Exploring the Klinkenberg effect at different scales.

Simulations of microflows usually require sophisticated numerical tools. Nevertheless in the slip regime, the hydrodynamic equation with slip boundary condition may be sufficient to account for the so-called Klinkenberg effect. We propose to visit this effect using a basic network of microchannels in which the Knudsen number is multiplied by two or four by introducing successive derivations to the channel. We derived an equivalent hydraulic conductivity up to second order. Theoretical results are compared both with the results of the Navier-Stokes equations with slip condition and with those obtained using a Bhatnagar-Gross-Krook-Hermite model developed especially for flows with a large spectrum of Knudsen numbers (typically 10^{-4}<K_{n}<10). A criterion is provided in order to distinguish the slip regime from the transitional one in this multiscale network.