We consider the effective hydrophobicity of a periodically grooved surface immersed in liquid,
with trapped shear-free bubbles protruding between the no-slip ridges at a π/2 contact angle.
Specifically, we carry out a singular-perturbation analysis in the limit ǫ ≪ 1 where the bubbles are
closely spaced, finding the effective slip length (normalised by the bubble radius) for longitudinal
flow along the the ridges as π/√
2ǫ − (12/π) ln 2 + (13π/24)√
2ǫ + o(
√
ǫ), the small parameter ǫ
being the planform solid fraction. The square-root divergence highlights the strong hydrophobic
character of this configuration; this leading singular term (along with the third term) follows from
a local lubrication-like analysis of the gap regions between the bubbles, together with general
matching considerations and a global conservation relation. The O(1) constant term is found by
matching with a leading-order solution in the “outer” region, where the bubbles appear to be
touching. We find excellent agreement between our slip-length formula and a numerical scheme
recently derived using a “unified-transform” method (D. Crowdy, IMA J. Appl. Math., 80 1902,
2015). The comparison demonstrates that our asymptotic formula, together with the diametric
“dilute-limit” approximation (D. Crowdy, J. Fluid Mech., 791 R7, 2016), provides an elementary
analytical description for essentially arbitrary no-slip fractions.