  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.
