Dynamics for shapes with misaligned eigenframes of b_m and c

Determine the orientation dynamics during low-Reynolds-number sedimentation for rigid particles whose translation-rotation coupling tensor about the center of mobility (b_m) and rotational mobility tensor (c) have non-coincident eigenframes, i.e., the eigenvectors of b_m are not aligned with the eigenvectors of c.

Background

Throughout the paper, the sedimentation orientation dynamics are formulated in terms of the mobility tensors: the translation-rotation coupling tensor about the center of mobility (b_m) and the rotational mobility tensor (c). For the helical ribbons studied experimentally and numerically, b_m and c share the same eigenframe, enabling a comprehensive description of fixed points, symmetry constraints, and bifurcation surfaces in the space of center-of-force offsets.

The authors note that other shapes—such as bent disks and other di-bilaterals—may have b_m and c that share only one eigenvector or none, and while certain symmetric offsets can preserve reversibility, more general offsets and fully misaligned eigenframes are expected to yield different dynamics. The general classification and analysis of sedimentation dynamics when the eigenframes of b_m and c are misaligned remains unresolved.