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.
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