  Capillary droplets form due to surface tension when two immiscible fluids are
mixed. We describe the motion of gravity-driven capillary droplets flowing
through narrow constrictions and obstacle arrays in both simulations and
experiments. Our new capillary deformable particle model recapitulates the
shape and velocity of single oil droplets in water as they pass through narrow
constrictions in microfluidic chambers. Using this experimentally validated
model, we simulate the flow and clogging of single capillary droplets in narrow
channels and obstacle arrays and find several important results. First, the
capillary droplet speed profile is nonmonotonic as the droplet exits the narrow
orifice, and we can tune the droplet properties so that the speed overshoots
the terminal speed far from the constriction. Second, in obstacle arrays, we
find that extremely deformable droplets can wrap around obstacles, which leads
to decreased average droplet speed in the continuous flow regime and increased
probability for clogging in the regime where permanent clogs form. Third, the
wrapping mechanism causes the clogging probability in obstacle arrays to become
nonmonotonic with surface tension $\Gamma$. At large $\Gamma$, the droplets are
nearly rigid and the clogging probability is large since the droplets can not
squeeze through the gaps between obstacles. With decreasing $\Gamma$, the
clogging probability decreases as the droplets become more deformable. However,
in the small-$\Gamma$ limit the clogging probability increases, since the
droplets are extremely deformable and cause clogs as they wrap around the
obstacles. The results from these studies are important for developing a
predictive understanding of capillary droplet flows through complex and
confined geometries.
