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Pulse dynamics in a power-law falling film

Published online by Cambridge University Press:  17 April 2014

M. Pradas ,
D. Tseluiko ,
C. Ruyer-Quil  and
S. Kalliadasis
M. Pradas
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
D. Tseluiko
Affiliation:
School of Mathematical Sciences, Loughborough University, Leicestershire LE11 3TU, UK
C. Ruyer-Quil
Affiliation:
Université de Savoie, CNRS – Laboratoire LOCIE, 73376 Le Bourget du Lac, France Institut Universitaire de France
S. Kalliadasis*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: s.kalliadasis@imperial.ac.uk
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Abstract

We examine the stability, dynamics and interactions of solitary waves in a two-dimensional vertically falling thin liquid film that exhibits shear-thinning effects. We use a low-dimensional two-field model that describes the evolution of both the local flow rate and the film thickness and is consistent up to second-order terms in the long-wave expansion. The shear-thinning behaviour is modelled via a power-law formulation with a Newtonian plateau in the limit of small strain rates. Our results show the emergence of a hysteresis behaviour as the control parameter (the Reynolds number) is increased which is directly related to the shear-thinning character of the liquid and can be quantified with both linear analysis arguments and a physical interpretation. We also study pulse interactions, observing that two pulses may attract or repel each other either monotonically or in an oscillatory manner. In large domains we find that for a given Reynolds number the final state depends on the initial condition, a consequence of the presence of multiple solutions.

JFM classification

Non-Newtonian Flows: Non-Newtonian Flows Nonlinear Dynamical Systems: Pattern formation Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 747 , 25 May 2014 , pp. 460 - 480
Copyright
© 2014 Cambridge University Press 

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References

Allgower, E. L. & Georg, K. 1987 Introduction to Numerical Continuation Methods. SIAM.Google Scholar
Amaouche, M., Djema, A. & Bourdache, L. 2009 A modified Shkadov’s model for thin film flow of a power law fluid over an inclined surface. C. R. Méc. 337, 48–52.CrossRefGoogle Scholar
Balmforth, N. J. 1995 Solitary waves and homoclinic orbits. Annu. Rev. Fluid Mech. 27, 335–373.CrossRefGoogle Scholar
Balmforth, N. J. & Craster, R. V. 1999 A consistent thin-layer theory for Bingham plastics. J. Non-Newtonian Fluid Mech. 65–81.Google Scholar
Balmforth, N. J. & Liu, J. J. 2004 Roll waves in mud. J. Fluid Mech. 519, 33–54.Google Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150–155.Google Scholar
Bewersdorff, H.-W. & Singh, R. 1988 Rheological and drag reduction characteristics of xanthan gum solutions. Rheol. Acta 27, 617–627.Google Scholar
Chang, H.-C. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 26, 103–136.CrossRefGoogle Scholar
Chang, H.-C. & Demekhin, E. 2002 Complex Wave Dynamics on Thin Films. Elsevier.Google Scholar
Chang, H.-C., Demekhin, E. & Kalaidin, E. 1995 Interaction dynamics of solitary waves on a falling film. J. Fluid Mech. 294, 123–154.CrossRefGoogle Scholar
Demekhin, E. A., Kalaidin, E. N., Kalliadasis, S. & Vlaskin, S. Yu. 2007 Three-dimensional localized coherent structures of surface turbulence. I. Scenarios of two-dimensional–three-dimensional transition. Phys. Fluids 19, 114103.Google Scholar
Denier, J. P. & Dabrowski, P. P. 2004 On the boundary-layer equations for power-law fluids. Proc. R. Soc. Lond. A 460, 3143.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.CrossRefGoogle 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
Duprat, C., Tseluiko, D., Saprykin, S., Kalliadasis, S. & Giorgiutti-Dauphiné, F. 2011 Wave interactions on a viscous film coating a vertical fibre: Formation of bound states. Chem. Eng. Process. 50, 519–524.Google Scholar
Elphick, C., Ierley, G. R., Regev, O. & Spiegel, E. A. 1991 Interacting localized structures with galilean invariance. Phys. Rev. A 44, 1110–1122.Google Scholar
Fernández-Nieto, E. D., Noble, P. & Vila, J.-P. 2010 Shallow water equations for non-Newtonian fluids. J. Non-Newtonian Fluid Mech. 165, 712–732.Google Scholar
Hulme, G. 1974 The interpretation of lava flow morphology. Geophys. J. R. Astron. Soc. 39, 361–383.Google Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films. (Springer Series on Applied Mathematical Sciences) , vol. 176. Springer.Google Scholar
Kapitza, P. L. 1948a Wave flow of thin layers of a viscous fluid: I. Free flow. Zh. Eksp. Teor. Fiz. 18, 3–18.Google Scholar
Kapitza, P. L. 1948b Wave flow of thin layers of a viscous fluid: II. Fluid flow in the presence of continuous gas flow and heat transfer. Zh. Eksp. Teor. Fiz. 18, 19–28.Google Scholar
Krauskopf, B., Osinga, H. M. & Galán-Vioque, J.(Eds.) 2007 Numerical Continuation Methods for Dynamical Systems. Springer.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.CrossRefGoogle Scholar
Malamataris, N. A., Vlachogiannis, M. & Bontozoglou, V. 2002 Solitary waves on inclined films: Flow structure and binary interactions. Phys. Fluids 14, 1082–1095.Google Scholar
Pego, R. L. & Weinstein, M. I. 1994 Asymptotic stability of solitary waves. Commun. Math. Phys. 164, 305–349.Google Scholar
Pradas, M., Kalliadasis, S., Nguyen, P.-K. & Bontozoglou, V. 2013 Bound-state formation in interfacial turbulence: direct numerical simulations and theory. J. Fluid Mech. 716, R2.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.CrossRefGoogle Scholar
Pradas, M., Kalliadasis, S. & Tseluiko, D. 2012 Binary interactions of solitary pulses in falling liquid films. IMA J. Appl. Maths 77, 408–419.Google Scholar
Rizwan-Sadiq, I. M. & Usha, R. 2010 Effect of permeability on the instability of a non-Newtonian film down a porous inclined plane. J. Non-Newtonian Fluid Mech. 165, 1171–1188.Google Scholar
Rousset, F., Millet, S., Botton, V. & Benhadid, H. 2007 Temporal stability of Carreau fluid flow down an incline. J. Fluids Eng. 129, 913–920.Google Scholar
Ruyer-Quil, C., Chakraborty, S. & Dandapat, B. S. 2012 Wavy regime of a power-law film flow. J. Fluid Mech. 692, 220–256.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.Google Scholar
Sandstede, B. & Scheel, A. 2000 Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D 145, 233–277.Google Scholar
Savva, N. & Pavliotis, G. A. 2010 Two-dimensional droplet spreading over random topographical substrates. Phys. Rev. Lett. 104, 084501.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
Sisoev, G. M., Dandapat, B. S., Matveyev, K. S. & Mukhopadhyay, A. 2007 Bifurcation analysis of the travelling waves on a falling power-law fluid film. J. Non-Newtonian Fluid Mech. 141, 128–137.Google Scholar
Sisoev, G. M. & Usha, R. 2009 Wave regimes on power-law fluid film flowing down a porous plane. Intl J. Non-Linear Mech. 45, 236–241.Google Scholar
Tseluiko, D. & Kalliadasis, S. 2014 Weak interaction of solitary pulses in active dispersive–dissipative nonlinear media. IMA J. Appl. Maths 79, 274–299.Google Scholar
Tseluiko, D., Saprykin, S., Duprat, C., Giorgiutti-Dauphiné, F. & Kalliadasis, S. 2010a Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: Experiments and theory. Physica D 239, 2000–2010.Google Scholar
Tseluiko, D., Saprykin, S. & Kalliadasis, S. 2010b Interaction of solitary pulses in active dispersive–dissipative media. Proc. Est. Acad. Sci. 59, 139–144.Google Scholar
Usha, R., Millet, S., Benhadid, H. & Rousset, F. 2011 Shear-thinning film on a porous substrate: Stability analysis of a one-sided model. Chem. Eng. Sci. 66, 5614–5627.CrossRefGoogle Scholar
Vlachogiannis, M. & Bontozoglou, V. 2001 Observations of solitary wave dynamics of film flows. J. Fluid Mech. 435, 191–215.Google Scholar
Vlachogiannis, M., Samandas, A., Leontidis, V. & Bontozoglou, V. 2010 Effect of channel width on the primary instability of inclined film flow. Phys. Fluids 22, 012106.Google Scholar
Weinstein, S. J. 1990 Wave propagation in the flow of shear-thinning fluids down an incline. AIChE J. 36, 1873–1889.Google Scholar
Yeong, K. K., Gavriilidis, A., Zapf, R. & Hessel, V. 2004 Experimental studies of nitrobenzene hydrogenation in a microstructured falling film reactor. Chem. Eng. Sci. 59, 3491–3494.Google Scholar