  In this article, we construct a novel 1D-model of microfluidic laminar flows
in tapered circular and rectangular channels assuming the flow in channels
fully developed. In the model, we take into account the inertance and dynamic
pressure terms. The model can be used for a wide range of flows: from the pure
capillary flow regime, where the capillary forces are the main driver of the
liquid in the channel, to the external pressure flow regime where the external
pressure applied to the liquid at the entrance to the channel is much larger
than the capillary pressure in the channel, so that the capillary force can be
ignored. We apply the model to rectangular Y-shape junctions, where the base
channel is connected to a reservoir and the end channels are exposed to
atmospheric air. We show that, in asymmetric Y-shape junctions, there can be a
time of meniscus arrest, where only one of the two channels with a smaller
radius fills, and, the other one, with a larger radius, is arrested. The time
of meniscus arrest decreases with an increase in the applied external pressure;
when this pressure becomes large enough, the meniscus arrest disappears.
However, if the ratio of the radii of the end channels is large, the meniscus
arrest does not appear because the flow resistance in the channel with the
smaller radius is much large than in the channel with the larger radius so that
the initial flow in the wider channel cannot be stopped by the differences in
the surface tension pressures in these channels. In this article, we also
investigate the applicability of the fully developed flow approximation assumed
in the model.
