CAIMS Prize Award

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Institute of Applied Mathematics, University of British Columbia

Supervisor:  B. Wetton

Thesis title:  Analysis and Computation of Immersed Boundaries, with Application to Pulp Fibres


Immersed fibres are a very useful tool for modeling moving, elastic interfaces that interact with a surrounding fluid. The "Immersed Boundary Method" is a computational tool based on the immersed fibre model which has been used successfully to study a wide range of applications including blood flow in the heart and arteries and motion of suspended particles.

This work centres around a linear analysis of an isolated fibre in two dimensions, which pinpoints a discrete set of solution modes associated solely with the fibre. We investigate the stability and stiffness characteristics of the fibre modes and then relate the results to the severe time step restrictions experienced in explicit and semi-implicit immersed boundary computations. A subset of the modes corresponding to tangential oscillations of the fibre are the main source of stiffness, which intensifies when the fibre force is increased or fluid viscosity is decreased -- this explains why computations are limited to unrealistically small Reynolds numbers.

We also investigate the effects of smoothing the fibre over a given thickness, which corresponds to the delta function approximation that is central to the discrete scheme. The results can be applied to explore the accuracy of various interpolation methods in an idealised setting.

The analysis is next extended to predict time step restrictions and convergence rates for various explicit and semi-implicit discretisations. The results are verified in numerical experiments.

Finally, we introduce a novel application of the Immersed Boundary Method to the motion of pulp fibres in a two-dimensional shear flow. The method is shown to reproduce both theoretical results and experimentally observed behaviour over a wide range of parameter values.