Chaotic saddles in a gravitational field: The case of inertial particles in finite domains

Abstract

The motion of inertial particles is investigated numerically in a time-periodic flow in the presence of gravity. The flow is restricted to a finite (or semi-infinite) vertical column, and the dynamics is therefore transiently chaotic. The long-term motion of the center of mass is a uniform settling. The settling velocity is found to differ from the one that would characterize a still fluid, and the distribution of an ensemble of settling particles spreads with a well-defined diffusion coefficient. The underlying chaotic saddle appears to have a height-dependent fractal dimension. The coarse-grained density of both the natural measure and the conditionally invariant measure (defined along the unstable manifold) of the saddle is smooth, and exhibits a local maximum as a function of the height. The latter density corresponds to the eigenfunction of the first eigenvalue of an effective Fokker-Planck equation subject to an absorbing boundary condition at the bottom. The transport coefficients can be determined as averages taken with respect to the conditionally invariant measure.