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 nonuniform
temperature field. We first determine the associated base flow by solving the long-time-scale
stretching flow of a flat sheet as a function of two dimensionless parameters: the normalized
stretching velocity α and a dimensionless width of the heating zone β. 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 that 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 β, 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.