The friction of a mesh-like super-hydrophobic surface

TL;DR

This paper develops an analytical model to quantify the slip length and friction of a mesh-like super-hydrophobic surface, revealing a logarithmic dependence on the solid area fraction, which advances understanding of their flow properties.

Contribution

It introduces a new analytical approach for calculating the friction of mesh-like super-hydrophobic surfaces with a thin mesh, expanding beyond previous geometries studied.

Findings

Effective slip length depends logarithmically on solid area fraction.

Analytical results compare favorably with simple estimates.

Provides a practical fit for the slip length in terms of mesh parameters.

Abstract

When a liquid droplet is located above a super-hydrophobic surface, it only barely touches the solid portion of the surface, and therefore slides very easily on it. More generally, super-hydrophobic surfaces have been shown to lead to significant reduction of viscous friction in the laminar regime, so it is of interest to quantify their effective slipping properties as a function of their geometric characteristics. Most previous studies have considered flows bounded by arrays of either long grooves, or isolated solid pillars on an otherwise flat solid substrate, and for which therefore the surrounding air constitutes the continuous phase. Here we consider instead the case where the super-hydrophobic surface is made of isolated holes in an otherwise continuous no-slip surface, and specifically focus on the mesh-like geometry recently achieved experimentally. We present an analytical method to calculate the friction of such a surface in the case where the mesh is thin. The results for the effective slip length of the surface are computed, compared to simple estimates, and a practical fit is proposed displaying a logarithmic dependance on the area fraction of the solid surface.