  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.
