Optimal Control of Thin-Film Flow on a Flexible Topography

TL;DR

This paper develops a mathematical framework for optimally controlling thin-film flows on flexible surfaces, including rupture and coalescence, using PDE-based models and numerical algorithms to achieve desired film profiles.

Contribution

It introduces a novel optimal control approach for thin-film dynamics with rupture and coalescence, combining nonlinear PDE modeling with efficient numerical methods.

Findings

Control improves film profile accuracy.

Accelerates convergence to steady state.

Reduces instabilities and stabilizes dewetting.

Abstract

This work presents a mathematical model for the optimal control of thin-film flows over a flexible topography influenced by an external force. Our thin-film model allows for the rupture of films as well as the coalescence of bubbles. The objective is to find the optimal distributed force acting on the topography that minimises the differences between actual and desired thin-film profiles. A nonlinear lubrication equation governing the fluid dynamics and appropriate functional settings for this model are presented. It is also shown that this system satisfies a global energy-dissipation law for a suitable energy functional. Optimality conditions are derived for the solution of the minimisation problem of a specified cost function across a time horizon. These conditions are formulated at a continuous level as a system of coupled, forward-backward PDEs, which are subsequently discretised for numerical investigation. To ensure computational efficiency and stability, first-order Implicit-Explicit (IMEX) time-stepping schemes are employed to handle the nonlinearities in the model, and a reduced gradient descent algorithm is applied to obtain a numerical approximation of the optimal control signal. Numerical results illustrate that controlling the thin film, even during rupture, achieves a precise film profile. This control strategy accelerates convergence towards a steady state, reduces instabilities, stabilises dewetting processes, and meets the desired profile specifications.