On the asymptotic solution of the Maxey-Riley equation

The Maxey-Riley [Phys. Fluids 26, 883 (1983)] particle equation of motion is considered without the history term and for an asymptotically small Stokes number. The equation admits a globally attractive invariant manifold identified as the Eulerian particle velocity field asymptotically close to the unperturbed fluid velocity field, thus suppressing the inconsequential initial transients. A recursive asymptotic scheme is obtained for the calculation of the invariant manifold in any order of accuracy. The dimension of the particle equation on the invariant manifold is reduced by half, which considerably facilitates the analysis of its motion in physical space. Structural stability theory provides comprehensive qualitative description of the particle motion.