  We present a computational framework to address the flow of two immiscible
viscous liquids which co-flow into a shallow rectangular container at one side,
and flow out into a holding container at the opposite side. Assumptions based
on the shallow depth of the domain are used to reduce the governing equations
to one of Hele-Shaw type. The distinctive feature of the numerical method is
the accurate modeling of the capillary effects. A continuum approach coupled
with a volume-of-fluid formulation for computing the interface motion and for
modeling the interfacial tension in Hele-Shaw flows is formulated and
implemented. The interface is reconstructed with a height-function algorithm.
The combination of these algorithms is a novel development for the
investigation of Hele-Shaw flows. The order of accuracy and convergence
properties of the method are discussed with benchmark simulations. A
microfluidic flow of a ribbon of fluid which co-flows with a second liquid is
simulated. We show that for small capillary numbers of O(0.01), there is an
abrupt change in interface curvature and focusing occurs close to the exit.
