Global stability of trajectories of inertial particles within domains of instability

• Inertial particles in a fluid converge to streamlines in certain ‘stable’ subsets of the fluid domain and repelled from the streamlines in ‘unstable’ subsets. We show that such a classification is incomplete by demonstrating that the relative velocity of inertial particles can decay and they can converge to streamlines in ‘unstable’ sets too.
• The relative velocity of inertial particles can decay in unstable sets due to the rapid rotation of the eigenvectors of the deformation gradient along an inertial particles trajectory.
• A second mechanism for the decay of the relative velocity is due to heteroclinic trajectories of saddle fixed points in the fluid.
• The global stability of certain invariant regions in fluid could have significant applications in manipulation of inertial particles in microfluidics.

Finite sized particles exhibit complex dynamics that differ from that of the underlying fluid flow. These dynamics such as chaotic motion, size dependent clustering and separation can have important consequences in many natural and engineered settings. Though fluid streamlines are global attractors for the inertial particles, regions of local instability can exist where the inertial particle can move away from the fluid streamlines. Identifying and manipulating the location of the so called stable and unstable regions in the fluid flow can find important applications in microfluidics. Research in the last two decades has identified analytical criteria that can partition the fluid domain into locally stable and unstable regions. In this paper, we identify two new mechanisms by which neutrally buoyant inertial particles could exhibit globally stable dynamics in the regions of the fluid flow that are thought to be locally unstable and demonstrate this with examples. The examples we use are restricted to the simpler case of time independent fluid flows.