The non-self-adjoint thin film problem: dynamics of fixed-volume pinned films*

The deformation of thin liquid films has been investigated for more than a century, owing to their importance in a wide range of natural phenomena and engineering applications. Theoretical studies addressed infinite and periodic domain, or cases that give rise to self-similar solutions. Yet, to the best of our knowledge, the simple case of a fixed-volume liquid film that is pinned on impermeable boundaries — arguably the most relevant for engineering applications such as adaptive optics — has been overlooked.

We here present a theoretical framework and experimental measurements for the dynamics of a thin film within an impermeable domain that is subjected to a normal force distribution at its interface. From a physical perspective, the mass conservation in the systems acts as an additional constraint which drives distinctly different dynamics. Mathematically, this manifests in a non-self-adjoint differential equation that requires more careful treatment in its solution. Under the long wave approximation, we obtain a time-dependent solution of the linearized system. We provide experimental validation, showing very good agreement with the model, by actuating a liquid film using dielectrophoretic forces, and imaging its topography in real time using holographic microscopy.