Abstract: An analytical solution of start-up flow of a Maxwell fluid flowing across an annular was obtained by the method of separation of variables.The fluid was visco-elastic and obeyed the Maxwell constitutive equation.The results showed that the start-up flow took on the characteristics of damping and periodically vibrating flow.The velocity profile across the annular-space presented different traits at different times.Reversed flow might occur when the elastic effect of the fluid was strong.The variation of velocity with time at different radii was different, with the vibration intensity at the central part of the annular space stronger than the wall part.The stronger the elasticity of the fluid, the [JP2]higher the vibration amplitude and the longer the steady time of flow.The effect of fluid elasticity on velocity profile decreased with time.The annular space could restrain the vibration of the flow.A smaller annular space could decrease the vibration amplitude and increase the vibration frequency,decreasing the average velocity at steady flow.