Abstract
The present paper is devoted to a consistent asymptotic derivation of model equations for two-dimensional nonlinear (2DNL) waves on a viscous dilatable liquid film. Various approximate equations describing the NL evolution in time - space of the thickness of the film are obtained . More precisely, for small amplitudes in the long wave approximation, we consider three cases : high Reynolds number boundary layer regime, moderate Reynolds number of Poiseuille type one, and low Reynolds number second Poiseuille type regime. These various situations arise from the smallness of the thin film aspect ratio parameter E and are distinguished from each other by selecting, as far as order of magnitude are concerned, some dimensionless parameters with respect to E . They are Reynolds, Fronde, Marangoni, Weber, Grashof numbers as well as a Bict one which is assumed, throughout, to be of order unity. For the high Reynolds number case we obtain an equation for the thickness h(t,x) of the film, which is similar to equation previously derived by Oron and Rosenau 1992. From this last evolution equation for h(t,x) a new limiting process, for small amplitude (long) waves, leads to a modified (by Boussinesq effects ) Kuramoto- Sivashinsky (KS) equation. The two other regimes do not lead to a unique consistent equation for h(t,x) unless we focus on small amplitude (long) waves; one finds a KS or a KS with Korteweg-de-Vries (KdV) dispersive term equations. The role of buoyancy is quite different in the various regimes considered; for the low Reynolds number regime the buoyancy coupling effect is “effectively” significant in the KS - KdV model equation.
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References
Benney, D.J. (1966): Long — Waves on Liquid Films, J. Math. and Phys. 45; 150–155
Chang, H.-Ch. (1994): Wave Evolution on a Falling film, Ann. Rev. Fluid Mech. 26; 103–136
Garazo, A.N. and Velarde, M.G. (1991): Dissipative KdV Description of Marangoni-Bénard Oscillatory Convection, Phys. Fluids, A3 (10); 2295–2300
Lin, S.P. and Wang, C.Y. (1985): “Modeling wavy film flows”, in Encyclopedia of Fluid Mechanics, Gulf, Houston, Vol. 1; p. 93
Oron, A. and Rosenau, Ph. (1992): Formation of Patterns Induced by Thermocapillarity. J.Phys. II France 2; 131–146
Prokopiuo, Th., Cheng, M. and Chang, H.-Ch. (1991): Long waves on inclined films at High Reynolds number, J. Fluid Mech., 222; 665–691
Shkadov, V. Ya. (1967): Wave conditions in the flow of thin layer of a viscous liquid under the action of gravity, Izv. Akad. Nauk SSSR, Mekh. Zhidk. i Gaza, 1; 43–50
Trifonov, Yu. Ya. and Tsvelodub, O. Yu. (1991): J. Fluid Mech., 229; 531–554
Zeytounian, R. Kh. (1993): Nonlinear Dynamics of Waves on the Free Surface of a Falling Liquid Film Revisited. Preprint
Zeytounian, R. Kh. (1994): Equations d'évolution pour les films dilatables, visqueux, minces. Projet de Note aux CRAS
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© 1994 Springer-Verlag
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Zeytounian, R.K. (1994). Long-waves on thin viscous liquid film derivation of model equations. In: Bois, PA., Dériat, E., Gatignol, R., Rigolot, A. (eds) Asymptotic Modelling in Fluid Mechanics. Lecture Notes in Physics, vol 442. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-59414-0_64
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DOI: https://doi.org/10.1007/3-540-59414-0_64
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