The evolution of engineering design processes from experience-based practices to numerical strategies has brought to the forefront the challenges inherent in the flows of complex fluids within computational domains. Isogeometric Analysis (IGA) stands as a pivotal response to these challenges. This transformation is essential not only for laying the groundwork for advanced simulations but also for exerting a substantial influence on structural design and the optimization landscape. This talk will introduce robust parameterization methodologies that rise to meet the demands of complex fluid flow simulations, particularly through the generation of high-quality domain parameterizations from CAD boundary representations. Initially, we will discuss barrier-type optimization-based methods and PDE-based parameterization techniques, which have demonstrated significant utility in industrial applications such as the mesh generation for twin-screw compressors. These methods are incorporated into the Geometry + Simulation Modules (G+Smo) library, underscoring the synergy between geometric design and simulation in isogeometric analysis. Subsequently, the focus will shift to the crucial role of boundary correspondence in capturing the intricacies of fluid dynamics in slender and elongated geometries. A novel application of Schwarz-Christoffel mapping will be presented, illustrating how this conformal mapping variant can maintain geometric integrity in parameterizations—a salient factor in the accurate simulation and optimization of fluid flow.