  We construct a unique local strong solution to the finitely extensible
nonlinear elastic (FENE) dumbbell model of Warner-type for an incompressible
polymer fluid (described by the Navier-Stokes-Fokker-Planck equations)
interacting with a flexible elastic shell. The latter occupies the flexible
boundary of the polymer fluid domain and is modeled by a beam equation coupled
through kinematic boundary conditions and the balance of forces. In the 2D case
for the co-rotational Fokker-Planck model we obtain global-in-time strong
solutions.
  A main step in our approach is the proof of local well-posedness for just the
solvent-structure system in higher-order topologies which is of independent
interest. Different from most of the previous results in the literature, the
reference spatial domain is an arbitrary smooth subset of $\mathbb{R}^3$,
rather than a flat one. That is, we cover viscoelastic shells rather than
elastic plates. Our result also supplements the existing literature on the
Navier-Stokes-Fokker-Planck equations posed on a fixed bounded domain.
