Abstract: A six dimensional dynamical system for Oldroyd B flow between two concentric cylinders is derived in the narrow gap limit by a truncation of the flow field and stress. The effects of polymer additives on the flow stability are analyzed. The results show that a small amount of polymer additives can delay the transition from laminar flow to turbulence and the effect of polymer additives on the flow stability of the fluids is closely related to the concentration of the additive therein.