a1 Department of Physics and School of Engineering and Applied Sciences, Harvard University, 29 Oxford Street, Cambridge, MA 02138, USA
a2 Institute for Defense Analyses, Virginia, USA
We analyse the buckling stability of a thin, viscous sheet when subject to simple shear, providing conditions for the onset of the dominant out-of-plane modes using two models: (i) an asymptotic theory for the dynamics of a viscous plate, and (ii) the full Stokes equations. In either case, the plate is stabilized by a combination of viscous resistance, surface tension and buoyancy relative to an underlying denser fluid. In the limit of vanishing thickness, plates buckle at a shear rate independent of buoyancy, where is the plate thickness, is the average surface tension between the upper and lower surfaces, and is the fluid viscosity. For thicker plates stabilized by an equal surface tension at the upper and lower surfaces, at and above onset, the most unstable mode has moderate wavelength, is stationary in the frame of the centreline, spans the width of the plate with crests and troughs aligned at approximately to the walls, and closely resembles elastic shear modes. The thickest plates that can buckle have an aspect ratio (thickness/width) of approximately 0.6 and are stabilized only by internal viscous resistance. We show that the viscous plate model can only accurately describe the onset of buckling for vanishingly thin plates but provides an excellent description of the most unstable mode above onset. Finally, we show that, by modifying the plate model to incorporate advection and make the model material-frame-invariant, it is possible to extend its predictive power to describe relatively short, travelling waves.
(Received March 03 2011)
(Reviewed October 02 2011)
(Accepted October 02 2011)
(Online publication November 24 2012)