In this paper, the velocity profile and pressure gradient of the unsteady state unidirectional flow of a Voigt fluid between the parallel surfaces are considered. The flow motion between the surfaces is induced by a prescribed arbitrary inlet volume flow rate which varies with time. Based on the flow conditions described, two basic flow situations 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.