Effective slip for Stokes flow between two grooved walls with an arbitrary phase shift

This work aims to determine how the effective slip length for a wall-bounded flow may depend on, among other geometrical parameters, the phase shift between patterns on the two walls. An analytical model is developed for Stokes flow through a channel bounded by walls patterned with a regular array of rectangular ribs and grooves, where the patterns on the two walls can be misaligned by any phase shift. This study incorporates several previous studies as limiting or special cases. It is shown that the phase shift can have qualitatively different effects on the flow rate and effective slip length, depending on the flow direction. In a narrow channel, increasing the phase shift may mildly decrease the flow rate and effective slip length for flow parallel to the grooves, but can dramatically increase the flow rate and effective slip length for flow transverse to the grooves. It is found that unless the channel height is much larger than the period of the wall pattern, the effect due to wall confinement has to be taken into account on evaluating the effective slip lengths. (paper)