In this study, the structure of the fluid flow and vortices are investigated for the circular Couette flow, Taylor vortex and wavy vortex regimes by changing the Womersley number and periods of oscillations of the inner cylinder rotation. The rotational velocity of the inner cylinder, ω, increases linearly with time from zero to a maximum value and then decreases to zero in a periodic manner. The Womersley number, Wo, varies between 0.27≤Wo≤12.14, and the flow is assumed laminar. The results consist of two separate parts as the inner cylinder rotates with 1) positive and 2) negative accelerations. It is observed that by increasing Wo, the flow cannot follow the given boundary condition and there is a time delay. Therefore, the values of the critical Taylor number become different from those of the steady-state conditions. For Wo=0.27 and when the inner cylinder rotates with positive acceleration, the primary critical Taylor number is 40% higher than that of steady state, while as the inner cylinder rotates with negative acceleration, this is only 7.2% higher than the corresponding steady-state condition. It should be noted that for two values of the Womersley numbers, Wo=12.14 and Wo=8.59, no flow instability occurs and no vortex appears until the second period of the inner cylinder oscillations. The reason is that the timescale of the dynamics of flow is lower than the timescale of the flow instability; thus, the flow is circular Couette without any vortex.