Where do inertial particles go in fluid flows?

We derive a general reduced-order equation for the asymptotic motion of finite-size particles in unsteady fluid flows. Our inertial equation   is a small perturbation of passive fluid advection on a globally attracting slow manifold. Among other things, the inertial equation implies that particle clustering locations in two-dimensional steady flows can be described rigorously by the QQ parameter, i.e., by one-half of the squared difference of the vorticity and the rate of strain. Use of the inertial equation also enables us to solve the numerically ill-posed problem of source inversion, i.e., locating initial positions from a current particle distribution. We illustrate these results on inertial particle motion in the Jung–Tél–Ziemniak model of vortex shedding behind a cylinder in crossflow.