Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-25T06:50:58.428Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability of film flow over inclined topography based on a long-wave nonlinear model

Published online by Cambridge University Press:  24 July 2013

D. Tseluiko ,
M. G. Blyth  and
D. T. Papageorgiou
D. Tseluiko*
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, UK
M. G. Blyth
Affiliation:
School of Mathematics, University of East Anglia, Norwich NR4 7TJ, UK
D. T. Papageorgiou
Affiliation:
Department of Mathematics and Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of a viscous liquid film flowing under gravity down an inclined wall with periodic corrugations is investigated. A long-wave model equation valid at near-critical Reynolds numbers is used to study the film dynamics, and calculations are performed for either sinusoidal or rectangular wall corrugations assuming either a fixed flow rate in the film or a fixed volume of fluid within each wall period. Under the two different flow assumptions, steady solution branches are delineated including subharmonic branches, for which the period of the free surface is an integer multiple of the wall period, and the existence of quasi-periodic branches is demonstrated. Floquet–Bloch theory is used to determine the linear stability of steady, periodic solutions and the nature of any instability is analysed using the method of exponentially weighted spaces. Under certain conditions, and depending on the wall period, the flow may be convectively unstable for small wall amplitudes but undergo transition to absolute instability as the wall amplitude increases, a novel theoretical finding for this class of flows; in other cases, the flow may be convectively unstable for small wall amplitudes but stable for larger wall amplitudes. Solutions with the same spatial period as the wall become unstable at a critical Reynolds number, which is strongly dependent on the period size. For sufficiently small wall periods, the corrugations have a destabilizing effect by lowering the critical Reynolds number above which instability occurs. For slightly larger wall periods, small-amplitude corrugations are destabilizing but sufficiently large-amplitude corrugations are stabilizing. For even larger wall periods, the opposite behaviour is found. For sufficiently large wall periods, the corrugations are destabilizing irrespective of their amplitude. The predictions of the linear theory are corroborated by time-dependent simulations of the model equation, and the presence of absolute instability under certain conditions is confirmed. Boundary element simulations on an inverted substrate reveal that wall corrugations can have a stabilizing effect at zero Reynolds number.

JFM classification

Instability: Absolute/convective instability Low-Reynolds-number flows: Boundary integral methods Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 729 , 25 August 2013 , pp. 638 - 671
Copyright
©2013 Cambridge University Press 

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

Alekseenko, S. V., Markovich, D. M., Evseev, A. R., Bobylev, A. V., Tarasov, B. V. & Karsten, V. M. 2008 Experimental investigation of liquid distribution over structured packing. AIChE J. 54, 1424–1430.CrossRefGoogle 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.Google Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150–155.Google Scholar
Bontozoglou, V. & Papapolymerou, G. 1997 Laminar film flow down a wavy incline. Intl J. Multiphase Flow 23, 69–79.Google Scholar
Brevdo, L. & Bridges, T. J. 1996 Absolute and convective instabilities of spatially periodic flows. Phil. Trans. R. Soc. Lond. A 354, 1027–1064.Google Scholar
Brevdo, L. & Bridges, T. J. 1997 Local and global instabilities of spatially developing flows: cautionary examples. Proc. R. Soc. Lond. A 453, 1345.CrossRefGoogle Scholar
Cao, Z., Vlachogiannis, M. & Bontozoglou, V. 2013 Experimental evidence for a short-wave global mode in film flow along periodic corrugations. J. Fluid Mech. 718, 304–320.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131.CrossRefGoogle Scholar
D’Alessio, S. J. D., Pascal, J. P. & Jasmine, H. A. 2009 Instability in gravity-driven flow over uneven surfaces. Phys. Fluids 21, 062105.Google Scholar
Decré, M. M. J. & Baret, J. C. 2003 Gravity-driven flows of viscous liquids over two-dimensional topographies. J. Fluid Mech. 487, 147–166.Google Scholar
Doedel, E. J. & Oldeman, B. E. 2009 AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations. Concordia University, available at http://cmvl.cs.concordia.ca/auto/.Google Scholar
Ern, A., Joubaud, R. & Lelièvre, T. 2011 Numerical study of a thin liquid film flowing down an inclined wavy plane. Physica D 240, 1714–1723.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 Rep. No. UR 823/28.Google Scholar
Georgantaki, A., Vatteville, J., Vlachogiannis, M. & Bontozoglou, V. 2011 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.Google 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.Google Scholar
Häcker, T. & Uecker, H. 2009 An integral boundary layer equation for film flow over inclined wavy bottoms. Phys. Fluids 21, 092105.Google Scholar
Heining, C., Bontozoglou, V., Aksel, N. & Wierschem, A. 2009 Nonlinear resonance in viscous films on inclined wavy planes. Intl J. Multiphase Flow 35, 78–90.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12, 1889–1898.Google Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2011 Falling Liquid Films. Springer.Google Scholar
Kalliadasis, S. & Thiele, U. (Eds) 2007 Thin Films of Soft Matter. Springer.Google Scholar
Kraus, A., Bar-Cohen, A. & Wative, A. A. 1971 Cooling Electronic Equipment. Wiley Online Library.Google Scholar
Lin, S. P. 1969 Finite-amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113–126.Google Scholar
Lin, S. P. 1974 Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417–429.Google Scholar
Lin, T.-S. & Kondic, L. 2010 Thin films flowing down inverted substrates: two-dimensional flow. Phys. Fluids 22, 052105.Google Scholar
Luo, H. & Pozrikidis, C. 2007 Gravity-driven film flow down an inclined wall with three-dimensional corrugations. Acta Mechanica 188, 209–225.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.Google 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 Rep. No. NL-UR 030/97.Google Scholar
Monkewitz, P. A., Huerre, P. & Chomaz, J. M. 1993 Global linear-stability analysis of weakly nonparallel shear flows. J. Fluid Mech. 251, 1–20.Google Scholar
Nakaya, C. 1975 Long waves on a thin fluid layer flowing down an inclined plane. Phys. Fluids 18, 1407.Google Scholar
Oron, A. & Gottlieb, O. 2002 Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 2622–2636.Google Scholar
Papageorgiou, D. T., Papanicolaou, G. C. & Smyrlis, Y.-S. 1993 Modulational stability of periodic solutions of the Kuramoto-Sivashinsky equation. In Singularities in Fluids, Plasmas and Optics (Heraklion, 1992), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 404. pp. 255–263. Kluwer.Google Scholar
Pozrikidis, C. 1988 The flow of a liquid film along a periodic wall. J. Fluid Mech. 188, 275–300.Google Scholar
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods with the Software Library Bemlib. Chapman & Hall/CRC.Google Scholar
Pozrikidis, C. 2003 Effect of surfactants on film flow down a periodic wall. J. Fluid Mech. 496, 105–127.Google Scholar
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.Google Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modelling film flows down inclined planes. Eur. Phys. J. B 6, 277–292.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modelling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results on the modelling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14, 170–183.Google Scholar
Sandstede, B. 2002 Stability of travelling waves. In Handbook of Dynamical Systems II (ed. Fiedler, B.). pp. 983–1055. North-Holland.Google Scholar
Sandstede, B. & Scheel, A. 2000 Absolute and convective instabilities of waves on unbounded domains. Physica D 145, 233–277.Google Scholar
Saprykin, S., Koopmans, R. J. & Kalliadasis, S. 2007 Free-surface thin-film flows over topography: influence of inertia and viscoelasticity. J. Fluid Mech. 578, 271–293.Google Scholar
Scheid, B., Oron, A., Colinet, P., Thiele, U. & Legros, J. C. 2002 Nonlinear evolution of non-uniformly heated falling liquid films. Phys. Fluids 14, 4130.Google Scholar
Scheid, B., Ruyer-Quil, C. & Manneville, P. 2006 Wave patterns in film flows: modelling and three-dimensional waves. J. Fluid Mech. 562, 183–222.Google Scholar
Scheid, B., Ruyer-Quil, C., Thiele, U., Kabov, O. A., Legros, J. C. & Colinet, P. 2004 Validity domain of the Benney equation including Marangoni effect for closed and open flows. J. Fluid Mech. 527, 303–335.Google Scholar
Scholle, M., Wierschem, A. & Aksel, N. 2004 Creeping films with vortices over strongly undulated bottoms. Acta Mechanica 168, 167–193.Google Scholar
Shkadov, V. Y. 1969 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Fluid Dyn. Res. 2, 29–34.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.Google Scholar
Thiele, U. 2011 On the depinning of a drop of partially wetting liquid on a rotating cylinder. J. Fluid Mech. 671, 121–136.Google 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.Google Scholar
Trifonov, Yu. Ya. 1999 Viscous liquid film flows over a periodic surface. Intl J. Multiphase Flow 24, 1139–1161.Google Scholar
Trifonov, Yu. Ya. 2004 Viscous film flow down corrugated surfaces. J. Appl. Mech. Tech. Phys. 45, 389–400.Google Scholar
Trifonov, Yu. Ya. 2007 Stability of a viscous liquid film flowing down a periodic surface. Intl J. Multiphase Flow 33, 1186–1204.Google Scholar
Tseluiko, D. & Blyth, M. G. 2009 Effect of inertia on electrified film flow over a wavy wall. J. Engng Maths 65, 229–242.Google Scholar
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2008a Effect of an electric field on film flow down a corrugated wall at zero Reynolds number. Phys. Fluids 20, 042103.Google Scholar
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J. M. 2008b Electrified viscous thin film flow over topography. J. Fluid Mech. 597, 449–475.CrossRefGoogle Scholar
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2009 Viscous electrified film flow over step topography. SIAM J. Appl. Maths 70, 845–865.Google Scholar
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2011 Electrified film flow over step topography at zero Reynolds number: an analytical and computational study. J. Engng Maths 69, 169–183.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. 2001 Observations of solitary wave dynamics of film flows. J. Fluid Mech. 435, 191–215.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
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.Google Scholar
Wierschem, A. & Aksel, N. 2003 Instability of a liquid film flowing down an inclined wavy plane. Physica D 186, 221–237.Google Scholar
Wierschem, A. & Aksel, N. 2004 Influence of inertia on eddies created in films creeping over strongly undulated substrates. Phys. Fluids 16, 4566.CrossRefGoogle Scholar
Wierschem, A., Bontozoglou, V., Heining, C., Uecker, H. & Aksel, N. 2008 Linear resonance in viscous films on inclined wavy planes. Intl J. Multiphase Flow 34, 580–589.Google Scholar
Wierschem, A., Lepski, C. & Aksel, N. 2005 Effect of long undulated bottoms on thin gravity-driven films. Acta Mechanica 179, 41–66.Google Scholar
Wierschem, A., Scholle, M. & Aksel, N. 2002 Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel. Exp. Fluids 33, 429–442.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.Google Scholar
Yih, C.-S. 1955 Stability of two-dimensional parallel flows for three-dimensional disturbances. Q. Appl. Maths 12, 434–435.Google Scholar
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.CrossRefGoogle Scholar