  Recent experimental observations have demonstrated interesting instability
phenomenon during thermal drawing of microstructured glass/polymer fibers, and
these observations motivate us to examine surface-tension-driven instabilities
in concentric cylindrical shells of viscous fluids. In this paper, we focus on
a single instability mechanism: classical capillary instabilities in the form
of radial fluctuations, solving the full Navier--Stokes equations numerically.
In equal-viscosity cases where an analytical linear theory is available, we
compare to the full numerical solution and delineate the regime in which the
linear theory is valid. We also consider unequal-viscosity situations (similar
to experiments) in which there is no published linear theory, and explain the
numerical results with a simple asymptotic analysis. These results are then
applied to experimental thermal drawing systems. We show that the observed
instabilities are consistent with radial-fluctuation analysis, but cannot be
predicted by radial fluctuations alone---an additional mechanism is required.
We show how radial fluctuations alone, however, can be used to analyze various
candidate material systems for thermal drawing, clearly ruling out some
possibilities while suggesting others that have not yet been considered in
experiments.
