Singular effective slip length for longitudinal flow over a dense bubble mattress

We derive accurate asymptotic expansions in the small-solid-fraction limit $\epsilon\ll1$ for the effective slip length characterising unidirectional liquid flow over a `bubble mattress' --- a periodically grooved surface, with trapped bubbles protruding between solid ridges. The slip length diverges in this limit: inversely with $\sqrt{\epsilon}$ for contact angles $\theta$ near $\pi/2$, and logarithmically for $0\le\theta<\pi/2$. The analysis of the velocity field entails matching `inner' expansions valid close to the solid segments with `outer' expansions valid on the scale of the periodicity, where the protruding bubbles appear to touch. For $\theta$ close to $\pi/2$, the inner-region geometry is narrow and the analysis there resembles lubrication theory; for smaller contact angles the inner region is resolved using a Schwarz-Christoffel mapping. In both cases the outer problem is solved using a mapping from a degenerate curvilinear triangle to an auxiliary half plane. The asymptotic analysis explicitly illustrates the logarithmic-to-algebraic transition, and yields a uniformly valid approximation for the slip length for arbitrary contact angles $0\le \theta\le\pi/2$. We demonstrate good agreement with a numerical solution (courtesy of Ms Elena Luca).