Effective slip for Stokes flow over a surface patterned with two- or three-dimensional protrusions

Effective slip lengths are obtained, using semi-analytic methods, for Stokes flows over a surface that is patterned with a periodic array of two-dimensional (2D) cylindrical or 3D spherical protrusions. The protruding surface can be perfect- or non-slipping, corresponding to a bubble mattress or a rough boundary. For longitudinal and transverse flows over cylindrical bumps and 3D flow over a square array of spherical bumps, the effective slip length is obtained as a function of the protrusion angle, the area fraction of surface covered by protrusions and the partial slip length of the protruding surface. The results are compared with analytical dilute limits in order to ascertain the range of validity of these limits. Phenomenological equations are also derived to enable a quick evaluation of the slip length for some particular values of the protrusion angle at which the slip length is maximum in magnitude.