Carbon fibers are widely used in many industrial applications due the fact of their excellent properties. Carbonaceous mesophases are liquid crystalline precursor materials that can be spun into high performance carbon fibers using the melt spinning process, which is a flow cascade consisting of pressure driven flow-converging die flow-free surface extensional spinline flow that modifies the precursor molecular orientation structure. Carbon fiber property optimization requires a better understanding of the principles that control the structure development during the fiber formation processes and the rheological processing properties. This paper presents the elastic and continuum theory of liquid crystalsand computer simulations of structure formation for pressure-driven flow of carbonaceous liquid crystalline precursors used in the industrial carbon fiber spinning process. The simulations results capture the formation of characteristic fiber macro-textures and provide new knowledge on the role of viscous and elastic effects in the spinning process.