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

TL;DR

This paper introduces a novel computational volume-of-fluid method for simulating co-flowing immiscible fluids in Hele-Shaw cells, accurately capturing capillary effects and interface dynamics.

Contribution

It develops a new numerical framework combining a volume-of-fluid approach with height-function interface reconstruction for Hele-Shaw flows, emphasizing capillary effects.

Findings

Accurate modeling of capillary effects in Hele-Shaw flows.

Observation of interface curvature change and focusing at small capillary numbers.

Validation of the method's convergence and accuracy through benchmark tests.

Abstract

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.