The investigation of the couple stress fluid flow behaviour between two parallel plates under sudden stoppage of the pressure gradient is considered. Initially, a flow of couple stress fluid is developed between the two parallel plates under a constant pressure gradient. Suddenly, the applied pressure gradient is stopped, and the resulting unsteady flow is studied. This type of flow is known as run-up flow in the literature. Now the flow is expected to come to rest in a long time. Usually, these types of problems are solved by using the Laplace transform technique. There are difficulties in obtaining the inverse Laplace transform; hence, many researchers adopt numerical inversions of Laplace transforms. In this paper, the problem is solved by using the separation of variables method. This method is easier than the transform method. The velocity field is analyti-cally obtained by applying the usual no-slip condition and hyper-stick conditions on the plates, and hence the volumetric flow rate is derived at subsequent times. The steady state solution before the withdrawal of the pressure gradient is matched with the initial condition on time. The rest time, i.e. the time taken by the fluid to come to rest after the pressure gradient is withdrawn is calculated. The graphs for the velocity field at different times and different couple stress parameters are drawn. In the special case when a couple stress parameter approaches infinity, couple stress fluid becomes a viscous fluid. Our results are in good agreement with this special case.