Frictional slip lengths for unidirectional superhydrophobic grooved surfaces

Abstract

The exact solutions due to Philip [ZAMP 23, 353 (1972)] for Stokes shear flow over a periodic array of no-shear slots embedded in a no-slip surface are generalized to account for an arbitrary pattern of no-shear slots in each period window. The slots, or grooves, in each period window run parallel to each other, and are of infinite length, but their widths and separations can be specified arbitrarily. Explicit solutions are found both for longitudinal and transverse flows over the composite grooved surface. Analytical expressions for the transverse and longitudinal slip lengths associated with the microstructured surfaces are then found as functions of the geometrical parameters characterizing the surface. The formulae are relevant to a wide class of flow geometries and are expected to provide a useful tool in the design, analysis, and optimization of the friction properties of grooved microstructured superhydrophobic surfaces. The results are used to show that introducing even a very small wetted region in a no-shear slot can have a significant influence on the effective slip length.