Modelling flows of complex fluids using the immersed boundary method

This thesis is concerned with fluid-structure interaction problems using the immersed boundary method (IBM). Fluid-structure interaction problems can be classified into two categories: a remeshing approach and a fixed-grid approach. Both approaches consider the fluid and structure separately and then couple them together via suitable interface conditions. A common choice of remeshing approach is the Arbitrary-Eulerian-Lagrangian (ALE) technique. Whilst the ALE method is a good choice if deformations are small, it becomes computationally very expensive if deformations are large. In such a scenario, one turns to a fixed-grid approach. However, the issue with a fixed-grid approach is the enforcement of the interface conditions. An alternative to the remeshing and fixed-grid approach is the IBM. The IBM considers the immersed elastic structure to be part of the surrounding fluid by replacing the immersed structure with an Eulerian force density. Therefore, the interface conditions are enforced implicitly. This thesis applies the finite element immersed boundary method (IBM) to both Newtonian and Oldroyd-B viscoelastic fluids, where the fluid variables are approximated using the spectral element method (hence we name the method the spectral element immersed boundary method (SE-IBM)) and the immersed boundary variables are approximated using either the finite element method or the spectral element method. The IBM is known to suffer from area loss problems, e.g. when a static closed boundary is immersed in a fluid, the area contained inside the closed boundary decreases as the simulation progresses. The main source of error in such a scenario can be found in the spreading and interpolation phases. The aim of using a spectral element method is to improve the accuracy of the spreading and interpolation phases of the IBM. We illustrate that the SE-IBM can obtain better area conservation than the FE-IBM when a static closed boundary is considered. Also, the SE-IBM obtains higher order convergence of the velocity in the L2 and H1 norms, respectively. When the SE-IBM is applied to a viscoelastic fluid, any discontinuities which occur in either the velocity gradients or the pressure, introduce oscillations in the polymeric stress components. These oscillations are undesirable as they could potentially cause the numerics to break down. Finally, we consider a higher-order enriched method based on the extended finite element method (XFEM), which we call the eXtended Spectral Element Method (XSEM). When XSEM is applied to the SE-IBM with a viscoelastic fluid, the oscillations present in the polymeric stress components are greatly reduced