Hostname: page-component-8448b6f56d-gtxcr Total loading time: 0 Render date: 2024-04-20T05:42:52.161Z Has data issue: false hasContentIssue false
  • English
  • Français

Shear-imposed falling film

Published online by Cambridge University Press:  21 July 2014

Arghya Samanta
Arghya Samanta*
Affiliation:
Department of Chemical Engineering, Indian Institute of Science, Bangalore, Karnataka 560012, India
*
†Present address: Linné FLOW Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden. Email address for correspondence: arghyar@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

The study of a film falling down an inclined plane is revisited in the presence of imposed shear stress. Earlier studies regarding this topic (Smith, J. Fluid Mech., vol. 217, 1990, pp. 469–485; Wei, Phys. Fluids, vol. 17, 2005a, 012103), developed on the basis of a low Reynolds number, are extended up to moderate values of the Reynolds number. The mechanism of the primary instability is provided under the framework of a two-wave structure, which is normally a combination of kinematic and dynamic waves. In general, the primary instability appears when the kinematic wave speed exceeds the speed of dynamic waves. An equality criterion between their speeds yields the neutral stability condition. Similarly, it is revealed that the nonlinear travelling wave solutions also depend on the kinematic and dynamic wave speeds, and an equality criterion between the speeds leads to an analytical expression for the speed of a family of travelling waves as a function of the Froude number. This new analytical result is compared with numerical prediction, and an excellent agreement is achieved. Direct numerical simulations of the low-dimensional model have been performed in order to analyse the spatiotemporal behaviour of nonlinear waves by applying a constant shear stress in the upstream and downstream directions. It is noticed that the presence of imposed shear stress in the upstream (downstream) direction makes the evolution of spatially growing waves weaker (stronger).

JFM classification

Nonlinear Dynamical Systems: Low-dimensional models Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 753 , 25 August 2014 , pp. 131 - 149
Copyright
© 2014 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., Nakoryakov, V. Y. & Pokusaev, B. G. 1985 Wave formation on a vertical falling liquid film. AIChE J. 31, 1446–1460.CrossRefGoogle Scholar
Benjamin, T. B. 1957 Wave formation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–573.Google Scholar
Brauner, N. & Maron, D. M. 1982 Characteristics of inclined thin films, waviness and the associated mass transfer. Intl J. Heat Mass Transfer 25, 99–110.Google Scholar
Chang, H. C. 1994 Wave evolution on a falling film. Annu. Rev. Fluid Mech. 42, 1553–1568.Google Scholar
Chang, H. C., Demekhin, E. A. & Kalaidin, E. 1995 Interaction dynamics of a solitary waves on a falling film. J. Fluid Mech. 294, 123–154.Google Scholar
Chang, H. C., Demekhin, E. A. & Kalaidin, E. 1996 Simulation of noise driven wave dynamics on a falling film. AIChE J. 250, 433–480.Google Scholar
Chang, H. C., Demekhin, E. A. & Kopelvitch, D. I. 1993 Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech. 250, 433–480.Google Scholar
Chang, H. C., Demekhin, E. A. & Sparikin, S. S. 2002 Noise-driven wave transition on a vertically falling film. J. Fluid Mech. 462, 253–283.Google Scholar
Doedel, E. J., Champneys, A. R., Fairgrieve, T. F., Kuznetsov, Y. A., Sandstede, B. & Wang, X.-J.2007 Auto07: continuation and bifurcation software for ordinary differential equations. Tech Rep. Department of Computer Science, Concordia University, Montreal, Canada (available by FTP from ftp.cs.concordia.ca in directory pub/doedel/auto).Google Scholar
Finlayson, B. A. 1972 The Method of Weighted Residuals and Variational Principles, with Application in Fluid Mechanics, Heat and Mass Transfer. Academic Press.Google Scholar
Gao, D., Morley, N. & Dhir, V. 2003 Numerical simulation of wavy falling film flow using VOF method. J. Comput. Phys. 192, 624–642.CrossRefGoogle Scholar
Kalliadasis, S., Ruyer-Quil, C., Scheid, B. & Velarde, M. G. 2012 Falling Liquid Films, 1st edn p. 205. Springer.Google Scholar
Kapitza, P. L. 1948 Wave flow of thin layers of a viscous fluid: I. Free flow; and II. Fluid flow in the presence of continuous gas flow and heat transfer. InCollected Papers of P. L. Kapitza (1965) (ed. ter Haar, D.), pp. 662–669. Pergamon.Google Scholar
Kelley, R. E., Goussis, D. A., Lin, S. P. & Hsu, F. K. 1989 The mechanism for surface wave instability in film flow down an inclined plane. Phys. Fluids A 1, 819–828.Google Scholar
Liu, J. & Gollub, J. P. 1994 Solitary wave dynamics of film flows. Phys. Fluids 6, 1702–1712.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.Google Scholar
Malamataris, N. A., Vlachogiannis, M. & Bontozoglou, V. 2002 Solitary waves on inclined films: flow structure and binary interactions. Phys. Fluids 14, 1082–1094.Google Scholar
Mudunuri, R. R. & Balakotaiah, V. 2006 Solitary waves on thin falling films in the very low forcing frequency limit. AIChE J. 52, 3995–4003.Google Scholar
Nguyen, L. & Balakotaiah, V. 2000 Modelling and experimental studies of wave evolution on free falling viscous films. Phys. Fluids 12, 2236–2256.Google Scholar
Nosoko, T. & Miyara, A. 2004 The evolution and subsequent dynamics of waves on a vertically falling liquid film. Phys. Fluids 16, 1118–1126.Google Scholar
Ooshida, T. 1999 Surface equation of falling film flows with moderate Reynolds number and large but finite Weber number. Phys. Fluids 11, 3247–3269.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
Ramaswamy, B., Chippada, S. & Joo, S. W. 1996 A full-scale numerical study of interfacial instabilities in thin-film flows. J. Fluid Mech. 325, 163–194.CrossRefGoogle Scholar
Richtmeyer, R. D. & Morton, K. W. 1967 Difference Methods for Initial Value Problems. Interscience.Google Scholar
Ruyer-Quil, C. & Manneville, P. 1998 Modeling film flows down inclined planes. Eur. Phys. J. B 6, 277–292.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.Google Scholar
Ruyer-Quil, C. & Manneville, P. 2005 On the speed of solitary waves running down a vertical wall. J. Fluid Mech. 531, 181–190.Google Scholar
Salamon, T. R., Armstrong, R. C. & Brown, R. A. 1994 Traveling waves on vertical films: numerical analysis using the finite element method. Phys. Fluids 6, 2202–2220.CrossRefGoogle Scholar
Samanta, A. 2013a Effect of surfactant on two-layer channel flow. J. Fluid Mech. 735, 519–552.Google Scholar
Samanta, A. 2013b Shear wave instability for a electrified falling film. Phys. Rev. E 88, 053002.CrossRefGoogle Scholar
Samanta, A., Goyeau, B. & Ruyer-Quil, C. 2013 A falling film on a porous medium. J. Fluid Mech. 716, 414–444.Google Scholar
Samanta, A., Ruyer-Quil, C. & Goyeau, B. 2011 Falling film down a slippery inclined plane. J. Fluid Mech. 684, 353–383.Google Scholar
Shkadov, V. Y. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Izv. Akad. Nauk SSSR Mekh. Zhidk. Gaza 1, 43–51 (translation in 1970 Fluid Dyn. 2, 29–34).Google Scholar
Smith, M. K. 1982 The instability of sheared liquid layers. J. Fluid Mech. 121, 187–206.Google Scholar
Smith, M. K. 1990 The mechanism for the long-wave instability in thin liquid films. J. Fluid Mech. 217, 469–485.Google Scholar
Wei, H. H. 2005a Effect of surfactant on the long-wave instability of a shear-imposed liquid flow down an inclined plane. Phys. Fluids 17, 012103.Google Scholar
Wei, H. H. 2005b Stability of a viscoelastic falling film with surfactant subjected to an interfacial shear. Phys. Rev. E 71, 066306.Google Scholar
Wei, H. H. 2007 Role of base flows on surfactant-driven interfacial instabilities. Phys. Rev. E 75, 036306.Google Scholar
Weinstein, S. J. & Ruschak, K. J. 2004 Coating flows. Annu. Rev. Fluid Mech. 36, 29–53.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. p. 339. Wiley-Interscience.Google Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321–334.Google Scholar

Samanta supplementary movie

The spatial evolution of an inlet disturbance at large times with forcing frequency f=3Hz when A=0.01, \theta=6.4, Re=58, We=35 and \tau=1.

Download Samanta supplementary movie(Video)
Video 8 MB

Samanta supplementary movie

The spatial evolution of a noise-driven flow at large times when \theta=90^{\circ},

Download Samanta supplementary movie(Video)
Video 7.9 MB