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