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