Hydrodynamic friction of fakir-like super-hydrophobic surfaces

TL;DR

This paper provides an analytical model for the slip length of super-hydrophobic fakir surfaces with circular posts, showing excellent agreement with numerical results and advancing understanding of drag reduction mechanisms.

Contribution

It introduces a theoretical approach to calculate the slip length for fakir-like super-hydrophobic surfaces with circular posts, extending prior numerical studies.

Findings

Analytical expression for slip length derived for fakir surfaces.

Excellent agreement with previous numerical computations.

Asymptotic analysis valid for small surface coverage.

Abstract

A fluid droplet located on a super-hydrophobic surface makes contact with the surface only at small isolated regions, and is mostly in contact with the surrounding air. As a result, a fluid in motion near such a surface experiences very low friction, and super-hydrophobic surfaces display strong drag-reduction in the laminar regime. Here we consider theoretically a super-hydrophobic surface composed of circular posts (so called fakir geometry) located on a planar rectangular lattice. Using a superposition of point forces with suitably spatially-dependent strength, we derive the effective surface slip length for a planar shear flow on such a fakir surface as the solution to an infinite series of linear equations. In the asymptotic limit of small surface coverage by the posts, the series can be interpreted as Riemann sums, and the slip length can be obtained analytically. For posts on a square lattice, our analytical results are in excellent quantitative agreement with previous numerical computations.