An analysis of bi-directional Stokes micropump comprising a periodic array of moving belts

In this study, we present an analysis of a Stokes micropump comprising a periodic array of parallel finite belts moved by rotating shafts. The geometry of the mechanical micropump is uniquely determined by the ratio of the length of the belts to the width between two neighboring belts (i.e., the aspect ratio a). The method of eigenfunction expansions with collocation is applied to solve the Stokes equation for the pumping rate, the stream function, and the velocity field as well as for the pressure gradient, which are all normalized by proper scales. It is found that with increasing a, the normalized pumping rate per unit micropump (or, simply abbreviated as a unit channel) first increases drastically and then decreases exponentially until it becomes a constant for large a, indicating that there exists a critical aspect ratio ( ac = 0.035) at which the maximum pumping rate ( qmax = 0.861) occurs, while the limiting value of q at large a is 0.5. The steady flow is driven by the moving belts against the established pressure gradient, and the pressure gradient at the centerline reaches its maximum value at the channel center and vanishes at distances from the micropump. Moreover, it is shown that the average flow velocity component perpendicular to the moving direction of the belts is relatively small, so that the flow field in the channel is approximately a unidirectional laminar flow, and therefore, the results are not necessarily limited to very low Reynolds numbers.