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

TL;DR

This paper rigorously derives a sixth-order thin-film equation as a reduced model for a fluid-structure interaction problem involving a viscous fluid and an elastic structure, valid in the thin fluid layer limit.

Contribution

It provides a rigorous mathematical derivation and justification of a nonlinear sixth-order thin-film equation from a coupled fluid-structure interaction model.

Findings

Strong convergence of rescaled displacements to the thin-film solution

Explicit expressions for fluid velocity and pressure in terms of the thin-film solution

Validation of the thin-film equation as a reduced model in the thin fluid limit

Abstract

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.