Comparison Between Theory and Numerical Approach

In this section, we present a numerical approach based on the Surface Evolver numerical program [24]. Note that the Surface Evolver does not describe the dynamics of the motion, but iteratively relocates the interface to lower the energy. In the case of SCF, no equilibrium location exists, but we can still use Evolver to predict the direction of motion in SCF (but not the precise velocity of the motion). 

In fact. Surface Evolver calculates the static shape of the advancing interface, which depends on the geometry and contact angles with the walls. This shape corresponds to a Laplace pressure. Assuming that the upstream condition is zero pressure (large reservoir), if the front end of the flow has a negative Laplace pressure, the flow will continue if it has a zero curvature, the flow is stopped, if it has a positive Laplace pressure, the flow will recede. On one hand, at the onset of SCF, the velocities are 

Moreover, the interface relaxation time, characterized by the Tomotika time, is very small. Let us recall that the Tomotika time—or capillary time— noted is the time taken by a distorted liquid-air interface to 

At the microscale, using our typical numerical values, we obtain T 10 — 10 seconds. The capillary time is much smaller than the time taken by the flow to fill even a small distance of the channel. In summary, even if the quasi-static approach does not account for the dynamics, it produces plausible results because the capillary number is much smaller than unity and the Tomotika time much smaller than the flow time scale in the channel. Hence Evolver produces a realistic succession of steady-state location of the interface, but does not accoimt for the flow velocity. 

SCF in two different types of composite channels is investigated first a confined, rectangular mlcrochannel, and second two different geometries of open channels, i.e. channels having a boimdary with the surrounding air. 

---