Asymptotic formulae for flow in superhydrophobic channels with longitudinal ridges and protruding menisci

TL;DR

This paper develops new analytical formulas for fluid flow in superhydrophobic channels with ridges and protruding menisci, extending previous shear flow results and validated against numerical solutions for a wide range of conditions.

Contribution

It introduces generalized asymptotic formulae for flow in channels with patterned walls, accommodating arbitrary protrusions and extending prior work without restrictions on meniscus protrusion.

Findings

Analytical formulas match numerical solutions well across various protrusion sizes.

Formulas remain accurate even when menisci touch the opposite wall.

Large validity range despite small ridge period assumption.

Abstract

This paper presents new analytical formulae for flow in a channel with one or both walls patterned with a longitudinal array of ridges and arbitrarily protruding menisci. Derived from a matched asymptotic expansion, they extend results by Crowdy (J. Fluid Mech., vol. 791, 2016, R7) for shear flow, and thus make no restriction on the protrusion into or out of the liquid. The slip length formula is compared against full numerical solutions and, despite the assumption of small ridge period in its derivation, is found to have a very large range of validity; relative errors are small even for periods large enough for the protruding menisci to degrade the flow and touch the opposing wall.