The flow between two concentric cylinders, while the inner cylinder is rotating at a velocity larger than a critical value and the outer cylinder is stationary, is known as the Couette Taylor flow. Under those conditions, the Taylor vortices are formed in the fluid domain as hydrodynamic instabilities. Researchers study the Couette-Taylor system as a preliminary model for obtaining information about vortex structures, descriptions of this structures such as shape, size, center, energy intensity, mass transfer, and finally investigation the transition to turbulent flow. The reason for this choice is the geometric simplicity and hydrodynamic instabilities. In this study, the radius of cylinders are R 1 =85.5 mm R 2 =100 mm and their height is 225 mm. The rotational velocity of the inner cylinder oscillates (increases from zero to a maximum value and then goes to zero again with different slopes). The slopes of the velocity oscillations for laminar case are: 0.1 rad/s 2 , 0.05 rad/s 2 , 0.010 rad/s 2 , 0.005 rad/s 2 , 0.001 rad/s 2 , 0.0005 rad/s 2 , 0.0001 rad/s 2 , 0.00005 rad/s 2 and for turbulent regime is 0.1 rad/s 2 . In this study, the numerical modeling has been accomplished using the ANSYS-FLUENT software and UDF. turbulence model is used to simulate the turbulent flow case. The vortex structures are studied in different slopes and periods in an unsteady state flow. For laminar regime, the results were validated with experimental data and less than 4% difference was found. The results show that, with decreasing the rotation slope, the vortex structures is closer to the steady state. In laminar regime, when the rotational velocity is increasing and the rotational slope is minimum, that is 0.00005 rad/s 2 , the primary critical Taylor number is 40.04% different from that of steady state and when the rotational velocity is decreasing, the difference is 7.24%, which indicates a more delay of formation of the vortices in the process of increasing the rotational velocity compared with decreasing the rotational velocity. For the turbulent regime, the critical Taylor number is different by 13.95% compared to that of steady state. Also for the rotational slopes of 0.1 rad/s 2 and 0.05 rad/s 2 , the vortices do not appear until the second period of the rotational oscillations of the inner cylinder. The reason for this phenomenon is that there is not enough time to adapt the flow to the applied changes, and the flow regime is laminar Couette flow at initial stages. Keywords: Couette-Taylor Flow, Critical Taylor Number, Slope, Oscillatory, Period