  We study the slippage on hierarchical fractal superhydrophobic surfaces, and
find an unexpected rich behavior for hydrodynamic friction on these surfaces.
We develop a scaling law approach for the effective slip length, which is
validated by numerical resolution of the hydrodynamic equations. Our results
demonstrate that slippage does strongly depend on the fractal dimension, and is
found to be always smaller on fractal surfaces as compared to surfaces with
regular patterns. This shows that in contrast to naive expectations, the value
of effective contact angle is not sufficient to infer the amount of slippage on
a fractal surface: depending on the underlying geometry of the roughness,
strongly superhydrophobic surfaces may in some cases be fully inefficient in
terms of drag reduction. Finally, our scaling analysis can be directly extended
to the study of heat transfer at fractal surfaces, in order to estimate the
Kapitsa surface resistance on patterned surfaces, as well as to the question of
trapping of diffusing particles by patchy hierarchical surfaces, in the context
of chemoreception.
