|Abstract:|| Free surface problems are ubiquitous in fluid dynamics applications: e.g., we can think of waves on a liquid, capillary rising, droplets movement. Simulating such phenomena is a challenging issue from the numerics, analytics, and even modeling point of view, in particular when the contact line (the three-phase interface among the wall, the fluid, and the medium beyond the free surface) is allowed to move. Indeed, different models for a moving contact line are proposed in the literature, but each of them brings some hindrance to the analysis and the discretization of the problem.
In the present talk, a time-dependent free-surface Navier-Stokes problem with moving contact line is addressed. A variational derivation of the contact line model justifies it, and proper spaces are introduced to have it mathematically well-defined. A discussion on its discretization via ALE is then carried on, showing how numerical results and energy estimates drove to the choice of piecewise linear finite elements in both pressure and velocity. A control problem is, then, introduced, having the free-surface problem as a constraint, and an optimization procedure is designed for its solution, based on a two-level Lagrangian approach.