Some exact properties of the effective slip over surfaces with hydrophobic patternings

Shear flows of viscous fluid layers over nonuniformly hydrophobic surfaces are characterized in the far-field by an effective slip velocity, which relates to the applied stress through some mobility tensor characterizing the surface. Here, we identify two methods to determine the mobility tensor for flat surfaces with arbitrary slip-length variations. A family of Cross Flow Identities is then analyzed, which equate mobility components of different unidirectional patternings. We also calculate an analytical mobility solution for a family of continuously varying patterns. We validate the results numerically and discuss implications in various limits. (C) 2013 American Institute of Physics. [http://dx.doi.org/10.1063/1.4790536]