Numerical Study of Laminar Boundary-Layer Flows over a Superhydrophobic Plate

Liquid flow in the laminar boundary layer over a flat plate with a superhydrophobic surface formed by a square array of pillars is studied theoretically. Assuming the water surface on top of the pillars to be in the Cassie-Baxter state, asymptotic analysis is first carried out to separate the micro-scale flow in a typical cell surrounding a pillar and the macro-scale development of the laminar boundary layer of Blasius type. The 3-D cell problem and the 2-D boundary layer problem are solved together iteratively, yielding the slip length and the entire flow field. Numerical results are presented to examine the effect of solid fraction, pillar-to-pillar spacing, and the speed of the ambient flow on drag reduction. It is shown that the slip length is practically constant, while the boundary layer thickness grows monotonically downstream so that hydrophobicity affects drag reduction primarily over the leading portion of a long surface.