Spreading and leveling of finite thin films with elastic interfaces: An eigenfunction expansion approach

Abstract The flow of a thin viscous liquid layer under an elastic film arises in natural processes, such as magmatic intrusions between rock strata, and industrial applications, such as coating surfaces with cured polymeric films. We study the linear dynamics of small perturbations to the equilibrium state of the film in a closed trough (finite film). Specifically, we are interested in the spreading (early-time) and leveling (late-time) dynamics as the film adjusts to equilibrium, starting from different initial perturbations. We consider both smooth and non-smooth spatially symmetric and localized initial conditions (perturbations). We find the exact series solutions for the film height, using the sixth-order complete orthonormal eigenfunctions associated with the posed initial-boundary-value problem derived in our previous work [Papanicolaou N C and Christov I C 2023 J. Phys.: Conf. Ser . 2675 012016]. We show that the evolution of the perturbations begins with the spreading of the localized perturbations, followed by their mutual interaction as they spread, and finally, interactions with the confining lateral boundaries of the domain as the perturbations level. In particular, we highlight how the leading eigenvalues of the problem determine the scalings of certain figures of merit with time.