Orientation dynamics of a settling spheroid in simple shear flow: bifurcations and stochastic alignment

Abstract

We investigate the orientation dynamics of a settling spheroid in simple shear flow, combining a deterministic dynamical-systems analysis with a stochastic Fokker-Planck treatment. The dynamics is governed by the competition between the Jeffery torque from the background shear and the inertial torque from settling. For configurations in which gravity lies in the shear plane, the azimuthal dynamics reduces to overdamped motion in a tilted periodic potential controlled by a single effective parameter $\mathcal{R}$ that combines the particle shape anisotropy and the settling strength. A saddle-node bifurcation on an invariant circle (SNIC) at $\mathcal{R}=1$ governs the transition from sustained rotational motion to steady equilibrium, with the rotation period diverging as $(1-\mathcal{R})^{-1/2}$. When gravity is parallel to the vorticity axis, the attractor is a periodic orbit for all settling strengths. The stochastic analysis reveals that noise plays a fundamentally different role depending on whether settling-induced potential barriers are present: in the classical Jeffery problem it diffuses over the orbit constant, whereas with settling it drives Kramers-type phase slips whose rate is exponentially sensitive to the P\'{e}clet number, defined as the ratio of diffusive to convective time scales. Langevin simulations confirm the predicted intermittent dynamics, with phase slips becoming progressively rarer as the barrier height or P\'{e}clet number increases. Asymptotic results in both the small- and large-$\mathrm{Pe}$ limits, together with numerical solutions of the Fokker-Planck equation at arbitrary $\mathrm{Pe}$, quantify the orientation moments across all regimes.