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In fluid mechanics the widely accepted boundary condition for viscous flows is the no-slip condition. When applied to fluid-structure interaction problems this condition states that the fluid velocity at the moving structure boundary is equal to the velocity of the structure boundary itself. Recent advances in mathematical analysis and experimental measurements have shown, however, that the no-slip condition is not adequate to capture certain important physical phenomena, such as, e.g., contact of smooth elastic or rigid solids interacting with an incompressible, viscous fluid. Problems of this type arise in, e.g., modeling closure of human heart valves, and in modeling collision of blood cells. The Navier slip condition, which captures the slip in the tangential component of velocity at the fluid-structure interface, was shown, however, to ``allow'' collision. Furthermore, the Navier slip condition has been shown to capture the correct physics of various phenomena arising from new technologies, such as the behavior of flows over hydrophobic surfaces (e.g., spray fabricated liquid repellant surfaces), and flows over ``rough'' boundaries, such as elastic tissue scaffolds. Despite a relatively rich literature on FSI problems with no-slip boundary condition, and numerous studies on the Navier slip condition at fixed, rigid fluid-solid interfaces, there are no analytical results or partitioned numerical schemes that deal with FSI between elastic bodies and incompressible, viscous fluids interacting through the Navier slip condition. In this talk we present a preliminary existence result and a partitioned numerical scheme for this class of problems. The existence proof is constructive, based on the time-discretization via operator splitting, used in the design of the numerical scheme. We effectively show that the numerical scheme converges to a weak solution of this FSI problem. (This is joint work with B. Muha (U of Zagreb) and M. Bukac (Notre Dame))