Torques on curved atmospheric fibres

Abstract

Small particles are transported over long distances in the atmosphere, with significant environmental impact. The transport of symmetric particles is well understood, but atmospheric particles, such as curved microplastic fibres or ash particles, are generally asymmetric. This makes the description of their transport properties uncertain. Here, we derive a model for how planar curved fibres settle in quiescent air. The model explains that fluid-inertia torques may align such fibres at oblique angles with gravity as seen in recent laboratory experiments, and shows that inertial alignment is a general and thus important factor for the transport of atmospheric particles.

Figures (3)

• Figure 1: ( a) Particle geometry and definition of particle-fixed coordinate system $\hbox{\boldmath$p$}, \hbox{\boldmath$q$}, \hbox{\boldmath$n$}$. The angles $\theta$ and $\psi$ are Euler angles ($Z_\phi Y_\theta Z_\psi$ convention, see supplemental material SM for details). We refer to $\theta$ as the tilt angle, and $\psi$ as the spin angle. For $\phi\!=\!\theta\!=\!\psi\!=\!0$, the system $\hbox{\boldmath$p$}, \hbox{\boldmath$q$}, \hbox{\boldmath$n$}$ aligns with the lab system $\hat{{\bf e}}_1, \ldots, \hat{{\bf e}}_3$. Also shown is the origin $\hbox{\boldmath$x$}_0$ of the particle-fixed coordinate system, the centre-of-mass $\hbox{\boldmath$x$}_{\rm com}$, and the two radii of the fibre, $b$ and $R$. ( b) Stacked video stills of microplastic fibre settling in air with an oblique tilt angle, data from Ref. tatsii2024shape. Gravity $\hat{\hbox{\boldmath$g$}}$ points down. ( c) Illustration of the model used in the Stokes limit, consisting of hydrodynamically interacting beads.
• Figure 2: Inertial torques for the semi-circular fibre. Shown are the angular dependencies of $T^{{(1)}}_p/T_p^{({1},{\rm max})}$, $T^{{(1)}}_q/T_q^{({1},{\rm max})}$, and $T^{{(1)}}_n/T_n^{({1},{\rm max})}$ of the inertial torque around the three axes of the particle-fixed coordinate system, obtained from Eq. (\ref{['eq:torque_re']}) for $\hat{\hbox{\boldmath$v$}} = -\hat{\bf e}_3$ ($\hat{v}_p = \sin\theta \cos\psi, \hat{v}_q =-\sin\theta\sin\psi$, and $\hat{v}_n = -\cos\theta$). Colourbar: from $-1$ (white) to $1$ (violet).
• Figure 3: Comparison between theory and experiment (Table \ref{['tab:exp_summary']}). ( a) Full slender-body theory (see text) for $v_g^\ast$ for semi-circular fibres versus the inertia parameter $\mathscr{RV}$ ($\Box$); experimental $\langle v_g\rangle$ (red $\triangle$). ( b) same, but for quarter circles. ( c) Theory for steady-state tilt angle $\theta^\ast$ for semi-circular fibres ($\Box$). Estimated theoretical uncertainties (see text) are shown when larger than the symbol size. Experiment (red $\triangle$) with errors from Table \ref{['tab:exp_summary']}. Dashed line corresponds to $\theta^\ast = \tfrac{\pi}{2}$. ( d) same, but for quarter circles (theoretical uncertainty not shown).