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Wake and wave resistance on viscous thin films

Published online by Cambridge University Press:  07 March 2016

René Ledesma-Alonso ,
Michael Benzaquen ,
Thomas Salez  and
Elie Raphaël
René Ledesma-Alonso*
Affiliation:
Laboratoire de Physico-Chimie Théorique, UMR CNRS 7083 Gulliver, ESPCI ParisTech, PSL Research University, 10 Rue Vauquelin, 75005 Paris, France
Michael Benzaquen
Affiliation:
Laboratoire de Physico-Chimie Théorique, UMR CNRS 7083 Gulliver, ESPCI ParisTech, PSL Research University, 10 Rue Vauquelin, 75005 Paris, France Capital Fund Management, 23 Rue de l’Université, 75007 Paris, France
Thomas Salez
Affiliation:
Laboratoire de Physico-Chimie Théorique, UMR CNRS 7083 Gulliver, ESPCI ParisTech, PSL Research University, 10 Rue Vauquelin, 75005 Paris, France
Elie Raphaël
Affiliation:
Laboratoire de Physico-Chimie Théorique, UMR CNRS 7083 Gulliver, ESPCI ParisTech, PSL Research University, 10 Rue Vauquelin, 75005 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

The effect of an external pressure disturbance, being displaced with a constant speed along the free surface of a viscous thin film, is studied theoretically in the lubrication approximation in one- and two-dimensional geometries. In the comoving frame, the imposed pressure field creates a stationary deformation of the interface – a wake – that spatially vanishes in the far region. The shape of the wake and the way it vanishes depend on both the speed and size of the external source and the properties of the film. The wave resistance, namely the force that has to be externally furnished in order to maintain the wake, is analysed in detail. For finite-size pressure disturbances, it increases with the speed, up to a certain transition value, above which a monotonic decrease occurs. The role of the horizontal extent of the pressure field is studied as well, revealing that for a smaller disturbance the latter transition occurs at a higher speed. Eventually, for a Dirac pressure source, the wave resistance either saturates for a one-dimensional geometry, or diverges for a two-dimensional geometry.

JFM classification

Low-Reynolds-number flows: Lubrication theory Interfacial Flows (free surface): Thin films Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 792 , 10 April 2016 , pp. 829 - 849
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Copyright
© 2016 Cambridge University Press 

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