We investigate an asymptotic model of DiPrima & Stuart (1972b, 1975) describing steady Taylor vortex flow between eccentric cylinders, under the assumption that the eccentricity ϵ, the clearance ratio δ and the Taylor vortex amplitude A satisfy ϵ, δ and A small. By solving a boundary value problem for the radial eigenfunctions we numerically obtain the flow field of DiPrima & Stuart and investigate its topology, after correcting higher-order terms to ensure that the flow preserves volume. We find regions of chaotic streamlines at all eccentricities and discuss the reason for their existence. We make an analogy between the full model and a modulated vortex flow field which qualitatively displays the same behaviour.For large eccentricities, we examine the flow field and the topology of its streamlines, especially where the two-dimensional flow contains a separated region of recirculation. In this case Taylor vortices give rise to transport of fluid particles in and out of the separated region. We find that the onset of Taylor vortices encourages recirculation in the inflow plane, whilst discouraging it in the outflow plane.