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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: rene.ledesma-alonso@espci.fr
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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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References

Alleborn, N. & Raszillier, H. 2004 Local perturbation of thin film flow. Arch. Appl. Mech. 73, 734–751.Google Scholar
Alleborn, N., Sharma, A. & Delgado, A. 2007 Probing of thin slipping films by persistent external disturbances. Can. J. Chem. Engng 85, 586–597.CrossRefGoogle Scholar
Baumchen, O., Benzaquen, M., Salez, T., McGraw, J. D., Backholm, M., Fowler, P., Raphael, E. & Dalnoki-Veress, K. 2013 Relaxation and intermediate asymptotics of a rectangular trench in a viscous film. Phys. Rev. E 88, 035001, 1–5.Google Scholar
Benzaquen, M., Chevy, F. & Raphael, E. 2011 Wave resistance for capillary gravity waves: finite-size effects. Europhys. Lett. 96, 34003, 1–5.Google Scholar
Benzaquen, M., Darmon, A. & Raphael, E. 2014a Wake pattern and wave resistance for anisotropic moving disturbances. Phys. Fluids. 26, 092106, 1–7.Google Scholar
Benzaquen, M., Fowler, P., Jubin, L., Salez, T., Dalnoki-Veress, K. & Raphael, E. 2014b Approach to universal self-similar attractor for the leveling of thin liquid films. Soft Matt. 10, 8608–8614.CrossRefGoogle ScholarPubMed
Blossey, R. 2012 Thin Liquid Films. Springer.CrossRefGoogle Scholar
Burghelea, T. & Steinberg, V. 2002 Wave drag due to generation of capillary-gravity surface waves. Phys. Rev. E 66, 051204, 1–13.Google Scholar
Campbell, C. S. 1989 Self-lubrication for long runout landslides. J. Geol. 97 (6), 653–665.Google Scholar
Carusotto, I. & Rousseaux, G. 2013 The Cerenkov effect revisited: from swimming ducks to zero modes in gravitational analogues. In Analogue Gravity Phenomenology, Lecture Notes in Physics, vol. 870, pp. 109–144. Springer.Google Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.Google Scholar
Darmon, A., Benzaquen, M. & Raphael, E. 2014 Kelvin wake pattern at large Froude numbers. J. Fluid Mech. 738 (R3), 1–8.Google Scholar
Decre, M. M. J. & Baret, J.-C. 2003 Gravity-driven flows of viscous liquids over two-dimensional topographies. J. Fluid Mech. 487, 147–166.Google Scholar
Glenne, B. 1987 Sliding friction and boundary lubrication of snow. J. Tribol. 109 (4), 614–617.Google Scholar
Havelock, T. H. 1918 Wave resistance: some cases of three-dimensional fluid motion. Proc. R. Soc. Lond. A 95, 354–365.Google Scholar
Huppert, H. E. 1982 The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
Huppert, H. E. & Simpson, J. E. 1980 The slumping of gravity currents. J. Fluid Mech. 99 (4), 785–799.CrossRefGoogle Scholar
Kelvin, Lord 1887 On ship waves. Proc. Inst. Mech. Engrs 38, 409–434.Google Scholar
Kistler, S. F. & Schweizer, P. M. 1997 Liquid Film Coating. Scientific Principles and Their Technological Implications. Chapman & Hall.Google Scholar
Kondic, L. 2003 Instabilities in gravity driven flow of thin fluid films. SIAM Rev. 45 (1), 95–115.CrossRefGoogle Scholar
Ledesma-Alonso, R., Legendre, D. & Tordjeman, P. 2013 AFM tip effect on a thin liquid film. Langmuir 29, 7749–7757.Google Scholar
Ledesma-Alonso, R., Tordjeman, Ph. & Legendre, D. 2014 Dynamics of a thin liquid film interacting with an oscillating nano-probe. Soft Matt. 10 (39), 7736–7752.Google Scholar
McGraw, J. D., Salez, T., Baumchen, O., Raphael, E. & Dalnoki-Veress, K. 2012 Self-similarity and energy dissipation in stepped polymer films. Phys. Rev. Lett. 109, 128303, 1–5.Google Scholar
Orchard, S. E. 1961 On surface leveling in viscous liquids and gels. Appl. Sci. Res. 11 (4), 451–464.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin films. Rev. Mod. Phys. 69 (3), 931–980.Google Scholar
Rabaud, M. & Moisy, F. 2013 Ship wakes: Kelvin or Mach angle? Phys. Rev. Lett. 110, 214503, 1–5.Google Scholar
Raphael, E. & de Gennes, P.-G. 1996 Capillary gravity waves caused by a moving disturbance: wave resistance. Phys. Rev. E 53, 3448–3455.Google Scholar
Richard, D. & Raphael, E. 1999 Capillary-gravity waves: the effect of viscosity on the wave resistance. Europhys. Lett. 48 (1), 49–52.Google Scholar
Wedolowski, K. & Napiorkowskia, M. 2013 Capillary-gravity waves on a liquid film of arbitrary depth: analysis of the wave resistance. Phys. Rev. E 88, 043014, 1–13.Google Scholar
Wedolowski, K. & Napiorkowskia, M. 2015 Dynamics of a liquid film of arbitrary thickness perturbed by a nano-object. Soft Matt. 11, 2639–2654.CrossRefGoogle ScholarPubMed