Longitudinal pressure-driven flows between superhydrophobic grooved surfaces: Large effective slip in the narrow-channel limit

The gross amplification of the fluid velocity in pressure-driven flows due to the introduction of superhydrophobic walls is commonly quantified by an effective slip length. The canonical duct-flow geometry involves a periodic structure of longitudinal shear-free stripes at either one or both of the bounding walls, corresponding to flat-meniscus gas bubbles trapped within a periodic array of grooves. This grating configuration is characterized by two geometric parameters, namely the ratio κ of channel width to microstructure period and the areal fraction Δ of the shear-free stripes. For wide channels, κ 1, this geometry is known to possess an approximate solution where the dimensionless slip length λ, normalized by the duct semiwidth, is small, indicating a weak superhydrophobic effect. We here address the other extreme of narrow channels, κ 1, identifying large O(κ-2) values of λ for the symmetric configuration, where both bounding walls are superhydrophobic. This velocity enhancement is associated with an unconventional Poiseuille-like flow profile where the parabolic velocity variation takes place in a direction parallel (rather than perpendicular) to the boundaries. Use of matched asymptotic expansions and conformal-mapping techniques provides λ up to O(κ-1), establishing the approximationλ∼κ-2Δ33+κ-1Δ2πln4+,which is in excellent agreement with a semianalytic solution of the dual equations governing the respective coefficients of a Fourier-series representation of the fluid velocity. No similar singularity occurs in the corresponding asymmetric configuration, involving a single superhydrophobic wall; in that geometry, a Hele-Shaw approximation shows that λ=O(1).