Relative trajectories of two charged sedimenting particles in a Stokes flow

Abstract

We study the dynamics of two charged point particles settling in a Stokes flow.We find what ranges of initial relative positions and what ranges of system parameters lead to formation of stable doublets.The system is parameterized by the ratio of radii, ratio of masses and the ratio of electrostatic to gravitational force.We focus on opposite charges.We find a new class of stationary states with the line of the particle centers inclined with respect to gravity and demonstrate that they are always locallyasymptotically stable. Stability properties of stationary states with the vertical line of the particle centers are also discussed.We find examples of systems with multiple stable stationary states.We show that the basin of attraction for each stable stationary state has infinite measure, so that particles can capture one another even when they are very distant, and even if their charge is very small. This behavior is qualitatively different from the uncharged case where there only exists a bounded set of periodic relative trajectories.We determine the range of ratios of Stokes velocities and ratio masses which give rise to non-overlapping stable stationary states (given the appropriate ratio of electrostatic to gravitational force). For non-overlapping stable inclined or vertical stationary states the larger particle is always above the smaller particle. The non-overlapping stable inclined stationary states existonly if the larger particle has greater Stokes velocity, but there are non-overlapping stable vertical stationary states where the larger particle has higher or lower Stokes velocity.