A Lego Block Approach to Flow in Complex Microfluidic Networks

Abstract

We present a new way to construct analytical solutions for flow in complex microfluidic channel networks, as well as planar disordered media. Using a combination of Schwarz-Christoffel maps and segmentation techniques inspired by integrated circuit analysis, we build a library of base building blocks which can be reassembled to model complex geometries, in the style of ``Lego Blocks''. Our approach requires minimal numerical computation, and can then generate analytical solutions for any combination of inlet and outlet flow rates. Moreover, our method can tackle multiply connected domains which are usually difficult to model using typical conformal transform approaches. The solutions are developed for microfluidic Hele-Shaw cell devices, but also apply to ideal flow and Darcy flow in complex geometries, or any other flow problem adequately modeled by Laplace's equation. We end by showing how the procedure can be used to model complex disordered media, fractal-like flow geometries, as well as problems of steady advection-diffusion in microfluidic mixers.

Figures (8)

• Figure 1: Schematic representation of our approach. A complex microfluidic circuit is subdivided into simpler individual junction elements, which can then be analyzed individually and recombined. Conversely, a library of smaller elements can serve as building blocks for complex circuits.
• Figure 2: Transforming the boundary value problem in the channel junction element (w domain) to an equivalent problem in a disk domain (z domain). No-flux conditions are prescribed on the walls, and fixed value of the real potential are set on each inlet and outlet.
• Figure 3: Truncating a channel junction with outlets extending to infinity to create a finite piece. a. Illustration of a junction and the cut lines. b. The same flow problem in the disk domain. c. Zoomed-in version of the flow domain (red square in subfigure b) showing the image of the cut line
• Figure 4: Assembling unit elements to generate flow in a multiply connected domain. a. Schwarz-Christoffel maps precomputed for three individual building blocks. b. Assembling the three unit blocks to generate a complex circuit. Value of the potential at each element junction was determined by solving the equivalent resistive circuit problem. Blue lines are streamlines while red lines are the level sets of the real potential.
• Figure 5: Model porous media obtained by assembling junction unit elements