Universal Self-Similar Attractor in the Bending-Driven Leveling of Thin Viscous Films

TL;DR

This paper investigates the long-term universal self-similar behavior in the bending-driven leveling of thin viscous films, deriving the Green's function and analyzing convergence to a universal attractor.

Contribution

It derives the Green's function for the linearized thin-film equation and demonstrates its role as a universal self-similar attractor at long times, including nonlinear extensions.

Findings

Rescaled perturbations converge to the Green's function attractor

Existence of a short-term self-similar regime for stepped initial conditions

Characterization of convergence time and energy evolution

Abstract

We study theoretically and numerically the bending-driven leveling of thin viscous films within the lubrication approximation. We derive the Green's function of the linearized thin-film equation and further show that it represents a universal self-similar attractor at long times. As such, the rescaled perturbation of the film profile converges in time towards the rescaled Green's function, for any summable initial perturbation profile. In addition, for stepped axisymmetric initial conditions, we demonstrate the existence of another, short-term and one-dimensional-like self-similar regime. Besides, we characterize the convergence time towards the long-term universal attractor in terms of the relevant physical and geometrical parameters, and provide the local hydrodynamic fields and global elastic energy in the universal regime as functions of time. Finally, we extend our analysis to the non-linear thin-film equation through numerical simulations.