  We present a mesoscopic model of the fluid-wall interactions for flows in
microchannel geometries. We define a suitable implementation of the boundary
conditions for a discrete version of the Boltzmann equations describing a
wall-bounded single phase fluid. We distinguish different slippage properties
on the surface by introducing a slip function, defining the local degree of
slip for mesoscopic molecules at the boundaries. The slip function plays the
role of a renormalizing factor which incorporates, with some degree of
arbitrariness, the microscopic effects on the mesoscopic description. We
discuss the mesoscopic slip properties in terms of slip length, slip velocity,
pressure drop reduction (drag reduction), and mass flow rate in microchannels
as a function of the degree of slippage and of its spatial distribution and
localization, the latter parameter mimicking the degree of roughness of the
ultra-hydrophobic material in real experiments. We also discuss the increment
of the slip length in the transition regime, i.e. at O(1) Knudsen numbers.
Finally, we compare our results with Molecular Dynamics investigations of the
dependency of the slip length on the mean channel pressure and local slip
properties (Cottin-Bizonne et al. 2004) and with the experimental dependency of
the pressure drop reduction on the percentage of hydrophobic material deposited
on the surface -- Ou et al. (2004).
