In this paper, the velocity profile and pressure gradient of the unsteady state unidirectional flow of a Maxwell fluid in a circular duct are considered. The flow motion in the duct is induced by a prescribed arbitrary inlet volume flowrate which varies with time. Based on the flow conditions described, two basic flow situation are solved; these are a suddenly started and a constant acceleration flow respectively. These two results are applied to a practical case that is a trapezoidal motion. In addition, oscillatory flow is also considered. The results show that the effect of the relaxation time, λ, of the viscoelastic fluid is only of importance when the motion is unsteady.
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