Mesoscopic modelling of heterogeneous boundary conditions for microchannel flows

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 hydrodynamical fields 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 dependence of the slip length on the mean channel pressure and local slip properties and with the experimental dependence of the pressure drop reduction on the percentage of hydrophobic material deposited on the surface.