In the present study, we investigated computationally the laminar Couette flow past a periodically rotating circular cylinder for the purpose of observing flow phenomena behind the cylinder and controlling vortex shedding. The cylinder was placed with its center on the centerline. The Reynolds number, based on the mean incoming velocity and the width between the two flat plates, was fixed at 250. The unsteady laminar flow was computed via a finite volume method with an implicit time-marching technique. The two main parameters which affect the flow development behind the cylinder are the period (T), and the peak rotation speed (A) of the cylinder under rotation. We studied the variations of vorticity distribution, lift and drag coefficients and vortex flow patterns by varying the two parameters.