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Linear and nonlinear stability of floating viscous sheets

Published online by Cambridge University Press:  19 August 2011

G. Pfingstag ,
B. Audoly  and
A. Boudaoud
G. Pfingstag*
Affiliation:
Saint-Gobain Recherche, 39 Quai Lucien Lefranc, BP 135, 93303 Aubervilliers CEDEX, France CNRS and Université Pierre et Marie Curie, Univ Paris 06, UMR 7190, Institut Jean le Rond d’Alembert, Paris, France Laboratoire de Physique Statistique, École Normale Supérieure, Université Pierre et Marie Curie, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France
B. Audoly
Affiliation:
CNRS and Université Pierre et Marie Curie, Univ Paris 06, UMR 7190, Institut Jean le Rond d’Alembert, Paris, France
A. Boudaoud
Affiliation:
Laboratoire de Physique Statistique, École Normale Supérieure, Université Pierre et Marie Curie, Université Paris Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France
*
†Email address for correspondence: [email protected]
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Abstract

We study the stability of a thin, Newtonian viscous sheet floating on a bath of denser fluid. We first derive a general set of equations governing the evolution of a nearly flat sheet, accounting for geometrical nonlinearities associated with moderate rotations. We extend two classical models by considering arbitrary external body and surface forces; these two models follow from different scaling assumptions, and are derived in a unified way. The equations capture two modes of deformation, namely viscous bending and stretching, and describe the evolution of thickness, mid-surface and in-plane velocity as functions of two-dimensional coordinates. These general equations are applied to a floating viscous sheet, considering gravity, buoyancy and surface tension. We investigate the stability of the flat configuration when subjected to arbitrary in-plane strain. Two unstable modes can be found in the presence of compression. The first one combines undulations of the centre-surface and modulations of the thickness, with a wavevector perpendicular to the direction of maximum applied compression. The second one is a buckling mode; it is purely undulatory and has a wavevector along the direction of maximum compression. A nonlinear analysis yields the long-time evolution of the undulatory mode.

JFM classification

Instability: Nonlinear instability Low-Reynolds-number flows: Slender-body theory Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 683 , 25 September 2011 , pp. 112 - 148
Copyright
Copyright © Cambridge University Press 2011

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