In this preprint, we propose a fully coupled partitioned scheme to embed fibers (modeled by beams) into three-dimensional incompressible Navier-Stokes flow. The discretization of the truly 1D/3D coupling constraints utilizes a mortar-type approach and a penalty regularization. We solve the coupled problem by either an Aitken acceleration scheme or by a Matrix-Free Newton-Krylov (MFNK) approach.