Paramagnetic microsphere suspensions placed in a rotating magnetic field aggregate to form rotating magnetic chains. These chains that are several tens of micrometers in length act as microrotors, and can be modeled as circular cylinders. The flow and associated micromixing around these cylinders rotating about their radial axis is studied for a very low Reynolds number, creeping-flow system.. Time-scales for momentum transfer are much smaller than boundary movement, hence a quasi-steady approximation can be used. The flow is derived at every instant from the case of a steady motion of a horizontally translating cylinder, with the rotation approximated to a series of incremental translations. A numerical simulation was used to determine the pathlines and material lines of point fluid elements, which were analyzed to understand the behavior of the microfluidic system. The results indicate the flow to be unsteady, with chaotic advection observed in the system. The flow is primarily two-dimensional with planar fluid movement limited to the immediate area around the rotating cylinder, with a small disturbance in the axial direction that is experienced up to many diameters away. Elliptic and star shaped pathlines, including periodic orbits, are observed depending on the fluid element's initial location. The trajectories and phase angles compare well with the limited experimental results, as well as with data from particle dynamics simulations. Material lines and streaklines display stretching and folding, which are indicative of the chaotic behavior of the system. The material lines have similar lengths for the same amount of rotation at different speeds, and the affect of rotational speeds appears to be primarily to change the time of mixing.