  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
stabilised 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 $\gamma/(\mu d)$ independent of buoyancy, where
2d is the plate thickness, $\gamma$ is the average surface tension between the
upper and lower surfaces and $\mu$ is the fluid viscosity. For thicker plates
stabilised 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 centre-line, spans the width of the plate with crests and
troughs aligned at approximately $45^\circ$ to the walls and closely resembles
elastic shear modes. The thickest plates that can buckle have an aspect ratio
(thickness/width) approximately $\EC$ and are stabilised 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.
