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We present a computational framework to address the flow of two immiscibleviscous liquids which co-flow into a shallow rectangular container at one side,and flow out into a holding container at the opposite side. Assumptions basedon the shallow depth of the domain are used to reduce the governing equationsto one of Hele-Shaw type. The distinctive feature of the numerical method isthe accurate modeling of the capillary effects. A continuum approach coupledwith a volume-of-fluid formulation for computing the interface motion and formodeling the interfacial tension in Hele-Shaw flows is formulated andimplemented. The interface is reconstructed with a height-function algorithm.The combination of these algorithms is a novel development for theinvestigation of Hele-Shaw flows. The order of accuracy and convergenceproperties of the method are discussed with benchmark simulations. Amicrofluidic flow of a ribbon of fluid which co-flows with a second liquid issimulated. We show that for small capillary numbers of O(0.01), there is anabrupt change in interface curvature and focusing occurs close to the exit.

A volume-of-fluid formulation for the study of co-flowing fluids governed by the Hele-Shaw equations