Folding flakes: the deformation of elastic sheets in a viscous fluid

Abstract

The production, handling and processing of microscale materials, which is routinely performed in a liquid environment, involves a rich variety of processes in the Stokes limit of vanishing inertia, e.g., the segregation of 2D materials by centrifugation, blood testing methods, microalgae harvesting, and the separation of microplastics from natural sediment. However, the question of how the shape of a particle couples to its dynamics remains largely unanswered, except for highly symmetric cases. In particular, little is known about the sedimentation of non-planar two-dimensional objects, rigid or flexible. In this thesis, we study the sedimentation of a non-planar U-shaped rigid disk and of an elastic disk which deforms into a U-shape under the influence of viscous and elastic forces. Using a combination of experiments and the theoretical framework introduced by H. Brenner, we show that a rigid U-shaped disk possessing two planes of symmetry exhibits rich reorientation dynamics, making its centre of mass follow complex quasi-periodic spirals. We show that inertialess settling of any particle in an unbounded fluid is always governed by no more than two orientational degrees of freedom, which precludes the possibility of chaos. We analyse the experimental data and extract the relevant entries of the mobility matrix with which we predict the long-term dynamics, showing that in this orientational phase plane, the trajectories are periodic orbits, orbiting four centre fixed points. The phase-plane orbits are associated with a repeating sequence of pitching and rolling motions of the disk. We extend the study to a U-shaped disk with only one plane of symmetry. We extract the relevant entries of the mobility matrix and show that breaking that one plane of symmetry results in two fixed points becoming stable spirals and two becoming unstable spirals. In the case of an elastic disk, viscous loading from the fluid deforms the disk which, in turn, alters the flow. This elastohydrodynamic coupling leads to a myriad of complex reorientation and sedimentation paths set by its initial orientation, and shape. We show that in certain configurations, the elastic disk behaves similarly to the rigid case, exhibiting rolling towards an upright U orientation and pitching towards an edge-on motion. However, we identify several behaviours unique to the elastic case, including complex curvature inversion, self-opening of one of its ends and reconfiguration of its spine. This leads us to hypothesise that the elastic disk exhibits a periodic sequence of rolling, pitching and spine reconfiguration. However, nonlinear regression aided by a simple machine-learning model indicates that such a period would require a tank thousands of times taller and hundreds of times wider than the radius of the disk, which is unlikely to be feasible to realise experimentally.

Date of Award	14 Jun 2024
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Anne Juel (Main Supervisor), Draga Pihler-Puzovic (Co Supervisor) & Matthias Heil (Co Supervisor)