Through asymptotic analysis and physical modeling, we obtain a quantitative description of the flow and structure of the glass fiber as it is being pulled. We examine in particular the issue of whether initially planar cross sections remain planar, and find that they do so approximately, but not exactly, even in the absence of such factors as gravity, surface tension, and inertia. We also develop a quantitative theory for the structure of the air flow near the fiber, which is an important ingredient in determining how fast the fiber cools. In particular, we develop a description for the structure of the boundary layer near a fiber which has accelerating surface velocity and shrinking radius which is comparable or thin compared with the thickness of the boundary layer. One novel feature is that the acceleration of the fiber surface velocity leads to a compression of the boundary layer as it evolves downstream.