  The use of thin liquid films has expanded beyond lubrication and coatings,
and into applications in actuators and adaptive optical elements. In contrast
to their predecessors, whose dynamics can be typically captured by modelling
infinite or periodic films, these applications are characterized by a finite
amount of liquid in an impermeable domain. The global mass conservation
constraint, together with common boundary conditions (e.g., pinning) create
quantitatively and qualitatively different dynamics than those of infinite
films. Mathematically, this manifests itself as a non-self-adjoint problem.
This work presents a combined theoretical and experimental study for this
problem. We provide a time-dependent closed-form analytical solution for the
linearized non-self-adjoint system that arises from these boundary conditions.
We highlight that, in contrast to self-adjoint problems, here special care
should be given to deriving the adjoint problem to reconstruct the solution
based on the eigenfunctions properly. We compare these solutions with those
obtained for permeable and periodic boundary conditions, representing common
models for self-adjoint thin-film problems. We show that while the initial
dynamics are nearly identical, the boundary conditions eventually affect the
film deformation as well as its response time. To experimentally illustrate the
dynamics and to validate the theoretical model, we fabricated an experimental
setup that subjects a thin liquid film to a prescribed normal force
distribution through dielectrophoresis, and used high-frame-rate digital
holography to measure the film deformation in real-time. The experiments agree
well with the model and confirm that confined films exhibit different behaviour
which could not be predicted by existing models.
