  Starting from a nonlinear 2D/1D fluid-structure interaction problem between a
thin layer of a viscous fluid and a thin elastic structure, on the vanishing
limit of the relative fluid thickness, we rigorously derive a sixth-order
thin-film equation describing the dynamics of vertical displacements of the
structure. The procedure is essentially based on quantitative energy estimates,
quantified in terms of the relative fluid thickness, and a uniform no-contact
result between the structure and the solid substrate. The sixth-order thin-film
equation is justified in the sense of strong convergence of rescaled structure
displacements to the unique positive classical solution of the thin-film
equation. Moreover, the limit fluid velocity and the pressure can be expressed
solely in terms of the solution to the thin-film equation.
