We simulate the startup flow of lyotropic Liquid-crystalline polymers (LCPs) in an eccentric cylinder geometry. The objectives are to explore the mechanisms for the generation of disclinations in a nonhomogeneous flow and to study the coupling between the flow and the polymer configuration. The Doi theory, generalized to spatially varying flows and approximated by the quadratic closure, is used to model the evolution of LCP configurations. This, along with the equations of motion for the fluid, is solved by a finite-element method. The flow modification by the polymer stress is mild for the parameters used, but the LCP exhibits complex orientational behavior in different regions of the flow domain. For relatively weak nematic strength, a steady state is reached in which the director is oriented either along or transverse to the streamline, depending upon local flow conditions and the deformation history. A pair of disclinations, with strength +/-1/2, are identified in the steady state, and the LCP configuration at the disclinations confirms the model of a structured defect core proposed by Greco and Marrucci (1992). For strong nematic strength, director tumbling occurs in the more rotational regions of the how field, giving rise to a polydomain structure. The boundary of the tumbling domain consists of two disclinations of +1/2 strength, a structure very similar to previous experimental observations of LCP domains.