Buckling of a thin-layer Couette flow

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 γ/(µd) independent of buoyancy, where 2d is the plate thickness, γ is the average surface tension between the upper and lower surfaces and µ 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 • to the walls and closely resembles elastic shear modes. The thickest plates that can buckle have an aspect ratio (thickness/width) approximately 0.6 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.

The viscous curtain: General formulation and finite-element solution for the stability of flowing viscous sheets

“…Benjamin and Mullin (1988) argue that thin shell models are inapplicable for this particular problem, and analyze the instability based on the equations for 3D viscous flows. Slim et al (2012) simplify the geometry by addressing an infinite viscous strip sheared by lateral walls; they modify the thin viscous shell model to include an advection term, and argue that the modified model captures the wrinkling instabilities more accurately. Bhattacharya et al (2013) analyze again the stability of a sheared annular floating viscous sheet, based on a thin shell model; while they obtain good agreement with their experiments in some range of the parameters, they also point out the shortcomings of the shell model which for very thin sheets predicts very short wavelength instabilities, and a vanishingly small buckling threshold.…”

All bent out of shape: buckling of sheared fluid layers

“…As reported in the following paper, Slim, Teichman & Mahadevan (2012) study a simpler model in which the plate is infinitely long, as shown in figure 1(b). The motion of the bounding walls at speeds ±U generates a linear shear velocity profile (brown arrows) that compresses material elements in a direction inclined at 45 • to the walls (black arrows), leading to buckling.…”

“…Benjamin & Mullin (1988) analysed the buckling of a laterally infinite viscous plate with uniform background shear using both a thin-plate model and a complete analytical solution of the three-dimensional Stokes equations. Slim et al (2012) improve on their work in two ways. First, they include the influence of the bounding walls, which constrain the form the buckling can take.…”

Vortex-induced buckling of a viscous drop impacting a pool

“…Theories have been formulated in great detail [16,17]. Buckles can appear during the rupture of bubbles at the surface of viscous pools [18,19] or thin liquid films subjected to shear in a Taylor-Couette-type device [20][21][22]. Buckles also appear during indentation of floating elastic films [23] or at the edges of partially wetting lenses [24].…”