This thesis consists of two main parts related to the modeling and computation of elastic, immersed fiber-like structures in non-Newtonian flows. We focus on the particular case of a flexible, microscopic swimmer which is modeled as an immersed sheet of finite extent in a two-dimensional, incompressible viscoelastic flow at zero Reynolds number. The swimmer is imposed a beating pattern or gait based on a given target curvature. In the first part, we present a comprehensive numerical investigation of such a swimmer in an Oldroyd-B fluid. An efficient semi-implicit version of the Immersed Boundary Method is employed to remove the impeding time-step limitation, induced by the strong interfacial forces needed to appropriately impose the swimmer's gait and inextensibility. Our study investigates in detail, for the first time, the important effects of the domain size, the stiffness parameters enforcing the constraints, and numerical resolution and dissipation. It is found that when the gait is accurately enforced, the mean propulsion speed of the swimmer always decreases monotonically with Deborah number De, which is a measure of the flow's viscoelasticity, i.e. viscoelasticity hinders locomotion. We observe that this monotonic ordering can be broken when the enforcement of the gait is sufficiently relaxed but without viscoelastic swimmer's mean speed surpassing that of the Newtonian swimmer. More importantly, our investigation reveals that the addition of dissipation in the polymeric stress invariably enhances locomotion and can lead to a speed-up with respect to the Newtonian swimmer. This result clarifies and reconciles several seemingly contradictory existing numerical and experimental results and identifies diffusive transport of momentum via the addition of polymeric stress dissipation as the main mechanism which can produce a swimmer's speed-up in the viscoelastic fluid.

The second part of of this thesis presents a novel multi-scale approach for the computation of the same type of finite swimmer but in a FENE (Finitely Extensible Nonlinear Elastic) fluid. The FENE model overcomes the main limitation of the Oldroyd B fluid which is the possibility of infinite extensions to the polymer chains (modeled as elastic dumbbells) but it requires a full resolution of the dynamics in the polymer's configuration space to evaluate the polymeric stress. When coupled with a flow, this becomes a highly dimensional multi-scale problem which is common to most models of complex fluids arising from kinetic theory and represents a formidable computational challenge. Here, we exploit that the flow affected by the non-inertial swimmer is highly localized and hence an adaptive multi-resolution approach can be effectively implemented. We combine this multi-resolution strategy with a robust, spectral method to solve cost-effectively for the polymer chain dynamics in configuration space. The overall adaptive, multi-resolution method offers a significant computational improvement over the direct approach, being about seven faster for this particular application.