  Superhydrophobic surfaces reduce drag by combining hydrophobicity and
roughness to trap gas bubbles in a micro- and nanoscopic texture. Recent work
has focused on specific cases, such as striped grooves or arrays of pillars,
with limited theoretical guidance. Here, we consider the experimentally
relevant limit of thin channels and obtain rigorous bounds on the effective
slip length for any two-component (e.g. low-slip and high-slip) texture with
given area fractions. Among all anisotropic textures, parallel stripes attain
the largest (or smallest) possible slip in a straight, thin channel for
parallel (or perpendicular) orientation with respect to the mean flow. For
isotropic (e.g. chessboard or random) textures, the Hashin-Strikman conditions
further constrain the effective slip. These results provide a framework for the
rational design of superhydrophobic surfaces.
