STL-to-Stokeslet Computation of Mobility Tensors and Sedimentation Dynamics for Shaped Particles

TL;DR

This work offers a user-friendly and efficient framework to compute the mobility tensor and resulting particle dynamics, and facilitates the design of steerable particles under diverse external forces, with relevance to colloidal transport, biological locomotion, diffusion, and self-assembly.

Abstract

Sedimentation is extremely common in nature, occurring throughout the atmosphere and oceans, and in every laboratory centrifuge. The shape and mass distribution of a particle uniquely determines its motion at low Reynolds number, and complex dynamics can emerge from even simple particle shapes. The dynamics are governed by the particle's hydrodynamic mobility tensor, which dictates the translational and rotational velocities given the forces and torques. However, to date the inference of the mobility tensor from the object shape has been cumbersome and tricky. Starting with an input file representing an object for a 3D printer, such as an STL file, here we present an efficient numerical framework to compute the mobility tensor by discretizing the particle surface into distributed point drag forces called stokeslets. We validate our results against analytical solutions of simple geometries and recent experimental measurements. With our calculated mobility tensors in hand, using standard transformation laws, we demonstrate the dramatic effect of shifting the center of mass from a center of symmetry: all initial orientations evolve into one, two, or three particular final motions dictated by the object. By providing a user-friendly and efficient framework to compute the mobility tensor and resulting particle dynamics, this work offers a broadly applicable tool for the soft-matter, fluid-mechanics, and biophysics communities, and facilitates the design of steerable particles under diverse external forces, with relevance to colloidal transport, biological locomotion, diffusion, and self-assembly.

Figures (4)

• Figure 1: Sedimentation of arbitrarily shaped particles.(a–d) Input geometries from STL files (left) and their stokeslet discretizations (middle) for a DNA-shaped helicoid, a chiral snowflake, a dolphin-like body, and a fractal ice grain nicolov2025dynamics, respectively. The right panel shows a zoomed-in view of the highlighted region, illustrating the local distribution of the stokeslets on the discretized surface. (e–h) Predicted CoM sedimentation trajectories under gravity for the four shapes, shown in the $x$--$y$ plane (left), in 3D (middle), and the corresponding force-vector trajectories on the unit sphere (right). Different colors denote different initial orientations. Chiral particles generate helical or spirograph-like trajectories with closed orbits on the unit sphere, while asymmetric shapes such as the dolphin and the ice grain exhibit biased or helical motion whose force-vector trajectories flow to a fixed point. Magenta points mark the center of reaction (CoR), about which the $[b]{\uuline{\hbox{B}}}$ is symmetric.Black points mark the CoM. For particles (a) and (b), the force center and CoM coincide, the $[b]{\uuline{\hbox{B}}}$ tensor is symmetric, and the rotation direction follows closed orbits. For particles (c) and (d), the CoM was defined assuming uniform density, the resulting $[b]{\uuline{\hbox{B}}}$ have a large antisymmetric component, and all initial states converge to the same final rotation axis.
• Figure 2: Validation of the STL-to-Stokeslet method for a sphere and a thin disk.(a) Sphere discretized with increasing numbers of stokeslets. The mobility error is defined in Eq. \ref{['error']} and is evaluated as a function of the stokeslet radius factor $\beta$. As the number of stokeslets increases, the error decreases and consistently exhibits a minimum near $\beta=0.25$. (b) Thin circular disk discretized with increasing numbers of Stokeslets. As in the spherical case, the mobility error decreases with increasing $N$. (c) Analytical prediction of the horizontal-to-vertical velocity ratio, $\rho(\alpha)$, for a thin disk as a function of the tilt angle $\alpha$. The inset illustrates the geometric definition of $\alpha$ and the decomposition of the sedimentation velocity into vertical and horizontal components. (d) Enlarged view of the highlighted region in (c) near the maximum of $\rho(\alpha)$. As $N$ increases, $\alpha$ and $\rho(\alpha)$ converge to the analytical predictions ($\alpha^\ast = 39.2^\circ$, $\rho^\ast(\alpha^\ast)=0.204$) happel2012low.
• Figure 3: Validation of the STL-to-Stokeslet method for complex geometries.(a--c) Comparison of our computational results with reference sedimentation trajectories reported in the recent literature for three particles: helical ribbons of different lengths and a U-shaped disk. Left panels: Predicted CoM trajectories in the $x$ -- $y$ plane for different initial orientations. Middle panels: 3D sedimentation trajectories for the same particles. Right panels: Corresponding force-direction trajectories on the unit sphere. Magenta and black points denote the CoM and CoR, respectively.
• Figure 4: Effect of CoM shift on sedimentation dynamics.(a) Schematic illustration of a double-paddle particle and its principal orthogonal body axes. $(\mathbf e_1,\mathbf e_2,\mathbf e_3)$. A CoM shift is introduced by displacing the mass center either along the body axis $\mathbf e_1$ ($\mathbf{d}_1$) or along a direction not aligned with any principal body axis ($\mathbf{d}_2$). (b) Sedimentation dynamics of the particle with no CoM shift. Left: CoM trajectories in the $x^*$--$y^*$ plane. Middle: 3D sedimentation trajectories. Right: Force-direction trajectories on the unit sphere. The unshifted particle exhibits closed orbits, consistent with chiral sedimentation behavior. (c) Sedimentation dynamics for CoM shifts along $\mathbf{d}_1$, with magnitude $|\mathbf{d}_1| = 2\%\, R$, where $R$ is the radius vector of the circumscribed sphere of the particle. (d) Sedimentation dynamics for CoM shifts along $\mathbf{d}_2$. Even a $0.2\%$ offset dramatically changes the trajectories, producing spiraling trajectories and the force direction paths converge to two fixed points on the unit sphere. Magenta and black points denote the CoM and CoR, respectively.