Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-19T12:20:28.343Z Has data issue: false hasContentIssue false
  • English
  • Français

Free-surface thin-film flows over topography: influence of inertia and viscoelasticity

Published online by Cambridge University Press:  26 April 2007

SERGEY SAPRYKIN ,
RUDY J. KOOPMANS  and
SERAFIM KALLIADASIS
SERGEY SAPRYKIN
Affiliation:
Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, UK
RUDY J. KOOPMANS
Affiliation:
Core R&D, Dow Benelux BV, 4530 AA Terneuzen, The Netherlands
SERAFIM KALLIADASIS*
Affiliation:
Department of Chemical Engineering, Imperial College London, London, SW7 2AZ, UK
*
†Author to whom correspondence should be addressed: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider viscoelastic flows over topography in the presence of inertia. Such flows are modelled by an integral-boundary-layer approximation of the equations of motion and wall/free-surface boundary conditions. Steady states for flows over a step-down in topography are characterized by a capillary ridge immediately before the entrance to the step. A similar capillary ridge has also been observed for non-inertial Newtonian flows over topography. The height of the ridge is found to be a monotonically decreasing function of the Deborah number. Further, we examine the interaction between capillary ridges and excited non-equilibrium inertia/viscoelasticity-driven solitary pulses. We demonstrate that ridges have a profound influence on the drainage dynamics of such pulses: they accelerate the drainage process so that once the pulses pass the topographical feature they become equilibrium ones and are no longer excited.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 578 , 10 May 2007 , pp. 271 - 293
Copyright
Copyright © Cambridge University Press 2007

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

REFRENCES

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.CrossRefGoogle Scholar
Bielarz, C. & Kalliadasis, S. 2003 Time-dependent free-surface thin film flows over topography. Phys. Fluids 15, 2512–2524.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids. Volume 1: Fluid Mechanics. Wiley.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. A. 2002 Complex Wave Dynamics on Thin Films. Elsevier.Google Scholar
Chang, H.-C., Demekhin, E. A. & Kalaidin, E. 1998 Generation and suppression of radiation by solitary pulses. SIAM J. Appl. Maths 58, 1246–1277.Google Scholar
Chang, H.-C., Demekhin, E. A. & Saprykin, S. 2002 Noise-driven wave transitions on a vertically falling film. J. Fluid Mech. 462, 255–283.CrossRefGoogle Scholar
Davis, J. M. & Troian, S. M. 2005 Generalized linear stability of noninertial coating flows over topographical features. Phys. Fluids 17, 072103.CrossRefGoogle Scholar
Fernandez-Parent, C. Lammers, J. H. & Decré, M. M. J. 1998 Flow of a gravity driven thin liquid film over one-dimensional topographies. Philips Res. Unclassified Rep. UR 823/98.Google Scholar
Fraysse, N. & Homsy, G. M. 1994 An experimental study of rivulet instabilities in centrifugal spin coating of viscous Newtonian and non-Newtonian fluids. Phys. Fluids 6, 1491–1504.CrossRefGoogle Scholar
Gramlich, C. M. Kalliadasis, S. Homsy, G. M. & Messer, C. 2002 Optimal levelling of flow over one-dimensional topography by Marangoni stresses. Phys. Fluids 14, 1841–1850.CrossRefGoogle Scholar
Kalliadasis, S. & Homsy, G. M. 2001 Stability of free-surface thin-film flows over topography. J. Fluid Mech. 448, 387–410.CrossRefGoogle Scholar
Kalliadasis, S., Bielarz, C. & Homsy, G. M. 2000 Steady free-surface thin film flows over topography. Phys. Fluids 12, 1889–1898.CrossRefGoogle Scholar
Kalliadasis, S., Kiyashko, A. & Demekhin, E. A. 2003 a Marangoni instability of a thin liquid film heated from below by a local heat source. J. Fluid Mech. 475, 377–408.CrossRefGoogle Scholar
Kalliadasis, S., Demekhin, E. A., Ruyer-Quil, C. & Velarde, M. G. 2003 b Thermocapillary instability and wave formation on a film falling down a uniformly heated plane. J. Fluid Mech. 492, 303–338.CrossRefGoogle Scholar
Kapitza, P. L. 1948 Wave flow of thin layers of a viscous fluid. I. Free flow. Sov. Phys., J. Exp. Theor. Phys. 18, 3–18.Google Scholar
Khayat, R. E. 2000 Transient two-dimensional coating flow of a viscoelastic fluid film on a substrate of arbitrary shape. J. Non-Newtonian Fluid Mech. 95, 199–233.CrossRefGoogle Scholar
Mazouchi, A. & Homsy, G. M. 2001 Free surface Stokes flow over topography. Phys. Fluids 13, 2751–2761.CrossRefGoogle Scholar
Messé, S. & Decré, M. J. 1997 Experimental study of a gravity driven water film flowing down inclined plates with different patterns. Philips Res. Uncliasified Rep. UR 030/97.Google Scholar
Oron, A. & Gottlieb, O. 2002 Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 2622–2636.CrossRefGoogle Scholar
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.CrossRefGoogle Scholar
Rosenau, P., Oron, A. & Hyman, J. M. 1992 Bounded and unbounded patters of the Benney equation. Phys. Fluids A 4, 1102–1104.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357–369.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results of the modeling of flows down inclined planes by weighted residual approximations. Phys. Fluids 14, 170–183.CrossRefGoogle Scholar
Ruyer-Quil, C. Scheid, B. Kalliadasis, S. Velarde, M. G. & Zeytounian, R. Kh. 2005 Thermocapillary long waves in a liquid film flow. Part 1. Low-dimensional formulation. J. Fluid Mech. 538, 199–222.CrossRefGoogle Scholar
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, 303–335.CrossRefGoogle Scholar
Scheid, B. Ruyer-Quil, C. Kalliadasis, S. Velarde, M. G. & Zeytounian, R. Kh. 2005 Thermocapillary long waves in a liquid film flow. Part 2. Linear stability and nonlinear waves. J. Fluid Mech. 538, 223–244.CrossRefGoogle Scholar
Shkadov, V. Ya 1967 Wave models in the flow of a thin layer of a viscous liquid under the action of gravity. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza 1, 43–50.Google Scholar
Shkadov, V. Ya 1968 Theory of wave flow of a thin layer of a viscous liquid. Izv. Akad. Nauk SSSR, Mekh. Zhid. i Gaza 2, 20–25.Google Scholar
Spaid, M. A. & Homsy, G. M. 1994 Viscoleastic free surface flows: spin coating and dynamic contact lines. J. Non-Newtonian Fluid Mech. 55, 249–281.CrossRefGoogle Scholar
Spaid, M. A. & Homsy, G. M. 1996 Stability of Newtonian and viscoelastic dynamic contact lines. Phys. Fluids 8, 460–478.CrossRefGoogle Scholar
Stillwagon, L. E. & Larson, R. G. 1988 Fundamentals of topographic substrate leveling. J. Appl. Phys. 63, 5251–5258.CrossRefGoogle Scholar
Stillwagon, L. E. & Larson, R. G. 1990 Leveling of thin films over uneven substrate during spin coating. Phys. Fluids 2, 1937–1944.CrossRefGoogle Scholar
Zhang, Y. L., Matar, O. K. & Craster, R. V. 2002 Surfactant spreading on a thin weakly viscoelastic film. J. Non-Newtonian Fluid Mech. 105, 53–78.CrossRefGoogle Scholar