Skip to main content Accessibility help
×
Home

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

  • D. Tseluiko (a1), M. G. Blyth (a2) and D. T. Papageorgiou (a3)

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.

Copyright

Corresponding author

Email address for correspondence: D.Tseluiko@lboro.ac.uk

References

Hide All
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, 14241430.
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.
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150155.
Bontozoglou, V. & Papapolymerou, G. 1997 Laminar film flow down a wavy incline. Intl J. Multiphase Flow 23, 6979.
Brevdo, L. & Bridges, T. J. 1996 Absolute and convective instabilities of spatially periodic flows. Phil. Trans. R. Soc. Lond. A 354, 10271064.
Brevdo, L. & Bridges, T. J. 1997 Local and global instabilities of spatially developing flows: cautionary examples. Proc. R. Soc. Lond. A 453, 1345.
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, 304320.
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131.
D’Alessio, S. J. D., Pascal, J. P. & Jasmine, H. A. 2009 Instability in gravity-driven flow over uneven surfaces. Phys. Fluids 21, 062105.
Decré, M. M. J. & Baret, J. C. 2003 Gravity-driven flows of viscous liquids over two-dimensional topographies. J. Fluid Mech. 487, 147166.
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/.
Ern, A., Joubaud, R. & Lelièvre, T. 2011 Numerical study of a thin liquid film flowing down an inclined wavy plane. Physica D 240, 17141723.
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.
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.
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, 10991104.
Häcker, T. & Uecker, H. 2009 An integral boundary layer equation for film flow over inclined wavy bottoms. Phys. Fluids 21, 092105.
Heining, C., Bontozoglou, V., Aksel, N. & Wierschem, A. 2009 Nonlinear resonance in viscous films on inclined wavy planes. Intl J. Multiphase Flow 35, 7890.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12, 18891898.
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2011 Falling Liquid Films. Springer.
Kalliadasis, S. & Thiele, U. (Eds) 2007 Thin Films of Soft Matter. Springer.
Kraus, A., Bar-Cohen, A. & Wative, A. A. 1971 Cooling Electronic Equipment. Wiley Online Library.
Lin, S. P. 1969 Finite-amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113126.
Lin, S. P. 1974 Finite amplitude side-band stability of a viscous film. J. Fluid Mech. 63, 417429.
Lin, T.-S. & Kondic, L. 2010 Thin films flowing down inverted substrates: two-dimensional flow. Phys. Fluids 22, 052105.
Luo, H. & Pozrikidis, C. 2007 Gravity-driven film flow down an inclined wall with three-dimensional corrugations. Acta Mechanica 188, 209225.
Malamataris, N. A. & Bontozoglou, V. 1999 Computer aided analysis of viscous film flow along an inclined wavy wall. J. Comput. Phys. 154, 372392.
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.
Monkewitz, P. A., Huerre, P. & Chomaz, J. M. 1993 Global linear-stability analysis of weakly nonparallel shear flows. J. Fluid Mech. 251, 120.
Nakaya, C. 1975 Long waves on a thin fluid layer flowing down an inclined plane. Phys. Fluids 18, 1407.
Oron, A. & Gottlieb, O. 2002 Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 26222636.
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. 255263. Kluwer.
Pozrikidis, C. 1988 The flow of a liquid film along a periodic wall. J. Fluid Mech. 188, 275300.
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods with the Software Library Bemlib. Chapman & Hall/CRC.
Pozrikidis, C. 2003 Effect of surfactants on film flow down a periodic wall. J. Fluid Mech. 496, 105127.
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 2750.
Ruyer-Quil, C. & Manneville, P. 1998 Modelling film flows down inclined planes. Eur. Phys. J. B 6, 277292.
Ruyer-Quil, C. & Manneville, P. 2000 Improved modelling of flows down inclined planes. Eur. Phys. J. B 15, 357369.
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, 170183.
Sandstede, B. 2002 Stability of travelling waves. In Handbook of Dynamical Systems II (ed. Fiedler, B.). pp. 9831055. North-Holland.
Sandstede, B. & Scheel, A. 2000 Absolute and convective instabilities of waves on unbounded domains. Physica D 145, 233277.
Saprykin, S., Koopmans, R. J. & Kalliadasis, S. 2007 Free-surface thin-film flows over topography: influence of inertia and viscoelasticity. J. Fluid Mech. 578, 271293.
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.
Scheid, B., Ruyer-Quil, C. & Manneville, P. 2006 Wave patterns in film flows: modelling and three-dimensional waves. J. Fluid Mech. 562, 183222.
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, 303335.
Scholle, M., Wierschem, A. & Aksel, N. 2004 Creeping films with vortices over strongly undulated bottoms. Acta Mechanica 168, 167193.
Shkadov, V. Y. 1969 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Fluid Dyn. Res. 2, 2934.
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, 381411.
Thiele, U. 2011 On the depinning of a drop of partially wetting liquid on a rotating cylinder. J. Fluid Mech. 671, 121136.
Tougou, H. 1978 Long waves on a film flow of a viscous fluid down an inclined uneven wall. J. Phys. Soc. Japan 44, 10141019.
Trifonov, Yu. Ya. 1999 Viscous liquid film flows over a periodic surface. Intl J. Multiphase Flow 24, 11391161.
Trifonov, Yu. Ya. 2004 Viscous film flow down corrugated surfaces. J. Appl. Mech. Tech. Phys. 45, 389400.
Trifonov, Yu. Ya. 2007 Stability of a viscous liquid film flowing down a periodic surface. Intl J. Multiphase Flow 33, 11861204.
Tseluiko, D. & Blyth, M. G. 2009 Effect of inertia on electrified film flow over a wavy wall. J. Engng Maths 65, 229242.
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.
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J. M. 2008b Electrified viscous thin film flow over topography. J. Fluid Mech. 597, 449475.
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, 845865.
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, 169183.
Tseluiko, D. & Papageorgiou, D. T. 2006 Wave evolution on electrified falling films. J. Fluid Mech. 556, 361386.
Vlachogiannis, M. & Bontozoglou, V. 2001 Observations of solitary wave dynamics of film flows. J. Fluid Mech. 435, 191215.
Vlachogiannis, M. & Bontozoglou, V. 2002 Experiments on laminar film flow along a periodic wall. J. Fluid Mech. 457, 133156.
Wang, C. Y. 1981 Liquid film flowing slowly down a wavy incline. AIChE J. 27, 207212.
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 2953.
Wierschem, A. & Aksel, N. 2003 Instability of a liquid film flowing down an inclined wavy plane. Physica D 186, 221237.
Wierschem, A. & Aksel, N. 2004 Influence of inertia on eddies created in films creeping over strongly undulated substrates. Phys. Fluids 16, 4566.
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, 580589.
Wierschem, A., Lepski, C. & Aksel, N. 2005 Effect of long undulated bottoms on thin gravity-driven films. Acta Mechanica 179, 4166.
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, 429442.
Wierschem, A., Scholle, M. & Aksel, N. 2003 Vortices in film flow over strongly undulated bottom profiles at low Reynolds numbers. Phys. Fluids 15, 426.
Yih, C.-S. 1955 Stability of two-dimensional parallel flows for three-dimensional disturbances. Q. Appl. Maths 12, 434435.
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321334.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

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

  • D. Tseluiko (a1), M. G. Blyth (a2) and D. T. Papageorgiou (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed