Accuracy, efficiency and robustness for rigid particle simulations in Stokes flow

Abstract: The thesis concerns simulation techniques for systems of nano- to micro-scaled rigid particles immersed in a viscous fluid, ubiquitous in nature and industry. With negligible fluid inertia, the set of PDEs known as the Stokes equations can be used to model the hydrodynamics. For a dynamic study, the PDEs have to be solved at any given instance of time, provided the particle configuration and any non-hydrodynamic interactions. The resulting particle velocities can then be used to update the particle coordinates, and the equations repeatedly solved anew. For any simulation result of a physical system to be reliable, it is crucial to control different error contributions, with two error types here particularly in focus: those related to solving the Stokes equations and those related to the update in time.The PDEs can be recast as boundary integral equations (BIEs) that hold on the particle surfaces. Hydrodynamic interactions are challenging: they are simultaneously long-ranged and expensive to resolve both in time and space for closely interacting particles. The latter is caused by strong lubrication forces resulting from bodies in relative motion. We approach two alternative and related techniques to BIEs that allow for more cost-effective simulations, namely the rigid multiblob method and the method of fundamental solutions. The former is a regularisation technique that allows for generally shaped particles in large systems, both with and without thermal fluctuations. We make two improvements: the basic error level is tied to the discretisation and set by solving a small optimisation problem off-line for each given particle shape, and the accuracy for closely interacting particles is improved by pair-corrections. With the method of fundamental solutions, we present a technique with linear or close to linear scaling in the number of particles, depending on if a so-called resistance or mobility problem is solved. For circles and spheres, the accuracy can be controlled to a target level independently of the particle separations. This is done by the introduction of a small set of image points for every pair of particles close to contact that well manage to represent lubrication forces.        In the model, particles can neither touch nor overlap, and our work on time-stepping is tied to the problem of contact avoiding. We develop a new strategy that guarantees contact free simulations in 3D, essential for studying the system of particles over long time spans.   Controlled accuracy in solutions to the Stokes equations can together with robust timestepping allow for simulations that can complement physical experiments of particle systems for a better understanding of their behaviour, to drive the development in fields such as materials science, biomedical engineering and environmental engineering.

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