Summary: Super-strength, lightweight materials used in bullet-proof vests, high-performance cables and tires, and stealth airplanes are built from liquid crystalline polymer (LCP) fibers. The remarkable strength properties are dominated by molecular alignment achieved as a result of the complex interactions at play in fiber processes. The fiber manufacturing process begins with a high temperature liquid phase of rigid rod macromolecules, whose orientation couples to the strong elongational free surface flow. The flow exits at a prescribed radius and velocity (\(v_0\)), tapers and cools as it evolves downstream, and solidifies along some free boundary, below which a take-up velocity (\(v_1 > v_0\)) is imposed at a fixed location. Our goal in this paper is a model for this process which realistically couples the hydrodynamics, the LCP dynamics, and the temperature field, along with the free surface and boundary conditions. Moreover, we aim for a model, by necessity complex, that provides nontrivial fiber process predictions and that admits a linearized stability analysis of steady fiber processes. We first generalize three-dimensional Doi-Edwards averaged kinetic equations to include temperature-dependent material behavior and a coupled energy equation. From this formulation we generalize previous isothermal hydrodynamic, isotropic viscoelastic, and anisotropic viscoelastic models, incorporating temperature-dependent material response. The model, its nontrivial boundary value solutions, and their linearized stability are presented, along with the translation of these mathematical results, to industrially relevant issues of fiber performance properties and bounds on stable spinning speeds.