Longitudinal flow in superhydrophobic channels with partially invaded grooves

Analytical expressions are derived for the longitudinal flow in a superhydrophobic microchannel where flat menisci in the Cassie state have partially invaded the grooves between no-slip blades. Using these solutions, the effective slip lengths are computed and compared with recent analytical results for unbounded shear flow over the same class of surfaces. Expressions for the first-order corrections to these effective slip lengths when the menisci are weakly curved are also derived. A mathematical connection to superhydrophobic channel flows where the flat menisci are still pinned to the tops of the pillars is also made, resulting in novel analytical expressions for those solutions too.