In the present study, the velocity profile and pressure gradient of the unsteady state unidirectional flow of a Voigt fluid in a circular duct with different prescribed volume flow rate are investigated. The flow motion in the duct is induced by a prescribed inlet volume flow rate which varies with time. Based on the flow conditions prescribed, two basic flow situations are solved; these are a suddenly started, and a constant accelerated, flow respectively. These two results are then applied to a practical case that is a trapezoidal motion which contains three phases of piston motion, the constant acceleration from the rest to a fixed velocity, then maintaining at this velocity, following with the constant deceleration to a stop. In addition, oscillatory flow is also considered.