We consider longitudinal shear flow over a $2L$-periodic array of grooves separated by solid slats of rectangular cross section. We address the case where the liquid partially invades the grooves, so a portion of the solid slat protrudes into the liquid. For flat menisci, the ratio $\lambda$ of the slip length to $L$ depends upon two geometric parameters: the ratio $\epsilon$ of the slat width to the period, and the aspect ratio $\mathrm{\Delta }$ of the protruding part of the solid slats. We analyze the singular limit $\epsilon \ll 1$ with arbitrary $\mathrm{\Delta }$ utilizing matched asymptotic expansions, with an outer region on the scale of the periodicity and an inner region on the scale of the protrusion. In the outer region, the flow is represented using a periodic array of source singularities. In the inner region, the flow is solved using a combination of Schwarz–Christoffel and Joukowsky mappings. Asymptotic matching yields $\lambda \approx (2/\pi )ln(2/\pi \epsilon A)$ wherein $A$, the scale factor in the Schwarz–Christoffel mapping, is a monotonically increasing function of $\mathrm{\Delta }$. We obtain simple approximations for $A$ for both small and large $\mathrm{\Delta }$. These lead to slip-length formulas which practically cover the entire range of relevant $\mathrm{\Delta }$ values. The agreement between the predictions of these formulas and the numerical results of Ng and Wang [Phys. Fluids 21, 013602 (2009)] is gratifying.

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• Low Reynolds number flows
• Shear flows
• Microbubbles