Longitudinal Shear Flow over a Superhydrophobic Grating with Partially Invaded Grooves and Curved Menisci

SIAM Journal on Applied Mathematics, Volume 84, Issue 3, Page 1186-1203, June 2024.
Abstract. We consider longitudinal shear flows over a superhydrophobic grating made up of a periodic array of grooves separated by infinitely thin slats, addressing the case where the liquid partially invades the grooves. We allow for curved menisci, specified via a depression angle at the contact line. We focus on the limit of small solid fractions where the length of the wetted portion of the slat is small compared with the period. Following an earlier analysis of the comparable flow over noninvaded grooves [O. Schnitzer, J. Fluid Mech., 820 (2017), pp. 580–603], this singular limit is treated using matched asymptotic expansions, with an outer region on the scale of a single period and an inner region on the scale of the wetted portion of the slat. The flow problem in both regions is solved using conformal mappings. Asymptotic matching yields a closed-form approximation for the slip length as a function of the solid fraction and depression angle.