Rigorous derivation of a linear sixth-order thin-film equation as a reduced model for thin fluid -- thin structure interaction problems

TL;DR

This paper rigorously derives a linear sixth-order thin-film equation as a simplified model for a fluid-structure interaction involving a thin viscous fluid layer and an elastic plate, with strong convergence and error estimates.

Contribution

It provides a rigorous derivation of a reduced sixth-order thin-film model from a 3D fluid-structure interaction problem, including error estimates and convergence analysis.

Findings

Reduced model is a linear sixth-order thin-film equation.

Quantitative error estimates for approximate solutions.

Strong convergence results relating the reduced model to the original problem.

Abstract

We analyze a linear 3D/3D fluid-structure interaction problem between a thin layer of a viscous fluid and a thin elastic plate-like structure with the aim of deriving a simplified reduced model. Based on suitable energy dissipation inequalities quantified in terms of two small parameters, thickness of the fluid layer and thickness of the elastic structure, we identify the right relation between the system coefficients and small parameters which eventually provide a reduced model on the vanishing limit. The reduced model is a linear sixth-order thin-film equation describing the out-of-plane displacement of the structure, which is justified in terms of weak convergence results relating its solution to the solutions of the original fluid-structure interaction problem. Furthermore, approximate solutions to the fluid-structure interaction problem are reconstructed from the reduced model and quantitative error estimates are obtained, which provide even strong convergence results.