  Recent experiments on mucociliary clearance, an important defense against
airborne pathogens, have raised questions about the topology of two-dimensional
(2D) flows. We introduce a framework for studying ensembles of 2D
time-invariant flow fields and estimating the probability for a particle to
leave a finite area (to clear out). We establish two upper bounds on this
probability by leveraging different insights about the distribution of flow
velocities on the closed and open streamlines. We also deduce an exact
power-series expression for the trapped area based on the asymptotic dynamics
of flow-field trajectories and complement our analytical results with numerical
simulations.
