Special Session 72: Fluid-structure interaction and free boundary problems

Local well-posedness of a model for fluidic elastic membranes

Helmut Abels University of Regensburg Germany Co-Author(s):    Yadong Liu, Andrea Poiatti

Abstract: We consider a system consisting of a surface Navier-Stokes equation for incompressible fluids on an elastic inextensible membrane, whose evolution is coupled to the flow of the fluid on the surface. This leads to a highly nonlinear quasilinear geometric evolution equation of parabolic-hyperbolic type. Using a suitable parametrization we linearize the system and obtain well-posedness of it in suitable $L^2$-type Sobolev spaces. With the aid of a suitable fixed-point argument we show existence of strong solutions locally in time.

Analysis and numerics of a fully averaged poroelastic plate model and its interaction with a fluid

Felix Brandt University of California, Berkeley USA Co-Author(s):    Sun\\v{c}ica \\v{C}ani\\`c, Andrew Scharf, Josip Tamba\\v{c}a

Abstract: In the first part of this talk, we derive a fully averaged poroelastic plate model. We discuss the existence of weak and mild solutions using a semigroup approach and propose a splitting scheme for the numerical analysis, comparing our model to existing poroelastic plate models. The second part of the talk addresses the interaction problem of the fully averaged plate model with a Stokes fluid. We first prove the existence of weak solutions using energy methods and then show the existence of a unique strong solution to a regularized version of the problem. This is achieved by establishing the sectoriality of the associated operator matrix with a non-diagonal domain, thereby paving the way for maximal $\mathrm{L}^p$-regularity of the corresponding Cauchy problem. Finally, we derive a finite element method for the numerical approximation of the coupled problem and demonstrate its effectiveness in capturing the interaction between the fluid and the poroelastic plate. This talk is based on joint work with S. \v{C}ani\`c, A. Scharf, and J. Tamba\v{c}a.

On the long time behavior of a family of several rigid bodies immersed in a viscous fluid

Eduard Feireisl Institute of Mathematics, Czech Academy of Sciences Czech Rep Co-Author(s):    Marco Bravin, Arnab Roy, Arghir Zarnescu

Abstract: We consider several rigid bodies immersed in a viscous Newtonian fluid contained in a bounded domain in $R^3$. We introduce a new concept of dissipative weak solution of the problem based on a combination of the approach proposed by Judakov with a suitable form of energy inequality. We show that global in time dissipative solutions always exist as long as the rigid bodies are connected compact sets. In addition, in the absence of external driving forces, the system always tends to a static equilibrium as time goes to infinity. The results hold independently of possible collisions of rigid bodies and for any finite energy initial data.

On 3D Navier-Stokes helical vortex filament

Francisco Gancedo Universidad de Sevilla Spain Co-Author(s):

Abstract: Vortex filaments are fundamental structures in three-dimensional incompressible fluid dynamics, capturing the concentration of vorticity along curves and providing a natural framework for the analysis of coherent structures. Understanding their evolution under the 3D Navier-Stokes equations remains a challenging problem, especially in regimes where geometric features of the flow play a significant role. In this talk, we consider vortex filaments with helical structure, which arise naturally in flows exhibiting symmetry and offer a tractable setting to investigate the interplay between geometry and diffusion. Our analysis applies to vortex filaments of arbitrary size and without vanishing swirl assumption. After introducing the mathematical formulation of the problem and the main analytical difficulties, we present new results concerning the existence, uniqueness, and evolution of such helical vortex filaments in the Navier-Stokes setting.

On the dynamics of a fluid-structure interaction problem

Rafael Granero Belinchon Universidad de Cantabria Spain Co-Author(s):

Abstract: In this talk we will present some new results for a fluid-structure problem. In particular, we will present the equations describing the dynamics. These new PDEs present a double nonlinear structure where the nonlinearity affects also to the time derivative of the solution. Furthermore, we will present their mathematical well-posedness theory.

Analysis of a Fluid-Poroviscoelastic Interaction Problem

Matthias Hieber TU Darmstadt Germany Co-Author(s):    Tim Binz and Arnab Roy

Abstract: In this talk we discuss a coupled viscoelastic Navier-Stokes-Biot system describing the interaction between an incompressible, viscous fluid and a poro-viscoelastic medium. The coupling is realized through Beavers-Joseph-Saffman type interface conditions. We establish the global wellposedness of this system in the strong sense for small initial data and derive in addition a Serrin-type blow-up criterion.

The vanishing viscosity limit in permeable domains

Anna L Mazzucato Penn State University USA Co-Author(s):

Abstract: I will discuss the zero-viscosity limit in domains that allow inflow and outflow for both the Navier-Stokes as well as the Boussinesq equations.

On the problem of compressible fluid-structure interaction with Navier boundary conditions

Sarka Necasova Czech Academy of Sciences Czech Rep Co-Author(s):    K. Bhandari, I. Djebour, S.Mitra, Y.Liu,

Abstract: In the talk, we will focus on the problem of compressible barotropic fluid with elastic structure and with Navier boundary conditions. We show the existence of weak solutions, a singular limit to the inviscid case. This part of the talk is based on collaboration with S. Mitra and Y. Liu. Moreover, we show the existence of a strong solution and the uniqueness of the strong solution. It is based on the work with K. Bhandari and I. Djebour.

Thermal effects in fluid structure interactions

Sebastian Schwarzacher Uppsala University/Charles University Sweden Co-Author(s):

Abstract: In this lecture we consider two different heat conducting fluids each modelled by the incompressible Navier-Stokes-Fourier system separated by a non-linear elastic Koiter shell. The motion of the shell changes the domain of definition of the two separated fluids. For this setting we discuss the existence of a weak solution. The heat capacity of the shell is given energetically. It allows to consider transmission laws ranging from insulation to superconductivity. A variational approach for fluid-structure interaction is used. This method seems particularly usefull for elastodynamics, for which also some results will be discussed in the lecture. The lecture is based on a joint work with Sourav Mitra.

The 3D Kutta-Joukowski effect

Franck Sueur University of Luxembourg France Co-Author(s):    Olivier Glass, David Meyer

Abstract: In this talk, I will present a joint work with Olivier Glass and David Meyer on the 3D counterpart of the well-known 2D Kutta-Joukowski effect, experimented by a rigid body moving in a perfect incompressible fluid.

Finite-time contact in fluid-elastic structure interactions

Krutika Tawri University of Washington USA Co-Author(s):    Nash Ward

Abstract: In this talk, we will consider a fluid-structure interaction problem involving a viscous, incompressible fluid flow, modeled by the 2D Navier-Stokes equations, through a thin deformable elastic tube, elastodynamics of which is modeled by 1D plate equations. The fluid and the structure are nonlinearly coupled at the fluid-structure interface. The fluid flow is driven by dynamic pressure data imposed at the inlet and the outlet of the tube. In this talk, we will impose the Navier-slip boundary condition at the fluid-structure interface and at the bottom rigid boundary of the fluid domain. We will first discuss the existence of weak solutions and reveal a `hidden' spatial regularity result for the structure displacement. Then we will discuss our recent result that establishes the existence of a finite time for the weak solutions at which the compliant upper boundary meets the lower boundary (i.e., the tube collapses), provided that there is a sufficient pressure drop across the channel. This resolves the ``no-collision'' paradox identified in the no-slip setting and thus validates the model to correctly capture near-contact dynamics.

Contact in fluid-plate interaction: formation and detachment

Srdan Trifunovic Faculty of Sciences, University of Novi Sad Yugoslavia Co-Author(s):

Abstract: In this talk, I will present a recent result on contact problem for the interaction between an elastic plate and a compressible viscous fluid located between the plate and a rigid bottom z = 0. First, by utilizing the vertical fluid dissipation, a new estimate is obtained $\ln\eta(t)\in L^1$ for any $t>0$ provided that $\ln\eta_0\in L^1$, ensuring that additional contact can form only on a set of a measure zero. Then, by utilizing the expanding capability of compressible fluid pressure, it is shown that all contact has to detach in finite time provided that the source force is not pushing down too much. Finally, it is shown that contact at any point can be detached in any given time with a strong enough source force localized around that point which is pulling the plate up. This is the first result where detachment of contact is proven.