Effective slip length for transverse shear flow over partially invaded grooves

We consider transverse shear flow over a $2L$ -periodic array of air-filled grooves, which are separated by solid slats. address the scenario where liquid partially invades grooves. Following Crowdy's solution longitudinal problem ( J. Fluid Mech. vol. 925, 2021, R2), we focus upon simplified geometry infinitely narrow For flat menisci, ratio $\lambda$ slip length to $L$ depends single geometric parameter, namely $H$ invasion depth . analyse singular limit $H\ll 1$ using matched asymptotic expansions, with an outer region on scale period and inner slat. In region, employ complex-variable formulation Stokes flows, obtaining complex velocity as divergent series. This formal representation is transformed admissible closed-form expression. equivalent that past finite line segment. Asymptotic matching yields $\lambda = [\ln (2/{\rm \pi} H)-1]/{\rm \pi}$