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Equilibrium configurations of a 3D fluid-beam interaction problem

Vincenzo Bianca, Edoardo Bocchi, Filippo Gazzola

Abstract

We study a fluid-structure interaction problem between a viscous incompressible fluid and an elastic beam with fixed endpoints in a static setting. The 3D fluid domain is bounded, nonsmooth and non simply connected, the fluid is modeled by the stationary Navier-Stokes equations subject to inflow/outflow conditions. The structure is modeled by a stationary 1D beam equation with a load density involving the force exerted by the fluid and, thereby, may vary its position. In a smallness regime, we prove the existence and uniqueness of the solution to the PDE-ODE coupled system.

Equilibrium configurations of a 3D fluid-beam interaction problem

Abstract

We study a fluid-structure interaction problem between a viscous incompressible fluid and an elastic beam with fixed endpoints in a static setting. The 3D fluid domain is bounded, nonsmooth and non simply connected, the fluid is modeled by the stationary Navier-Stokes equations subject to inflow/outflow conditions. The structure is modeled by a stationary 1D beam equation with a load density involving the force exerted by the fluid and, thereby, may vary its position. In a smallness regime, we prove the existence and uniqueness of the solution to the PDE-ODE coupled system.

Paper Structure

This paper contains 9 sections, 8 theorems, 156 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geometric and physical frameworks
  3. Fluid boundary-value problem
  4. Existence and uniqueness of weak solutions
  5. Regularity
  6. Fluid-structure interaction problem
  7. Preliminary results
  8. Proof of Theorem \ref{['main-theo']}
  9. The symmetric case

Key Result

Theorem 3.2

Let $h\in C_0^{1,1}([-1,1])$ be small enough to ensure eq_bound_infty. For any $\gamma\geq0$, eq_main admits a weak solution $(u, p)$ in $\Omega_h$; there exists $\overline{\gamma}>0$ such that the weak solution is unique for $\gamma\in [0, \overline{\gamma})$. Moreover, any weak solution is a stron

Figures (5)

  • Figure 1: The fluid domain $\Omega=P\setminus\mathcal{B}$ and a cross-section $\mathcal{B}_y$.
  • Figure 2: Examples of fluid domains $\Omega$ (left) and $\Omega_h$ (right).
  • Figure 3: The Zhangjiajie Bridge.
  • Figure 4: The curve $\mathbb{C}^{h,\alpha}$ in the plane $\mathcal{P}_\alpha$.
  • Figure 5: The depicted boundary conditions for the endpoints is clamped (left) and hinged (right).

Theorems & Definitions (19)

  • Definition 3.1
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • Lemma 4.2
  • proof
  • Remark 4.3
  • Proposition 4.4
  • proof
  • Lemma 4.5
  • ...and 9 more