  Super Hydrophobic (SH) coatings are widely used to mitigate drag in various
applications. Numerous studies have demonstrated that the beneficial wall-slip
effect produced by these materials disappears in laminar flow regimes. The main
mechanisms considered to be behind the decrease in performance are
Marangoni-induced stresses and air/liquid interface deformation. In the present
study, a new mechanism is proposed to explain the loss of performances of
SH-surfaces in laminar flow regimes. Here we consider the flow of air inside
the plastron and the associated momentum loses induced by roughness elements
with different geometric characteristics. The effects of air motion within the
plastron is coupled to the outer fluid with a homogenised boundary condition
approach. To this end, numerical simulations at the scale of the roughness
element were conducted as a function of the porosity and the tortuosity of the
domain to determine the slip velocity at the air-liquid interface.
  The homogenised boundary condition is then implemented in a theoretical model
for the outer flow to compute drag on SH-spheres at low $Re$ numbers.
Experiments of laminar SH falling spheres indicate that high values of the
tortuosity and low values of porosity lead to a loss of performances when
considering drag reduction. As anticipated, a 3D printed sphere with low
tortuosity and similar porosity demonstrated near-optimal drag reductions. A
comparative study between the predicted values and experiments shows that the
homogenised model is able to accurately predict the drag on SH surfaces for
values of the porosity and tortuosity estimated from microscopy images of the
SH textured surface.
