  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.
