THANAFSI was implemented through four work packages, three related to the research line (1) and one related to (2).
(1a) The project kicked off by addressing the equilibrium configurations of a 2D wind–bridge model in a wind tunnel. First, unique solvability of the stationary Navier–Stokes equations for any fixed position of the immersed bodies at small Reynolds numbers was proved, establishing a priori bounds depending on the body’s vertical displacement. Using a new solenoidal extension, the study then established existence and uniqueness of the coupled fluid–structure interaction, ensuring a unique equilibrium for 2D wind–bridge systems. As a related aspect, the lift force behavior was subsequently studied, proving continuity properties for admissible body and flow classes. A zero-lift result in asymmetric configurations provided direct relevance to wind-tunnel experiments. Finally, a new stability measure was introduced to define stability of structure immersed in laminar flows, and the existence of an optimal body shape minimizing this measure was established.
(1b) The second package examined a general 3D configuration with multiple bodies immersed in a viscous fluid, extending the 2D single-body case of (1a). Zero-velocity conditions were imposed on all but one solid boundary, where given data were prescribed. After defining collision regimes and quasi-contact regions, the variational structure of the Stokes problem was exploited through reduced energy functionals in terms of vector potentials and gradient functions. A double minimization of such functionals yielded explicit approximate velocity fields competing with the exact solution. As an outcome of a rigorous asymptotic analysis combined with error estimates, the explicit behavior of the Stokes solution and energy was established as the collision distances vanish, accounting for boundary data dependence.
(1c) The third package addressed equilibrium configurations of a 3D wind–bridge model in a wind tunnel, where the bridge is modeled by an elastic beam. The study first established existence, uniqueness, and a priori bounds for the fluid system corresponding to a given beam displacement. A delicate regularity analysis at the walls–beam contact ensured well-defined load densities (local lift force) for the beam equation, and in order to handle the domain’s limited smoothness, integrability was improved by working in non-Hilbertian spaces. The coupled fluid–structure interaction was then analyzed for both hinged and clamped boundary conditions of the beam. Under a smallness assumption on the inflow–outflow magnitudes, existence and uniqueness of the equilibrium were established.
(2a) The fourth package focused on the dynamics of a viscous fluid in a Hele–Shaw cell, mathematically described by the one-phase 2D Muskat problem with contact points between the fluid surface and the container walls. After introducing potential and fixed-domain formulations, the basic a priori energy–dissipation balance was established. Higher-order energy and novel dissipation terms were derived, with a focus on additional dissipation and trace estimates. Key elliptic estimates were then obtained by exploiting the Neumann problem satisfied by the velocity potential, allowing spatial regularity results without restrictions on the contact angles. Finally, a global-in-time a priori higher-order bound and exponential decay were established for solutions initially close to equilibrium.
