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 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  and Gharib  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.
KeywordsDispersion Relation Phase Velocity Weber Number Liquid Sheet Soap Film
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