Stokes shear flow over a grating: Implications for superhydrophobic slip

Abstract

A semianalytical model based on the method of eigenfunction expansions and domain decomposition is developed for Stokes shear flow over a grating composed of a periodic array of parallel slats, with finite slippage on solid surfaces and infinite slippage on the bottom of troughs mimicking a no-shear liquid-gas interface penetrating into the space between slats. The model gives the macroscopic slip lengths for flow parallel or normal to the slats in terms of the microscopic slip length of the liquid-solid interface, area fraction of the no-shear liquid-gas interface, and depth of the liquid-gas interface in the grooves. When the no-shear interface lies flat on the top of the slats, the macroscopic slip lengths are the maximum and can be estimated with reasonably good accuracy by simple formulas. However, the slip lengths, particularly the transverse one, are very sensitive to penetration of the no-shear interface into the grooves. They can be reduced by a large factor when the interface just slightly gets into the grooves. On comparing with some molecular-dynamics simulation measures, it is pointed out that the applied pressure, which has to be less than the capillary pressure in the superhydrophobic state, can be correlated with the penetration depth of the no-shear interface.