Periodic Reporting for period 1 - THANAFSI (Theoretical analysis of fluid-structure interaction problems and applications)
Période du rapport: 2023-09-01 au 2025-08-31
The THANAFSI project investigates fluid–structure interaction problems that arise in physical situations where fluids and solids mutually influence each other’s behavior, both in static and dynamic configurations. Understanding these interactions is crucial for tackling challenges of both scientific and societal relevance through advanced mathematical modeling. THANAFSI focuses on two research lines: (1) stability of suspension bridges under wind action, and (2) dynamics of a free-surface viscous fluid contained in a Hele-Shaw cell. Concerning the first line, the project incorporates nonlinear effects and asymmetric configurations in 2D and 3D wind-bridge models to more accurately describe real situations. By studying the equilibria of such systems, the continuity properties of the lift force exerted by the fluid on the structure and its repulsive nature near boundaries, the research advances the mathematical understanding of the stability of suspension bridges, eventually supporting the design of reliable infrastructure (UN SDG Target 9.1). The second research line of the project examines how the interaction between a free-surface viscous fluid and solid boundaries of the cell affects the dynamics. The goal is to achieve a complete understanding of the contact-angle problem at the interface and to explore the enhanced dissipation due to the contact and the long-time behavior of the dynamics. The outcomes of this study deepen the mathematical comprehension of wave-structure interactions, representing a key first step toward the development of efficient wave-energy converters (UN SDG Target 7.3).
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.
(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.
THANAFSI delivered significant advances in the mathematical study of fluid–structure interactions. For 2D and 3D wind–bridge systems, the project established existence, uniqueness, and a priori bounds for equilibria under small Reynolds numbers, as well as the continuous dependence of the lift force on variations in the body and inflow–outflow geometries. A zero-lift asymmetric configuration and a new stability measure were identified. These results open opportunities for future research aimed at determining optimal body shapes that minimize structural instability. The repulsive nature of the lift force near boundaries was rigorously analyzed, yielding explicit singular behavior of the solution and the associated energy in general multiple-collision configurations, through a variational method of independent interest to the mathematical fluid–structure interaction community. The analysis paves the way to the rigorous justification of helical trajectories and drift of particles along undulations observed in experiments. It also represents the first step toward a complete understanding of the multi-body configuration, a necessary step for the study of interactions in particle clouds. For the 3D beam–fluid interaction problem, the project established existence and uniqueness of equilibria under both hinged and clamped boundary conditions, and provided a rigorous regularity analysis ensuring a well-defined local lift force even in non-smooth geometrical settings. This result represents a key first step toward the study of the interaction in the evolutionary case and the stability of the equilibrium state. For the fluid dynamics in a Hele–Shaw cell, global-in-time a priori estimates showing the dissipative nature of the contact points and the long-time behavior of the dynamics were established for initial data close to the equilibrium state. These results were supported by a suitable energy–dissipation framework and by key elliptic estimates handling the contact-angle issue without imposing angle restrictions. This achievement constitutes a major step toward the establishment of global existence, uniqueness and decay results. Moreover, the analysis of the surface–wall contact problem advances the mathematical comprehension of wave–structure interactions and will support future research on full models for wave-energy converters.