Thin Film Dynamics
Thin films1 possess two radically distinct typical scales associated with their transverse and their longitudinal dimensions. Two distinct dynamics are thus associated to these length scales: transverse or longitudinal dispersive waves linked to the film
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J.M. Chomaz Ecole Polytechnique, Palaiseau, France M. Costa University of Naples "Federico II", Naples, Italy
Abstract Thin films 1 possess two radically distinct typical scales associated with their transverse and their longitudinal dimensions. Two distinct dynamics are thus associated to these length scales: transverse or longitudinal dispersive waves linked to the film thickness, and longitudinal quasi-two-dimensional (2D) motion scaling on the film length. The physics of both waves and 2D motion are studied here. The response of a film to a localized impulse is computed, and the behaviour is interpreted in the light of group-velocity notions. When air is blown on the film, the waves turn into instability modes, as demonstrated by a simple pressure argument in the limit of small density ratios. The different behavior observed in the case of a water jet and in the case of air blowing on a film is explained by introducing the equivalent of group velocity for instability waves, which naturally leads to discriminate between the absolute and the convective type of instability. In the long-wave limit, waves become similar to the elastic waves propagating on a stretched membrane. In recent experiments, Couder [7] and Gharib [13] use soap films as a two-dimensional fluid. In the present paper, we show that the necessary condition for the film to comply to Navier-Stokes equations is that the typical flow velocity be small compared to the Marangoni elastic wave velocity.
1 even if traditionally the term film refers to soap film and, for pure water, the term sheet is more generally used, we are going to use indifferently both, since soap films are now produced by a nozzle through an expansion as water sheets and since both are subject to similar instabilities and wavy motions.
H. C. Kuhlmann et al. (eds.), Free Surface Flows © Springer-Verlag Wien 1998
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J.M. Chomaz and M. Costa
Introduction
Water films are a matter of study since the prior papers of Squire [27] and Taylor [28] due to their natural beauty, their theoretical interest and their variety of applications ranging from atomization and sprays in combustion to curtain coating processes. Water films sustain waves originating from the interaction of the capillary waves developing on each of its interfaces. A large part of thin film dynamics can be understood from this dispersive-wave point of view and from the over-simplified stretched solid membrane model. Inertial effects in the surrounding fluid (taken into account by the Bernoulli equation) turn to be destabilizing when the fluid moves faster than the wave. This subtle destabilizing effect of the surrounding fluid (which may also be demonstrated on the solid membrane model) eventually leads to the breaking of the film. The above traditional applications of films refer to transverse motions in the film. When soap is added to water, the dependence of surface tension with the superficial soap concentration makes the film elastic and therefore reduce its tendency to break. In this case, large scale motion
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