# Wrinkling instability of an inhomogeneously stretched viscous sheet

## TL;DR

This paper analyzes the stability of an inhomogeneously stretched viscous sheet, identifying conditions for wrinkling instability and how to suppress it, with applications to ultrathin glass sheet production.

## Contribution

It develops a theoretical framework for predicting wrinkling in stretched viscous sheets under non-uniform heating, including stability criteria and suppression strategies.

## Key findings

- Instability occurs upstream and downstream of the heating zone.
- Surface tension can suppress wrinkling when sufficiently large.
- Transition from stationary to oscillatory modes with increasing heating zone width.

## Abstract

Motivated by the redrawing of hot glass into thin sheets, we investigate the shape and stability of a thin viscous sheet that is inhomogeneously stretched in an imposed non-uniform temperature field. We first determine the associated base flow by solving the long-timescale stretching flow of a flat sheet as a function of two dimensionless parameters: the normalized stretching velocity $\alpha$, and a dimensionless width of the heating zone $\beta$. This allows us to determine the conditions for the onset of an out-of-plane wrinkling instability stated in terms of an eigenvalue problem for a linear partial differential equation governing the displacement of the midsurface of the sheet. We show the sheet can become unstable in two regions that are upstream and downstream of the heating zone where the minimum in-plane stress is negative. This yields the shape and growth rates of the most unstable buckling mode in both regions for various values of the stretching velocity and heating zone width. A transition from stationary to oscillatory unstable modes is found in the upstream region with increasing $\beta$ while the downstream region is always stationary. We 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. Finally, 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.