Highly singular slip length for longitudinal shear flow over a dense bubble mattress

Abstract

Compound surfaces, consisting of periodic arrays of solid patches and free surfaces, exhibit hydrodynamic slipperiness which is quantified by their slip length. The limit of small solid fractions, where the slip length diverges, is of fundamental interest. This paper addresses longitudinal liquid flows over a periodic array of grooves which are partially invaded by the liquid. Assuming that the slats separating the grooves are infinitely thin, the solid fraction $\epsilon$ is set by the invasion depth. Inspired by the singular small-solid-fraction limit for non-invaded grooves (Schnitzer, Phys. Rev. Fluids, vol. 1, 2016, 052101R), we consider the idealised geometry of $90^{\circ }$ protrusion angles, where an integral force balance in the limit $\epsilon \to 0$ implies a slip length that scales as $\epsilon ^{-1}$ . The problem exhibits a nested structure, where the liquid domain is conceptually decomposed into four distinct regions: a unit-cell region on the scale of the period, where the wetted portion of the slat appears as a point singularity; two regions on the scale of the wetted slat, where the flow essentially varies in one dimension; and a transition region about the tip of the slat. Analysing these regions using matched asymptotic expansions and conformal mappings yields the ratio of slip length to semi-period as $2\epsilon ^{-1} - (10/{\rm \pi} )\ln 2 + \cdots$ .