We present a systematic study of capillary filling for multi-phase flows byusing mesoscopic lattice Boltzmann models describing a diffusive interfacemoving at a given contact angle with respect to the walls. We compare thenumerical results at changing the density ratio between liquid and gas phasesand the ratio between the typical size of the capillary and the interfacewidth. It is shown that numerical results yield quantitative agreement with theWashburn law when both ratios are large, i.e. as the hydrodynamic limit of ainfinitely thin interface is approached. We also show that in the initial stageof the filling process, transient behaviour induced by inertial effects and``vena contracta'' mechanisms, may induce significant departure from theWashburn law. Both effects are under control in our lattice Boltzmann equationand in good agreement with the phenomenology of capillary filling.
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