Effective slip for flow in a rotating channel bounded by stick-slip walls

Abstract

This paper aims to look into how system rotation may modify the role played by boundary slip in controlling flow through a rotating channel bounded by stick-slip walls. A semianalytical model is developed for pressure-driven flow in a slit channel that rotates about an axis perpendicular to its walls, which are superhydrophobic surfaces patterned with periodic alternating no-shear and no-slip stripes. The cases where the flow is driven by a pressure gradient parallel or normal to the stripes are considered. The effects of the no-shear area fraction on the velocities and effective slip lengths for the primary and secondary flows are investigated as functions of the rotation rate and the channel height. It is mathematically proved that the secondary flow rate is exactly the same in the two cases, irrespective of whether the primary flow is parallel or normal to the wall stripes. For any rotation speed, there is an optimal value of the no-shear area fraction at which the primary flow rate is maximum. This is a consequence of two competing effects: the no-shear part of the wall may serve to reduce the wall resistance, thereby enhancing the flow especially at low rotation, but it also weakens the formation of the near-wall Ekman layer, which is responsible for pumping the flow especially at high rotation. Wall slip in a rotating environment is to affect flow in the Ekman layer, but not flow in the geostrophic core.