The planar singular solutions of Stokes and Laplace equations are derived and applied to a number of transport problems associated with porous surfaces. The velocity, pressure, concentration, and temperature slip coefficients are determined exactly for the semi-infinite periodic arrays of spheres and these are compared with the predictions of two approximate continuum theories formulated by Brinkman [Appl. Sci. Res. Sect. A 1, 27 (1947)] and Chang and Acrivos [J. Appl. Phys. 59, 375 (1986)] to assess the utility of such theories in accurately predicting various overall properties related to the porous surfaces. It is found that in general these theories provide fairly accurate estimates of these properties even when the length scales based on the relevant macroscopic properties such as the permeability are much smaller than the length scales characterizing the microstructure of the porous media.
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