Contact-free annular seals are used in turbomachinery to reduce leakage flows between areas of different fluid pressures, whilst allowing shaft rotation. In the last fifteen years, a new type of seal was developed, which uses purposefully inserted structural stator flexibility. Thus, the structure is able to slightly deform and can consequently adapt to rotor motion: the inherent risk of rotor-to-stator contact is lowered. This allows for a smaller sealing clearance, therefore lowering leakage and increasing overall machine efficiency. The influence of structural flexibility on leakage performance of these new designs is well investigated within the literature. However, profound research of the influence on non-linear dynamics is scarce. Thus, the main objective of this work is the investigation of the influence of seal flexibility on non-linear rotordynamics. In that respect, two points are of central interest: firstly, the modelling of the fluid force acting on the rotor, with the force resulting from the fluid flow through the sealing clearance. Secondly, the combined mass imbalance excitation and fluid force excitation of the rotor, which generally leads to quasi-periodic oscillations. To adress these issues, the work is divided into three parts: in the first part, the rotordynamic model and models for turbulent incompressible lubrication flow are presented. On the basis of simulations of the used bulk-flow model, simplified ODE models are built for describing the fluid forces. These allow for an efficient incorporation into the rotordynamic model. The second part is dedicated to the theory and derivation of numerical tools. For the computation of quasi-periodic stationary solutions, a Fourier-Galerkin method based on a torus manifold approach is derived. For lubrication flow simulation, the bulk-flow equations are discretised in space and the resulting equations are solved as DAE system. In the third part, the basic phenomena of the models are discussed. Apart from representative results of the fluid flow and force simulations, the solution stability and solution curves of the flexible seal-rotor system are investigated for a wide variety of parameters. The main observations are an increase in stable operation range due to the addition of flexibility and complex non-linear dynamic behaviour involving amongst others resonance phenomena, quasi-periodic solutions as well as Hopf, fold, Neimark-Sacker and Bautin bifurcations.