  The sedimentation of a heavy Stokes particle in a laminar plane or
axisymmetric flow is investigated by means of asymptotic methods. We focus on
the occurrence of Stommel's retention zones, and on the splitting of their
separatrices. The goal of this paper is to analyze under which conditions these
retention zones can form, and under which conditions they can break and induce
chaotic particle settling. The terminal velocity of the particle in still fluid
is of the order of the typical velocity of the flow, and the particle response
time is much smaller than the typical flow time-scale. It is observed that if
the flow is steady and has an upward streamline where the vertical velocity has
a strict local maximum, then inertialess particle trajectories can take locally
the form of elliptic Stommel cells, provided the particle terminal velocity is
close enough to the local peak flow velocity. These structures only depend on
the local flow topology and do not require the flow to have closed streamlines
or stagnation points. If, in addition, the flow is submitted to a weak
time-periodic perturbation, classical expansions enable one to write the
particle dynamics as a hamiltonian system with one degree of freedom, plus a
perturbation containing both the dissipative terms of the particle motion
equation and the flow unsteadiness. Melnikov's method therefore provides
accurate criteria to predict the splitting of the separatrices of the elliptic
cell mentioned above, leading to chaotic particle trapping and chaotic
settling. The effect of particle inertia and flow unsteadiness on the
occurrence of simple zeros in Melnikov's function is discussed. Particle motion
in a plane cellular flow and in a vertical pipe is then investigated to
illustrate these results.
