Fluid-inertia torques from particle-shape symmetry

TL;DR

This work determines how particle-shape symmetry constrains hydrodynamic torque on particles settling in a quiescent fluid at small but finite ${\rm Re}_{\rm p}$. By combining symmetry analysis with matched asymptotic expansions, it derives the allowed tensor structures and separates singular (outer) and regular (inner) contributions to the ${\rm Re}_{\rm p}$ corrections, showing that breaking reflection or rotation symmetries increases the number of independent torque parameters. The results extend classical results by Brenner and Cox to unsteady motion and provide a general framework for parameterizing inertial torques for complex shapes, with implications for atmospheric ice crystals and curved fibres. Across symmetry classes (e.g., ${\rm O}_{h}$, ${\rm D}_{2h}$, ${\rm C}_{\infty v}$, ${\rm C}_{2v}$), the authors enumerate nonzero coupling tensors and verify with nearly spherical shapes that the symmetry predictions hold, though some terms vanish at a given order due to geometric or center-of-mass considerations. These insights illuminate how symmetry-breaking affects orientation and transient dynamics, and offer a foundation for improved modeling of non-spherical particles in weakly inertial regimes.

Abstract

Numerical simulation of particle motion in fluids at low particle Reynolds numbers is often based on empirical force and torque models obtained by fitting force and torque from ab-initio computations for simple particle shapes such as spheres, spheroids, or cylindrical disks and fibres. To do the same for more complex particles shapes, one needs to first know how particle shape constrains the dependence of force and torque on flow velocity, its gradient, and on particle orientation. Here we use symmetry analysis and perturbation theory to determine the form of the hydrodynamic torque on a particle settling in a quiescent fluid at low but non-zero particle Reynolds numbers, for particle shapes with different point-group symmetries. The symmetry conclusions are verified by comparing with explicit calculations for nearly spherical particles.

Figures (2)

• Figure 1: Particle with $C_{2v}$ shape symmetry moving through a quiescent fluid with velocity $\hbox{\boldmath$v$}$. The particle has the shape of a bent disk miara2024dynamics. It has one $C_2$-rotation axis $\hbox{\boldmath$n$}$ and two reflection symmetries $\hbox{\boldmath$p$} \to -\hbox{\boldmath$p$}$ and $\hbox{\boldmath$q$} \to -\hbox{\boldmath$q$}$. Here $\hbox{\boldmath$p$}, \hbox{\boldmath$q$}, \hbox{\boldmath$n$}$ are the basis vectors of a coordinate system that rotates with the particle. The basis vectors of the laboratory coordinate system are denoted by $\hat{\bf e}_1,\hat{\bf e}_2, \hat{\bf e}_3$, and the velocity is $\hbox{\boldmath$v$} = -v \hat{\bf e}_3$. Also shown are the Euler angles $\theta$, $\psi$ that specify particle orientation. We use the $Z_\phi Y_\theta Z_\psi$ convention, as described in the supplemental material of Ref. candelier2024torques.
• Figure 2: Particle-shape symmetries considered in Section \ref{['sec:symmetries']} (see that Section for the names of the six point groups). In panel ( a), the orientation of the body-centered coordinate system $\hbox{\boldmath$p$}$, $\hbox{\boldmath$q$}$, $\hbox{\boldmath$n$}$ is shown (same for other panels). Point-group symmetries represented as in Figs. 7.2 and 7.3 in Ref. Ashcroft.