Boundary elements method for microfluidic two-phase flows in shallow channels

TL;DR

This paper develops a boundary element method for simulating two-phase microfluidic flows in shallow channels, incorporating stabilization for surface tension effects, and validates it against analytical solutions and experiments.

Contribution

It introduces a stabilized boundary element approach for two-phase flows in microchannels using depth-averaged Brinkman equations, enabling efficient simulation of complex flow phenomena.

Findings

Validated against analytical solutions and experiments.

Successfully simulated Saffman-Taylor instability.

Demonstrated stable numerical scheme at low capillary numbers.

Abstract

In the following work we apply the boundary element method to two-phase flows in shallow microchannels, where one phase is dispersed and does not wet the channel walls. These kinds of flows are often encountered in microfluidic Lab-on-a-Chip devices and characterized by low Reynolds and low capillary numbers. Assuming that these channels are homogeneous in height and have a large aspect ratio, we use depth-averaged equations to describe these two-phase flows using the Brinkman equation, which constitutes a refinement of Darcy's law. These partial differential equations are discretized and solved numerically using the boundary element method, where a stabilization scheme is applied to the surface tension terms, allowing for a less restrictive time step at low capillary numbers. The convergence of the numerical algorithm is checked against a static analytical solution and on a dynamic test case. Finally the algorithm is applied to the non-linear development of the Saffman-Taylor instability and compared to experimental studies of droplet deformation in expanding flows.