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Time-periodic weak solutions for the interaction of an incompressible fluid with a linear Koiter type shell under dynamic pressure boundary conditions

Claudiu Mîndrilă, Sebastian Schwarzacher

Abstract

In many occurrences of fluid-structure interaction time-periodic motions are observed. We consider the interaction between a fluid driven by the three dimensional Navier-Stokes equation and a two dimensional linearized elastic Koiter shell situated at the boundary. The fluid-domain is a part of the solution and as such changing in time periodically. On a steady part of the boundary we allow for the physically relevant case of dynamic pressure boundary values, prominent to model inflow/outflow. We provide the existence of at least one weak time-periodic solution for given periodic external forces that are not too large. For that we introduce new approximation techniques and a-priori estimates.

Time-periodic weak solutions for the interaction of an incompressible fluid with a linear Koiter type shell under dynamic pressure boundary conditions

Abstract

In many occurrences of fluid-structure interaction time-periodic motions are observed. We consider the interaction between a fluid driven by the three dimensional Navier-Stokes equation and a two dimensional linearized elastic Koiter shell situated at the boundary. The fluid-domain is a part of the solution and as such changing in time periodically. On a steady part of the boundary we allow for the physically relevant case of dynamic pressure boundary values, prominent to model inflow/outflow. We provide the existence of at least one weak time-periodic solution for given periodic external forces that are not too large. For that we introduce new approximation techniques and a-priori estimates.
Paper Structure (21 sections, 12 theorems, 159 equations, 1 figure)

This paper contains 21 sections, 12 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The model
  3. Main results
  4. Mathematical strategy and technical novelties
  5. Overview of the paper
  6. Preliminaries
  7. Function spaces on variable domains
  8. Some analytical tools
  9. Weak solutions
  10. A divergence-free extension operator
  11. Geometric setting
  12. The decoupled problem
  13. Formal derivation of decoupled solutions
  14. A-priori estimates
  15. Proof of the main results
  16. ...and 6 more sections

Key Result

Theorem 1.16

Let $\Omega$ be a given reference domain with properties described as above. Then there exists a constant $\tilde{C}$ depending on $\Gamma_p,\Gamma_D,\kappa,L,|M|$ and $c_0$ such that if $\left(\mathbf{f},g,P\right)\in L_{\text{per}}^{2}\left(I;L^2(\mathbb{R}^{3})\right)\times L_{\text{per}}^{2}\lef then there exists at least one weak time-periodic solution $\left(\mathbf{u},\eta\right)$ as it is

Figures (1)

  • Figure 1: The domain $\Omega_{\eta(t)}$

Theorems & Definitions (25)

  • Theorem 1.16
  • Remark 1.20: On the size of $\tilde{C}$ and C
  • Remark 1.21: On the size of $\tilde{C}$ and C
  • Remark 1.22: Case $\Gamma_p=0$
  • Remark 1.23
  • Theorem 1.24
  • Lemma 2.2: Trace operator
  • proof
  • Theorem 2.3
  • Theorem 2.7: Kakutani-Glicksberg-Fan
  • ...and 15 more