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 sequence consisting of capillary, diverging, porous media, converging, and extensional flows 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 crystals and computer simulations of structure formation for pressure-driven capillary flow of carbonaceous mesophase precursors used in the industrial carbon fiber spinning process. The simulation results capture the non-Newtonian rheology of mesophase and the formation of characteristic fiber macro-textures.