Local Interactions and the Global Organization of a Two-Phase Flow in a Branching Tree

The transport of liquid plugs in a microfluidic branching tree is studied experimentally. The global flow pattern can be either symmetric or asymmetric, with daughter plugs dividing in synchrony or asynchrony as a function of the driving flow rate and the network geometry. For trees with narrowing channels, the plugs always reach the exits even at low flow rates. In contrast, only one path is opened in networks with widening channels when the flow rate is low. This behavior is explained by a comparison of the pressure drop necessary to drive viscocapillary motion of plugs in straight channels with the nonlinear pressure variations as a plug passes a bifurcation. A model is built, which predicts that only narrowing networks can be fully filled, while widening networks can never be fully invaded by a two-phase flow.

Figure 1

Image of a narrowing network with 5 generations. Generation (bifurcation) numbers are labeled with Arabic (Roman) numerals. A constant flow rate is applied through the network inlet, at generation 0. Sixteen holes are made at the exits to fix the boundary condition at atmospheric pressure. The four paths along which measurements are made are marked.

Figure 2

Traveled distance along four paths as a function of time in the $\mathcal{N}$ network. Driving flow rates: (a) 2, (b) 5, and (c) $20\mu \mathrm{l}/\min $. The shaded and nonshaded areas represent successive generations.

Figure 3

Traveled distance in the $\mathcal{W}$ network. Driving flow rates: (a) 2, (b) 4, and (c) $20\mu \mathrm{l}/\min $.

Figure 4

Micrographs showing a liquid plug entering a bifurcation (a) and after touching the opposite wall (b). The radius of curvature at the front interface changes value and sign during this passage. (c) The variation in $P_{\mathrm{cap}}$ versus the distance of the front interface from the facing tip. The calculation is performed by using Eq. (2) and for bifurcation II in the $\mathcal{N}$ network. (d),(e) Mesoscale building block for exploring long-range interactions in the tree.

Figure 5

Comparison of threshold pressures at every bifurcation to the pressures imposed in the straight channels by the driving flow rates $Q=2$ and $20\mu \mathrm{l}/\min $. Labels on the $x$ axis denote the bifurcation number for the threshold and the generation number for the viscocapillary pressure. (a) $\mathcal{N}$ network and (b) $\mathcal{W}$ network.