Justification of a nonlinear sixth-order thin-film equation as the reduced model for a fluid–structure interaction problem

Abstract

Starting from a nonlinear 2D/1D fluid–structure interaction problem between a thin layer of a viscous fluid and a thin elastic structure, in 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 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 limiting fluid velocity and pressure can be expressed solely in terms of the solution to the thin-film equation.