Fluid Dynamics and Thin Films
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Abstract
Abstract
Abstract. This work presents a mathematical model for the optimal control of thin-film flows over a flexible substrate influenced by an external force. The objective is to find the optimal distributed force acting on the topography that minimizes 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 minimization 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 discretized for numerical investigation. To ensure computational efficiency and stability, first-order implicit-explicit 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, stabilizes dewetting processes, and meets the desired profile specifications.
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@article{Alrashidy2026Optimal,
title = {Optimal Control of Thin-Film Flow on a Flexible Topography},
author = {Sattam S. Alrashidy and Anna Kalogirou and Dante Kalise and Kristoffer G. van der Zee},
journal = {SIAM Journal on Applied Mathematics},
year = {2026},
doi = {10.1137/25m1752572},
url = {https://doi.org/10.1137/25m1752572}
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