Stokes flow through a microchannel obstructed by a vertical plate

Two-dimensional Stokes flow through a microchannel obstructed by a vertical plate is investigated on the basis of Stokes approximation. The plate translates along the center-line of the channel and plane Poiseuille flow exists upstream and downstream from the plate. The Stokes flow is analyzed analytically using the eigenfunction expansion and the point collocation method. The stream function and the pressure distribution of the flow field are obtained for an arbitrary translating velocity of the plate and arbitrary magnitude of the Poiseuille flow. The force exerted on the plate and the pressure drop induced by the plate are calculated as functions of blockage factor. From the results, the drift velocity of the plate, for which the force exerted on the plate vanishes and the plate translates freely in the Poiseuille flow, is determined. The drift velocity is slightly lower than the mean velocity of the Poiseuille flow component projected by the plate, and induced pressure drop due to the drifting plate in the Poiseuille flow is quite small.