Similarity and singularity in adhesive elastohydrodynamic touchdown

TL;DR

This paper models the touchdown of an elastic sheet adhering to a wall, revealing a fundamental stable similarity solution in elastohydrodynamic adhesion influenced by viscous, elastic, and adhesive forces.

Contribution

It introduces a combined elastohydrodynamic and adhesion model with numerical and scaling analysis, identifying the stable similarity solution governing touchdown dynamics.

Findings

Only the fundamental similarity solution is observed in simulations.

The model predicts experimentally verifiable singular flow behaviors.

The analysis extends rupture theories to elastohydrodynamic adhesion contexts.

Abstract

We consider the touchdown of an elastic sheet as it adheres to a wall, which has a dynamics that is limited by the viscous resistance provided by the squeeze flow of the intervening liquid trapped between the two solid surfaces. The dynamics of the sheet is described mathematically by elastohydrodynamic lubrication theory, coupling the elastic deformation of the sheet, the microscopic van der Waals adhesion and the viscous thin film flow. We use a combination of numerical simulations of the governing partial differential equation and a scaling analysis to describe the self-similar solution of the touchdown of the sheet as it approaches the wall. An analysis of the equation satisfied by the similarity variables in the vicinity of the touchdown event shows that an entire sequence of solutions are allowed. However, a comparison of these shows that only the fundamental similarity solution is observed in the time-dependent numerical simulations, consistent with the fact that it alone is stable. Our analysis generalizes similar approaches for rupture in capillary thin film hydrodynamics and suggests experimentally verifiable predictions for a new class of singular flows linking elasticity, hydrodynamics and adhesion.