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Relative periodic solutions in conducting liquid films flowing down vertical fibres

Published online by Cambridge University Press:  28 June 2019

Zijing Ding Open the ORCID record for Zijing Ding [Opens in a new window]  and
Ashley P. Willis Open the ORCID record for Ashley P. Willis [Opens in a new window]
Zijing Ding*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge CB3 0WA, UK
Ashley P. Willis
Affiliation:
School of Mathematics and Statistics, University of Sheffield S3 7RH, UK
*
†Email address for correspondence: [email protected]
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Abstract

The dynamics of a conducting liquid film flowing down a cylindrical fibre, subjected to a radial electric field, is investigated using a long-wave model (Ding et al., J. Fluid Mech., vol. 752, 2014, p. 66). In this study, to account for the complicated interactions between droplets, we consider two large droplets in a periodic computational domain and find two distinct types of travelling wave solutions, which consist of either two identical droplets (type I) or two slightly different droplets (type II). Both are ‘relative’ equilibria, i.e. steady in a frame moving at their phase speed, and are stable in smaller domains when the electric field is weak. We also study relative periodic orbits, i.e. temporally recurrent dynamic solutions of the system. In the presence of the electric field, we show how these invariant solutions are linked to the dynamics, where the system can evolve into one of the steady travelling wave states, into an oscillatory state, or into a ‘singular structure’ (Wray et al., J. Fluid Mech., vol. 735, 2013, pp. 427–456; Ding et al., J. Fluid Mech., vol. 752, 2014, p. 66). We find that the oscillation between two similarly sized large droplets in the oscillatory state is well represented by relative periodic orbits. Varying the electric field strength, we demonstrate that relative periodic solutions arise as the dynamically important solution once the type-I or type-II travelling wave solutions lose stability. Oscillation can be either enhanced or impeded as the electric field’s strength increases. When the electric field is strong, no relative periodic solutions are found and a spike-like singular structure is observed. For the case where the electric field is not present, the oscillation is instead caused by the interaction between a large droplet and a nearby much smaller droplet. We show that this oscillation phenomenon originates from the instability of the type-I travelling wave solution in larger domains, and that the oscillatory state can again be represented by an exact relative periodic orbit. The relative periodic orbit solution is also compared with experimental study for this case. The present study demonstrates that the relative periodic solutions are better at capturing the wave speed and oscillatory dynamics than the travelling wave solutions in the unsteady flow regime.

JFM classification

Instability: Nonlinear instability Interfacial Flows (free surface): Thin films
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 835 - 855
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Copyright
© 2019 Cambridge University Press 

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References

Blyth, M. G., Tseluiko, D., Lin, T.-S. & Kalliadasis, S. 2018 Two-dimensional pulse dynamics and the formation of bound states on electrified falling films. J. Fluid Mech. 855, 210–235.Google Scholar
Budanur, N., Short, K., Farazmand, M. & Willis, A. Relative periodic orbits form the backbone of turbulent pipe flow. J. Fluid Mech. 833, 274–301.Google Scholar
Chandler, G. & Kerswell, R. 2013 Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554–595.Google Scholar
Chang, C. & Demekhin, E. 1999 Mechanism for drop formation on a coated vertical fibre. J. Fluid Mech. 380, 233–255.Google Scholar
Chantry, M., Willis, A. & Kerswell, R. 2014 Genesis of streamwise-localized solutions from globally periodic travelling waves in pipe flow. Phys. Rev. Lett. 112, 164501.Google Scholar
Conroy, D., Matar, O., Craster, R. & Papageorgiou, D. 2011 Breakup of an electrified perfectly conducting, viscous thread in an AC field. Phys. Rev. E 83, 066314.Google Scholar
Craster, R. & Matar, O. 2005 Electrically induced pattern formation in thin leaky dielectric films. Phys. Fluids 17 (3), 032104.Google Scholar
Craster, R. & Matar, O. 2006 On viscous beads flowing down a vertical fibre. J. Fluid Mech. 553, 85–105.Google Scholar
Craster, R. & Matar, O. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.Google Scholar
Cvitanović, P., Davidchack, R. & Sminos, E. 2010 On the state space geometry of the Kuramoto–Sivashinsky flow in a periodic domain. SIAM J. Appl. Dyn. Syst. 9, 1–33.Google Scholar
Cvitanović, P. & Gibson, J. 2010 Geometry of the turbulence in wall-bounded shear flows: periodic orbits. Phys. Scr. T 142, 014007.Google Scholar
De Ryck, A. & Quéré, D. 1996 Inertial coating of a fibre. J. Fluid Mech. 311, 219–237.Google Scholar
Ding, Z., Xie, J., Wong, T. N. & Liu, R. 2014 Dynamics of liquid films on vertical fibres in a radial electric field. J. Fluid Mech. 752, 66.Google Scholar
Ding, Z. & Wong, T. N. 2017 Three-dimensional dynamics of thin liquid films on vertical cylinders with Marangoni effect. Phys. Fluids 29, 1.Google Scholar
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.Google Scholar
Duprat, C., Ruyer-Quil, C., Kalliadasis, S. & Giorgiutti-Dauphiné, F. 2007 Absolute and convective instabilities of a viscous film flowing down a vertical fiber. Phys. Rev. Lett. 98, 244502.Google Scholar
Frenkel, A. 1992 Nonlinear thoery of strongly undulating thin films flowing down vertical cylinders. Eur. Phys. Lett. 18, 583–588.Google Scholar
Kalliadasis, S. & Chang, C. 1994 Drop formation during coating of vertical fibres. J. Fluid Mech. 261, 135.Google Scholar
Kliakhandler, I., Davis, S. & Bankhoff, S. 2001 Viscous beads on vertical fibre. J. Fluid Mech. 429, 381–390.Google Scholar
Lin, T.-S., Tseluiko, D., Blyth, M. G. & Kalliadasis, S. 2018 Continuation methods for time-periodic traveling-wave solutions to evolution equations. Appl. Math. Lett. 86, 291–297.Google Scholar
Liu, R., Chen, X. & Ding, Z. 2018 Absolute and convective instabilities of a film flow down a vertical fiber subjected to a radial electric field. Phys. Rev. E 97, 013109.Google Scholar
Novbari, E. & Oron, A. 2009 Energy integral method model for the nonlinear dynamics of an axisymmetric thin liquid film falling on a vertical cylinder. Phys. Fluids 21, 062107.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Papageorgiou, D. 2018 Film flows in the presence of electric fields. Annu. Rev. Fluid Mech. 51, 155.Google Scholar
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.Google Scholar
Quéré, D. 1999 Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31, 347–384.Google Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277–292.Google Scholar
Ruyer-Quil, C., Treveleyan, P., Giorgiutti-Dauphiné, F. & Kalliadasis, S. 2008 Modelling film flows down a fibre. J. Fluid Mech. 603, 431–462.Google Scholar
Sadeghpour, A., Zeng, Z. & Ju, Y. 2017 Effects of nozzle geometry on the fluid dynamics of thin liquid films flowing down vertical strings in the Rayleigh–Plateau regime. Langmuir 33, 6292–6299.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 27–64.Google Scholar
Schäffer, E., Thurn-Albrecht, T., Russell, T. & Steiner, U. 2000 Electrically induced structure formation and pattern transfer. Nature 403, 874.Google Scholar
Shkadov, V. Y., Beloglazkin, A. & Gerasimov, S. 2008 Solitary waves in a viscous liquid film flowing down a thin vertical cylinder. Mosc. Univ. Mech. Bull. 63, 122–128.Google Scholar
Sisoev, G., Craster, R., Matar, O. & Gerasimov, S. 2006 Film flow down a fibre at moderate flow rates. Chem. Engng Sci. 61, 7279–7298.Google Scholar
Trifonov, Y. Y. 1992 Steady-state travelling waves on the surface of a viscous liquid film falling down on vertical wires and tubes. AIChE J. 38, 821–834.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.Google Scholar
Wang, Q. & Papageorgiou, D. 2011 Dynamics of a viscous thread surrounded by another viscous fluid in a cylindrical tube under the action of a radial electric field: breakup and touchdown singularities. J. Fluid Mech. 683, 27–56.Google Scholar
Wray, A., Matar, O. & Papageorgiou, D. 2012 Nonlinear waves in electrified viscous film flow down a vertical cylinder. IMA J. Appl. Maths 77, 430–440.Google Scholar
Wray, A., Papageorgiou, D. & Matar, O. 2013a Electrified coating flows on vertical fibres: enhancement or suppression of interfacial dynamics. J. Fluid Mech. 735, 427–456.Google Scholar
Wray, A., Papageorgiou, D. & Matar, O. 2013b Electrostatically controlled large-amplitude, non-axisymmetric waves in thin film flows down a cylinder. J. Fluid Mech. 736, R2.Google Scholar
Zeng, Z., Sadeghpour, A., Warrier, G. & Ju, Y. 2017 Experimental study of heat transfer between thin liquid films flowing down a vertical string in the Rayleigh–Plateau instability regime and a counterflowing gas stream. Intl J. Heat Mass Transfer 108, 830–840.Google Scholar

Ding and Willis supplementary movie 1

Relative periodic orbit at L=4, Eb=3

Download Ding and Willis supplementary movie 1(Video)
Video 8.1 MB

Ding and Willis supplementary movie 2

Relative periodic orbit 1 at L=12, Eb=0

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Video 4.7 MB

Ding and Willis supplementary movie 3

Relative periodic orbit 2 at L=12, Eb=0

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Video 4.7 MB

Ding and Willis supplementary movie 4

Relative periodic orbit 3 at L=12, Eb=0

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Video 4.8 MB

Ding and Willis supplementary movie 5

Relative periodic orbit of flow regime c

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Video 1.6 MB