Universal self-similar attractor in the bending-driven levelling of thin viscous films

Abstract

We study theoretically and numerically the bending-driven levelling of thin viscous films within the lubrication approximation. We derive 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. We also 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 nonlinear thin-film equation through numerical simulations.

Fluid-Elastic Interactions Near Contact at Low Reynolds Number

“…As the sheet approaches its fully relaxed state h ∼ σ , the dynamics become linear and self-similar. Scaling estimates and detailed analysis of Equation 20 in this limit (Pedersen et al 2021) find that the radius of the blister grows as t 1/6 , and its height decreases as t −1/3 to conserve volume. Thermal fluctuations of the fluid speed up the dynamics significantly (both the prefactor and the power law exponent) when they dominate over bending (Carlson 2018, Pedersen et al 2019) (see Figure 7f ).…”