The oscillating flow of the viscoelastic fluid in cylindrical pipes has been applied in many fields, such as industries of petroleum, chemistry, and bioengineering. It is studied using the fractional derivative Maxwell model in this paper. The exact solution is obtained utilizing a simpler and more reasonable technique. According to this velocity solution, the time-velocity profile of one kind of viscoelastic fluid is analyzed. From analysis, it is found that the flow behaves like the Newton fluid when the oscillating frequency is low, and the flow reversal occurs when the oscillating frequency is high. Moreover, two series approximations for the velocity are obtained and analyzed for different model parameters. In one series approximation, the velocity is parabolic in profile, while in the other series approximation, the velocity presents three characteristics: (1) it is independent of radius and at the centerline is smaller than that of steady Poiseuille flow, (2) the phase lags about $90deg$ with respect to the imposed pressure gradient, and (3) the Richardson annular effect is found near the wall.

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