Exact solution for the motion of a rigid particle with $\boldsymbol{S_4}$ and $\boldsymbol{C_{2v}}$ symmetry settling under gravity in a viscous fluid

Abstract

We provide an exact and complete solution for the dynamics of a rigid particle of uniform density with $C_{2v}$ and $S_4$ symmetry, settling under gravity in a viscous fluid at a Reynolds number much smaller than unity. The $S_4$ symmetry renders the problem exactly integrable, with all orbits labelled by a single conserved quantity $0\le C\le 1$. We show that, for $0<C<1$, there are two different time scales, which lead to quasi-periodic evolution, with a significant time-dependent horizontal displacement. We obtain the tilt $θ$ and spin $ψ$ Euler angles as periodic Jacobi elliptic functions of time, and the azimuthal angle $ϕ$ through an incomplete elliptic integral of the third kind with a complex characteristic, which splits $ϕ$ into a uniform drift and a strictly periodic modulation. The orientation period and the drift rate (corresponding to a constant angular velocity around the gravity direction) follow in a closed form. The vertical centre-of-mass displacement is obtained in terms of periodic in time incomplete elliptic integrals, and the periodic orbit-averaged settling velocity reduces to a single ratio of complete elliptic integrals. The horizontal component of the motion in the laboratory frame of reference is obtained exactly and algebraically in terms of all three Euler angles, so it is quasi-periodic. The horizontal component of the centre-of-mass position traces rosette-like, in general open curves confined by two concentric `envelope' circles. We determine very simple exact expressions for radii of both envelopes and demonstrate that they tend to infinity for a family of shapes with the rotation-translation coupling decreasing to zero. We also explain the origin of cusps at the rosette-like trajectory and provide a commensurability condition selecting strictly periodic rosettes.