling down an inclined plane. Phys. Rev. E 53, 3573–3578.Google ScholarPubMed
Griffing, E. M., Bankoff, S. G., Miksis, M. J. & Schluter, R. A. 2006 Electrohydrodynamics of thin flowing films. J. Fluids Engng 128.CrossRefGoogle Scholar
Gu, F., Liu, C. J., Yuan, X. G. & Yu, G. C. 2004 CFD simulation of liquid film flow on inclined plates. Chem. Engng Technol. 27, 1099–1104.CrossRefGoogle Scholar
Howell, P. D. 2003 Surface-tension-driven flow on a moving curved surface. J. Engng Math. 45, 283–308.CrossRefGoogle Scholar
Jensen, O. E., Chini, G. P. & King, J. R. 2004 Thin-film flows near isolated humps and into corners. J. Engng Math. 50, 289–309.CrossRefGoogle Scholar
Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12, 1889–1898.CrossRefGoogle Scholar
Kalliadasis, S. & Homsy, G. M. 2001 Stability of free-surface thin-film flows over topography. J. Fluid Mech. 448, 387–410.CrossRefGoogle Scholar
Kim, H., Bankoff, S. G. & Miksis, M. J. 1992 The effect of an electrostatic field on film flow down an inclined plane. Phys. Fluids A 4, 2117–2130.CrossRefGoogle Scholar
Kim, H., Bankoff, S. G. & Miksis, M. J. 1994 The cylindrical electrostatic liquid-film radiator for heat rejection in space. Trans. ASME: J. Heat Transfer 116, 986–992.Google Scholar
Kistler, S. F. & Schweizer, P. M. 1997 Liquid Film Coating: Scientific Principles and their Technological Implications. Chapman & Hall.CrossRefGoogle Scholar
Lin, S. P. 1974 Finite amplitude sideband stability of a viscous film. J. Fluid Mech. 63, 417–429.Google Scholar
Lister, J. R. 1990 Buoyancy-driven fluid fracture: similarity solutions for the horizontal and vertical propagation of fluid-filled cracks. J. Fluid Mech. 217, 213–239.Google Scholar
Liu, J. & Gollub, J. P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712.CrossRefGoogle Scholar
Liu, J., Paul, J. D. & Gollub, J. P. 1993 Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101.Google Scholar
Malamataris, N. A. & Bontozoglou, V. 1999 Computer aided analysis of viscous film flow along an inclined wavy wall. J. Comput. Phys. 154, 372–392.CrossRefGoogle Scholar
Malamataris, N. A., Vlachogiannis, M. & Bontozoglou, V. 2002 Solitary waves on inclined films: Flow structure and binary interactions. Phys. Fluids 14, 1082–1094.CrossRefGoogle Scholar
Mazouchi, A. & Homsy, G. M. 2001 Free surface Stokes flow over topography. Phys. Fluids 13 (10), 2751–2761.CrossRefGoogle Scholar
Messé, S. & Decré, M. M. J. 1997 Experimental study of a gravity driven water film flowing down inclined plates with different patterns. Phillips Research Unclassified Report No. NL-UR 030/97.Google Scholar
Papageorgiou, D. T. & Petropoulos, P. G. 2004 Generation of interfacial instabilities in charged electrified viscous liquid films. J. Engng Maths 50, 223–240.Google Scholar
Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2004 Large-amplitude capillary waves in electrified fluid sheets. J. Fluid. Mech. 508, 71–88.Google Scholar
Pozrikidis, C. 1988 The flow of a liquid film along a periodic wall. J. Fluid Mech. 188, 275–300.CrossRefGoogle Scholar
Ramaswamy, B., Chippada, S. & Joo, S. W. 1996 A full-scale numerical study of interfacial instabilities in thin-film flows. J. Fluid Mech. 325, 163–194.CrossRefGoogle Scholar
Roy, R. V., Roberts, A. J. & Simpson, M. E. 2002 A lubrication model of coating flows over a curved substrate in space. J. Fluid Mech. 454, 235–261.CrossRefGoogle Scholar
Schwartz, L. W. & Weidner, D. E. 1995 Modelling of coating flows on curved surfaces. J. Engng Math. 29, 91–103.Google Scholar
Serifi, K., Malamataris, N. A. & Bontozoglou, V. 2004 Transient flow and heat transfer phenomena in inclined wavy films. Intl J. Therm. Sci. 43, 761–767.CrossRefGoogle Scholar
Shetty, S. & Cerro, R. L. 1993 Flow of a thin film over a periodic surface. Intl J. Multiphase Flow 19, 1013–1027.Google Scholar
Spence, D. A., Sharp, P. W. & Turcotte, D. L. 1987 Buoyancy-driven crack propagation: a mechanism for magma migration. J. Fluid Mech. 174, 135–153.CrossRefGoogle Scholar
Stillwagon, L. E. & Larson, R. G. 1988 Fundamentals of topographic substrate leveling. J. Appl. Phys. 63, 5251–5258.Google Scholar
Stillwagon, L. E. & Larson, R. G. 1990 Leveling of thin films over uneven substrates during spin coating. Phys. Fluids A 2, 1937–1944.Google Scholar
Stocker, R. & Hosoi, A. E. 2005 Lubrication in a corner. J. Fluid Mech. 544, 353–377.CrossRefGoogle Scholar
Tougou, H. 1978 Long waves on a film flow of a viscous fluid down an inclined uneven wall. J. Phys. Soc. Japan 44, 1014–1019.CrossRefGoogle Scholar
Trifonov, Yu. Ya. 1999 Viscous liquid film flows over a periodic surface. Intl J. Multiphase Flow 24, 1139–1161.CrossRefGoogle Scholar
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2008 Effect of an electric field on the creeping flow of a liquid film down a corrugated wall. (Submitted).Google Scholar
Tseluiko, D. & Papageorgiou, D. T. 2006 Wave evolution on electrified falling films. J. Fluid Mech. 556, 361–386.Google Scholar
Vlachogiannis, M. & Bontozoglou, V. 2002 Experiments on laminar film flow along a periodic wall. J. Fluid Mech. 457, 133–156.Google Scholar
Wang, C. Y. 1981 Liquid film flowing slowly down a wavy incline. AIChE J 27, 207–212.Google Scholar
Wang, C. Y. 1984 Thin film flowing down a curved surface. Z. Angew. Math. Phys. 35, 532–544.Google Scholar
Wierschem, A. & Aksel, N. 2004 a Hydraulic jumps and standing waves in gravity-driven flows of viscous liquids in wavy open channels. Phys. Fluids 16, 3868–3877.Google Scholar
Wierschem, A. & Aksel, N. 2004 b Influence of inertia on eddies created in films creeping over strongly undulated substrates. Phys. Fluids 16, 4566–4574.Google Scholar
Wierschem, A., Scholle, M. & Aksel, N. 2003 Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers. Phys. Fluids 15, 426–435.Google Scholar
Yih, C.-H. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar
Yoshimura, P. N., Nosoko, T. & Nagata, T. 1996 Enhancement of mass transfer into a falling laminar liquid film by two-dimensional surface waves: Some experimental observations and modeling. Chem. Engng Sci. 51, 1231–1240.Google Scholar
Zhao, L. & Cerro, R. L. 1992 Experimental characterization of viscous film flows over complex surfaces. Intl J. Multiphase Flow 18, 495–516.CrossRefGoogle Scholar