  We present a systematic study of capillary filling for a binary fluid by
using mesoscopic a lattice Boltzmann model describing a diffusive interface
moving at a given contact angle with respect to the walls. We compare the
numerical results at changing the ratio the typical size of the capillary, H,
and the wettability of walls. Numerical results yield quantitative agreement
with the Washburn law in all cases, provided the channel lenght is sufficiently
larger then the interface width. We also show that in the initial stage of the
filling process, transient behaviour induced by inertial effects are under
control in our lattice Boltzmann equation and in good agreement with the
phenomenology of capillary filling. Finally, at variance with multiphase LB
simulations, velocity and pressure profiles evolve under the sole effect of
capillary drive all along the channel.
