An elastic plate on a thin viscous film

TL;DR

This paper models the steady-state behavior of an elastic plate on a thin viscous film, analyzing the effects of elasticity, viscosity, and surface tension through numerical and asymptotic methods, revealing boundary-layer phenomena.

Contribution

It introduces a coupled mathematical model combining a Landau-Levich equation and a beam equation, with new insights into boundary-layer effects in elastocapillary problems.

Findings

Boundary-layer effects occur near the plate ends.

The model captures the interplay of elasticity, viscosity, and surface tension.

Asymptotic analysis reveals behavior in relevant parameter limits.

Abstract

We consider the steady-state analysis of a pinned elastic plate lying on the free surface of a thin viscous fluid, forced by the motion of a bottom substrate moving at constant speed. A mathematical model incorporating elasticity, viscosity, surface tension, and pressure forces is derived, and consists of a third-order Landau-Levich equation for the thin film, and a fifth-order beam equation for the plate. A numerical and asymptotic analysis is presented in the relevant limits of the elasticity and Capillary numbers. We demonstrate the emergence of boundary-layer effects near the ends of the plate, which are likely to be a generic phenomenon for singularly perturbed elastocapillary problems.