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The compressible flow between a slider and a moving rough surface is examined asymptotically and numerically in the limit of very small gap height. The roughness is assumed to consist of a well-defined periodic pattern of grooves whose amplitude and wavelength are of the order of the gap height. A two-scale homogenization analysis is employed to determine a non-linear elliptic partial differential equation governing the leading order pressure in the gap on the scale of the slider. The equation involves coefficient functions that are determined numerically by averaging Stokes flows on the scale of the patterned roughness. Numerical results are obtained for the cases of transverse and longitudinal roughness for both no-slip and slip-flow boundary conditions, and for the limiting case of large bearing number. Comments and a brief analysis are given on the reduction of the governing equation for pressure in the conventional limit of long wavelength of the surface roughness. [ABSTRACT FROM AUTHOR]