We present a numerical investigation of the sedimentation dynamics of thin, deformed circular disks sedimenting freely under gravity in an otherwise quiescent, 3D viscous fluid at low Reynolds number. The sharp edge of such an object creates a singularity in the fluid pressure and the velocity gradient. At low Reynolds number, this means a significant contribution of the total drag comes from the vicinity of the disk edge; for a flat disk at zero Reynolds number, 30\% of the total drag is generated by the outermost 5\% of the disk radius. This implies that any under-resolution of the pressure and the velocity gradient leads to critical errors in the sedimentation velocity and rotation, but such singularities have a severe impact on the convergence rate of standard finite-element (FE) discretisations. To address this, we develop a novel augmented FE method which allows analytic (singular) functions of unknown amplitude to be subtracted from the full solution in a sub-domain around the disk edge, rendering the FE remainder part of the solution regular and thus restoring the standard FE convergence rate. The amplitudes of the singular functions are determined via PDE-constrained minimisation of a suitably chosen functional which captures the key signatures of the singularity. This method is then utilised in a fluid-structure interaction framework to examine the behaviour of two distinct classes of disk shape, namely cylindrically and conically deformed disks with one and two planes of symmetry, respectively. We find periodic orbits in the trajectories of the disks with two planes of symmetry, about a centre which corresponds to steady helical motion of the centre of mass. This behaviour is driven by the presence of two saddle points in a suitably chosen phase-space. They correspond to a 'U' shaped orientation and the inverted 'U' configuration. We explore the limiting behaviour as the curvature tends to zero towards the special case of the flat disk, and find a power-law decay of the norm of the coupling tensor, which quantifies the torque induced by translation of the particle. We then explore disks with one plane of symmetry, and find that the loss of symmetry introduces one stable and one unstable orientation, with the more tightly curved end point downwards and upwards, respectively. Finally, we examine the limiting behaviour as the asymmetry tends to zero, i.e. towards the special case of the cylindrically deformed disk. We discover a critical level of asymmetry at which there is a bifurcation in phase space; the two fixed points split into pairs of sources and sinks, into/away from which neighbouring trajectories converge. The basins of attraction of the two attractors is separated by a newly formed saddle point. A second bifurcation occurs at a slightly lower level of asymmetry, whereby the fixed points become spiral nodes; neighbouring trajectories spiral into/away form these points in phase space. At the limiting case, there is a third bifurcation; the stable and unstable fixed points become neutrally stable centres, and the spiral trajectories form closed periodic orbits.
- rigid-body dynamics
- semi-analytic finite element method
- fluid dynamics
- sedimentation dynamics
The Sedimentation Dynamics of Thin, Rigid Disks
Vaquero-Stainer, C. (Author). 1 Aug 2023
Student thesis: Phd