  Super-hydrophobic array of grooves containing trapped gas (stripes), have the
potential to greatly reduce drag and enhance mixing phenomena in microfluidic
devices. Recent work has focused on idealized cases of stick-perfect slip
stripes, with limited guidance. Here, we analyze the experimentally relevant
situation of a pressure-driven flow past striped slip-stick surfaces with
arbitrary local slip at the gas sectors. We derive analytical formulas for
maximal (longitudinal) and minimal (transverse) directional effective slip
lengths that can be used for any surface slip fraction (validated by numerical
calculations). By representing eigenvalues of the slip length-tensor, they
allow us to obtain the effective slip for any orientation of stripes with
respect to the mean flow. Our results imply that flow past stripes is
controlled by the ratio of the local slip length to texture size. In case of a
large (compared to the texture period) slip at the gas areas, surface
anisotropy leads to a tensorial effective slip, by attaining the values
predicted earlier for a perfect local slip. Both effective slip lengths and
anisotropy of the flow decrease when local slip becomes of the order of texture
period. In the case of small slip, we predict simple surface-averaged,
isotropic flows (independent of orientation). These results provide a framework
for the rational design of super-hydrophobic surfaces and devices.

Fluid Dynamics;Soft Condensed Matter