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Two-dimensional pulse dynamics and the formation of bound states on electrified falling films

  • M. G. Blyth (a1), D. Tseluiko (a2), T.-S. Lin (a3) and S. Kalliadasis (a4)

Abstract

The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling waves on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model, which is valid in the presence of weak inertia, and second using the Stokes equations. For obtuse angles, gravity is destabilising and solitary pulses exist even in the absence of an electric field. For acute angles, spatially non-uniform solutions exist only beyond a critical value of the electric field strength; moreover, solitary-pulse solutions are present only at sufficiently high supercritical electric-field strengths. The electric field increases the amplitude of the pulses, can generate recirculation zones in the humps and alters the far-field decay of the pulse tails from exponential to algebraic with a significant impact on pulse interactions. A weak-interaction theory predicts an infinite sequence of bound-state solutions for non-electrified flow, and a finite set for electrified flow. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. In addition, the electric field is shown to trigger a switch from absolute to convective instability, thereby regularising the dynamics, and this is confirmed by time-dependent simulations of the long-wave model.

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Corresponding author

Email address for correspondence: m.blyth@uea.ac.uk

References

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Balmforth, N. J. 1995 Solitary waves and homoclinic orbits. Annu. Rev. Fluid Mech. 27, 335373.
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554573.
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150155.
Blyth, M. G. 2008 Effect of an electric field on the stability of contaminated film flow down an inclined plane. J. Fluid Mech. 595, 221237.
Chang, H. C. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26, 103136.
Chang, H. C. & Demekhin, E. A. 2002 Complex Wave Dynamics on Thin Films. Springer.
Chang, H.-C., Demekhin, E. A. & Kopelevich, D. I. 1995 Stability of a solitary pulse against wave packet disturbances in an active medium. Phys. Rev. Lett. 75, 17471750.
Chang, H. C., Demekhin, E. A. & Kopelevich, D. I. 1996 Local stability theory of solitary pulses in an active medium. Physica D 97, 353375.
Charogiannis, A. & Markides, C. N. 2016 Application of planar laser-induced fluorescence for the investigation of interfacial waves and rivulet structures in liquid films flowing down inverted substrates. Intl Phenom. Heat Transfer 4 (4), 232252.
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 11311198.
Demekhin, E. A., Kalaidin, E. N., Kalliadasis, S. & Vlaskin, S.Yu 2010 Three-dimensional localized coherent structures of surface turbulence: model validation with experiments and further computations. Phys. Rev. E 82, 036322.
Duprat, C., Giorgiutti-Dauphiné, F., Tseluiko, D., Saprykin, S. & Kalliadasis, S. 2009 Liquid film coating a fiber as a model system for the formation of bound states in active dispersive-dissipative nonlinear media. Phys. Rev. Lett. 103, 234501.
Edmunds, D. E. & Evans, W. D. 1987 Spectral Theory and Differential Operators. Oxford.
Elphick, C., Ierley, G. R., Regev, O. & Spiegel, E. A. 1991 Interacting localized structures with Galilean invariance. Phys. Rev. A 44, 11101122.
Elphick, C., Meron, E. & Spiegel, E. A. 1990 Patterns of propagating pulses. SIAM J. Appl. Maths 50 (2), 490503.
Glendinning, P. & Sparrow, C. 1984 Local and global behavior near homoclinic orbits. J. Stat. Phys. 35, 645696.
Gomes, S. N., Papageorgiou, D. T. & Pavliotis, G. A. 2017 Stabilizing non-trivial solutions of the generalized Kuramoto–Sivashinsky equation using feedback and optimal control. IMA J. Appl. Maths 82 (1), 158194.
Gonzales, A. & Castellanos, A. 1996 Nonlinear electrohydrodynamic waves on films falling down an inclined plane. Phys. Rev. E 53, 35733578.
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473537.
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films, Series on Applied Mathematical Sciences, vol. 176. Springer.
Kawahara, T. 1983 Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation. Phys. Rev. Lett. 51, 381383.
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, 986992.
Lighthill, M. J. 1958 An Introduction to Fourier Analysis and Generalised Functions. Cambridge University Press.
Lin, T.-S., Pradas, M., Kalliadasis, S., Papageorgiou, D. T. & Tseluiko, D. 2015 Coherent structures in nonlocal dispersive active-dissipative systems. SIAM J. Appl. Maths 75, 538563.
Liu, J. & Gollub, J. P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 17021712.
Nosoko, T. & Miyara, A. 2004 The evolution and subsequent dynamics of waves on a vertically falling liquid film. Phys. Fluids 16, 11181126.
Park, C. D. & Nosoko, T. 2003 Three-dimensional wave dynamics on a falling film and associated mass transfer. AIChE J. 49, 27152727.
Pego, R. L. & Weinstein, M. I. 1992 Eigenvalues, and instabilities of solitary waves. Phil. Trans. R. Soc. Lond. A 340, 4794.
Pozrikidis, C. 2002 A Practical Guide to Boundary Element Methods. Chapman & Hall/CRC.
Pradas, M., Tseluiko, D. & Kalliadasis, S. 2011 Rigorous coherent-structure theory for falling liquid films: viscous dispersion effects on bound-state formation and self-organization. Phys. Fluids 23, 044104.
Reck, D. & Aksel, N. 2015 Recirculation areas underneath solitary waves on gravity-driven film flows. Phys. Fluids 27, 112107.
Rohlfs, W., Pischke, P. & Scheid, B. 2017 Hydrodynamic waves in films flowing under an inclined plane. Phys. Rev. Fluids 2, 044003.
Ruyer-Quil, C., Kofman, N., Chasseur, D. & Mergui, S. 2014 Dynamics of falling liquid films. Eur. Phys. J. E 37, 117.
Schäffer, E., Thurn-Albrecht, T., Russell, T. P. & Steiner, U. 2000 Electrically induced structure formation and pattern transfer. Nature 403, 874877.
Tomlin, R. J., Papageorgiou, D. T. & Pavliotis, G. A. 2017 Three-dimensional wave evolution on electrified falling films. J. Fluid Mech. 822, 5479.
Tseluiko, D., Blyth, M. G. & Papageorgiou, D. T. 2013 Stability of film flow over inclined topography based on a long-wave nonlinear model. J. Fluid Mech. 729, 638671.
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2008 Effect of an electric field on film flow down a corrugated wall at zero Reynolds number. Phys. Fluids 20, 042103.
Tseluiko, D. & Kalliadasis, S. 2014 Weak interaction of solitary pulses in active dispersive–dissipative nonlinear media. IMA J. Appl. Maths 79, 274299.
Tseluiko, D. & Papageorgiou, D. T. 2006 Wave evolution on electrified falling films. J. Fluid Mech. 556, 361386.
Tseluiko, D. & Papageorgiou, D. T. 2010 Dynamics of an electrostatically modified Kuramoto–Sivashinsky–Korteweg–de Vries equation arising in falling film flows. Phys. Rev. E 82, 016322.
Tseluiko, D., Saprykin, S., Duprat, C., Giorgiutti-Dauphiné, F. & Kalliadasis, S. 2010 Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: experiments and theory. Physica D 239, 20002010.
Vellingiri, R., Tseluiko, D. & Kalliadasis, S. 2015 Absolute and convective instabilities in counter-current gas–liquid film flows. J. Fluid Mech. 763, 166201.
Vlachogiannis, M. & Bontozoglou, V. 2001 Observations of solitary wave dynamics of film flows. J. Fluid Mech. 435, 191215.
Wray, A. W., Matar, O. K. & Papageorgiou, D. T. 2017 Accurate low-order modeling of electrified falling films at moderate Reynolds number. Phys. Rev. Fluids 2, 063701.
Yih, C.-S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321334.
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Two-dimensional pulse dynamics and the formation of bound states on electrified falling films

  • M. G. Blyth (a1), D. Tseluiko (a2), T.-S. Lin (a3) and S. Kalliadasis (a4)

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