Buckling instability of a thin-layer rectilinear Couette flow

TL;DR

This paper investigates the buckling instability of thin viscous sheets under shear, deriving conditions for buckling onset and analyzing mode characteristics using asymptotic and full Stokes models.

Contribution

It introduces a viscous plate model that accurately predicts buckling onset in thin sheets and extends to describe traveling wave modes with modifications.

Findings

Buckling occurs at a shear rate independent of buoyancy for vanishingly thin plates.

Most unstable modes have moderate wavelength and resemble elastic shear modes.

The viscous plate model effectively predicts buckling onset and mode shapes for thin plates.

Abstract

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.