Nonlinear instability of a thin film flowing down a smoothly deformed surface

Here, the nonlinear flow instability of a thin film flowing down an inclined wall with smooth deformations is investigated. A nonlinear evolution equation is obtained under the small wavenumber approximation. This equation describes the film free surface deformation including the effects of inertia, viscosity, surface tension, and deformation of the wall. The equation has a forcing term that corresponds to periodic time-dependent perturbations hitting on the free surface. These time-dependent perturbations are tested against the presence of the free surface waves due to wall deformations. In particular, a wall sinusoidal profile is investigated. It is found that when the wall wavelength is small enough, the valleys of the surface response to the wall profile become very deep, showing a kind of resonance, which has an important stabilizing influence in the evolution of the superposed time-dependent perturbations.

[1]  Yu. Ya. Trifonov,et al.  Viscous liquid film flows over a periodic surface , 1999 .

[2]  R. Khayat,et al.  Influence of inertia, gravity, and substrate topography on the two-dimensional transient coating flow of a thin Newtonian fluid film , 2001 .

[3]  Y. Trifonov.  Viscous Film Flow down Corrugated Surfaces , 2004 .

[4]  David Quéré,et al.  FLUID COATING ON A FIBER , 1999 .

[5]  V. Bontozoglou,et al.  Experiments on laminar film flow along a periodic wall , 2002, Journal of Fluid Mechanics.

[6]  F. Busse,et al.  Instability of a thin film flowing on a rotating horizontal or inclined plane. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  J. Baret,et al.  Gravity-driven flows of viscous liquids over two-dimensional topographies , 2003, Journal of Fluid Mechanics.

[8]  C. Wang,et al.  Thin film flowing down a curved surface , 1984 .

[9]  N. Alleborn,et al.  Local perturbation of thin film flow , 2004 .

[10]  S. H. Davis,et al.  Instabilities of three-dimensional viscous falling films , 1992, Journal of Fluid Mechanics.

[11]  Geoffrey F. Hewitt,et al.  Thin film flow over structured packings at moderate Reynolds numbers , 2005 .

[12]  Chao-Yang Wang,et al.  Liquid film flowing slowly down a wavy incline , 1981 .

[13]  G. Homsy,et al.  Steady free-surface thin film flows over topography , 2000 .

[14]  N. Aksel,et al.  Comparison of different theoretical approaches to experiments on film flow down an inclined wavy channel , 2002 .

[15]  Jeffrey M. Davis,et al.  Generalized linear stability of noninertial coating flows over topographical features , 2005 .

[16]  R. L. Cerro,et al.  Flow of a thin fiom over a periodic surface , 1993 .

[17]  R. L. Cerro,et al.  Experimental characterization of viscous film flows over complex surfaces , 1992 .

[18]  J. M. Floryan,et al.  Instabilities of a liquid film flowing down a slightly inclined plane , 1987 .

[19]  D. J. Benney.  Long Waves on Liquid Films , 1966 .

[20]  M. Meyer,et al.  Study of a laminar falling film flowing over a wavy wall column: Part I. Numerical investigation of the flow pattern and the coupled heat and mass transfer , 2001 .

[21]  Hirohumi Tougou,et al.  Long Waves on a Film Flow of a Viscous Fluid down an Inclined Uneven Wall , 1978 .

[22]  B. Gjevik.  Occurrence of Finite‐Amplitude Surface Waves on Falling Liquid Films , 1970 .

[23]  G. J. Roskes.  Three‐Dimensional Long Waves on a Liquid Film , 1970 .

[24]  G. Sivashinsky,et al.  On Irregular Wavy Flow of a Liquid Film Down a Vertical Plane , 1980 .

[25]  S. Bankoff,et al.  Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers , 1991, Journal of Fluid Mechanics.

[26]  Vasilis Bontozoglou,et al.  Laminar film flow down a wavy incline , 1997 .

[27]  S. P. Lin.  Instability of a Liquid Film Flowing down an Inclined Plane , 1967 .

[28]  M. Meyer,et al.  Study of a laminar falling film flowing over a wavy wall column: Part II. Experimental validation of hydrodynamic model , 2001 .

[29]  R. Usha,et al.  Long waves on a viscoelastic film flow down a wavy incline , 2004 .

[30]  A. Ovcharova.  Controlling the Free‐Surface Profile of Film Flow over Complex Topography , 2004 .

[31]  Chia-Shun Yih,et al.  Stability of Liquid Flow down an Inclined Plane , 1963 .

[32]  Nuri Aksel,et al.  Instability of a liquid film flowing down an inclined wavy plane , 2003 .

[33]  S. Kalliadasis,et al.  Time-dependent free-surface thin film flows over topography , 2003 .

[34]  V. Bontozoglou,et al.  Regular Article: Computer Aided Analysis of Viscous Film Flow along an Inclined Wavy Wall , 1999 .

[35]  N. Aksel,et al.  Creeping films with vortices over strongly undulated bottoms , 2004 .

[36]  S. Bankoff,et al.  Nonlinear instability of a fluid layer flowing down a vertical wallunder imposed time-periodic perturbations , 1997 .

[37]  S. V. Alekseenko,et al.  Wave formation on a vertical falling liquid film , 1985 .

[38]  C. Pozrikidis,et al.  The flow of a liquid film along a periodic wall , 1988, Journal of Fluid Mechanics.