  We study a nonlinear, moving boundary fluid-structure interaction problem
between an incompressible, viscous Newtonian fluid, modeled by the 2D
Navier-Stokes equations, and an elastic structure modeled by the shell or plate
equations. The fluid and structure are coupled via the {\em Navier slip
boundary condition} and balance of contact forces at the fluid-structure
interface. The slip boundary condition is more realistic than the classical
no-slip boundary condition in situations, e.g., when the structure is "rough",
and in modeling dynamics near, or at a contact. Cardiovascular tissue and
cell-seeded tissue constructs, which consist of grooves in tissue scaffolds
that are lined with cells, are examples of "rough" elastic interfaces
interacting with and incompressible, viscous fluid. The problem of heart valve
closure is an example of a fluid-structure interaction problem with a contact.
We prove the existence of a weak solution to this class of problems by
designing a constructive proof based on the time discretization via operator
splitting. This is the first existence result for fluid-structure interaction
problems involving elastic structures satisfying the Navier slip boundary
condition
