We study the paths of fluid particles in velocity fields modelling rigidly rotating velocity fields that occur in the concentric Taylor problem. We set up velocity fields using the model of Davey, DiPrima & Stuart (1968) based on small-gap asymptotics. This allows a numerical study of the Lagrangian properties of steady flow patterns in a rotating frame. The spiral and Taylor vortex modes are integrable, implying that in these cases almost all particle paths are confined to two-dimensional surfaces in the fluid. For the case of Taylor vortices the motion on these surfaces is quasi-periodic, whereas for spirals the particles propagate up or down the cylinder on these surfaces.

The non-axisymmetric modes we consider are wavy vortices, spirals, ribbons and twisted Taylor vortices. All of these flows have the property that they are steady flows when examined in a rotating frame of reference. For all non-axisymmetric modes with the exception of spirals, we observe the existence of regions of chaotic mixing within the fluid. We discuss mixing of the fluid by these flows with reference to the pattern of stagnation points and some of the periodic trajectories within the fluid and on the boundary.