Similarity and singularity in adhesive elastohydrodynamic touchdown

Abstract

We consider the dynamics of an elastic sheet as it starts to adhere to a wall, a process that is limited by the viscous squeeze flow of the intervening liquid. Elastohydrodynamic lubrication theory allows us to derive a partial differential equation coupling the elastic deformation of the sheet, the microscopic van der Waals adhesion, and viscous thin film flow. We use a combination of numerical simulations of the governing equation and a scaling analysis to describe the self-similar touchdown of the sheet as it approaches the wall. An analysis of the equation in terms of similarity variables in the vicinity of the touchdown event 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.