(369f) Wrinkling Instabilities in Thin Inhomogenously Stretched Viscous Sheets
Here, we present a mathematical framework to characterizeÂ the shape and stability of a thin viscous fluid sheet that is inhomogeneously stretched in an imposed non-uniform temperature field. We determine that the sheet can become unstable in two regions that are upstream and downstream of the heating zone, and where the minimum in-plane stress is negative. We characterize the shape and growth rates of the most unstable modes in both regions for various operating conditions, and show that the behavior of the instability changes from a stationary mode to a pair of unstable traveling modes upon varying the heating zone width, corresponding to a transition from a stationary bifurcation to a Hopf bifurcation.
Finally, we investigate the effect of surface tension and show that the wrinkling instability can be entirely suppressed when the surface tension is large enough relative to the magnitude of the in-plane stress. We present an operating diagram that indicates regions of the parameter space that result in a required outlet sheet thickness upon stretching, while simultaneously minimizing or suppressing the out-of-plane buckling; a result that is relevant for the glass redraw method used to create ultrathin glass sheets.