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We perform a three-dimensional study of steady state viscous fingers thatdevelop in linear channels. By means of a three-dimensional Lattice-Boltzmannscheme that mimics the full macroscopic equations of motion of the fluidmomentum and order parameter, we study the effect of the thickness of thechannel in two cases. First, for total displacement of the fluids in thechannel thickness direction, we find that the steady state finger iseffectively two-dimensional and that previous two-dimensional results can berecovered by taking into account the effect of a curved meniscus across thechannel thickness as a contribution to surface stresses. Secondly, when a thinfilm develops in the channel thickness direction, the finger narrows withincreasing channel aspect ratio in agreement with experimental results. Theeffect of the thin film renders the problem three-dimensional and resultsdeviate from the two-dimensional prediction.

Three-dimensional aspects of fluid flows in channels. II. Effects of Meniscus and Thin Film regimes on Viscous Fingers