Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of weak solutions for incompressible fluid-Koiter shell interactions with Navier slip boundary condition

Claudiu Mîndrilă, Arnab Roy

Abstract

We study a three-dimensional fluid-structure interaction problem describing the motion of an incompressible, viscous fluid coupled with a deformable elastic shell of Koiter type that forms part of the fluid boundary. The fluid motion is governed by the incompressible Navier--Stokes equations posed on a time-dependent domain, while the shell evolution is described by a nonlinear elastic model. At the fluid-structure interface, we impose Navier slip boundary conditions, allowing for tangential slip penalized by friction. Our main result establishes the global-in-time existence of weak solutions up to the first possible self-intersection of the shell, for arbitrarily large initial data with finite energy. The analysis is carried out in a fully three-dimensional setting and addresses the major mathematical challenges arising from the moving domain, the geometric nonlinearity of the shell, and the reduced regularization induced by the slip boundary condition. The proof relies on a careful construction of suitable approximation schemes, novel compactness arguments adapted to the slip framework, and a new extension operator for divergence-free test functions compatible with the fluid-shell coupling. As a further contribution, we provide a direct approach to the strong convergence of second-order spatial derivatives of the shell displacement, which allows us to treat nonlinear Koiter shell models within the same framework.

Existence of weak solutions for incompressible fluid-Koiter shell interactions with Navier slip boundary condition

Abstract

We study a three-dimensional fluid-structure interaction problem describing the motion of an incompressible, viscous fluid coupled with a deformable elastic shell of Koiter type that forms part of the fluid boundary. The fluid motion is governed by the incompressible Navier--Stokes equations posed on a time-dependent domain, while the shell evolution is described by a nonlinear elastic model. At the fluid-structure interface, we impose Navier slip boundary conditions, allowing for tangential slip penalized by friction. Our main result establishes the global-in-time existence of weak solutions up to the first possible self-intersection of the shell, for arbitrarily large initial data with finite energy. The analysis is carried out in a fully three-dimensional setting and addresses the major mathematical challenges arising from the moving domain, the geometric nonlinearity of the shell, and the reduced regularization induced by the slip boundary condition. The proof relies on a careful construction of suitable approximation schemes, novel compactness arguments adapted to the slip framework, and a new extension operator for divergence-free test functions compatible with the fluid-shell coupling. As a further contribution, we provide a direct approach to the strong convergence of second-order spatial derivatives of the shell displacement, which allows us to treat nonlinear Koiter shell models within the same framework.
Paper Structure (33 sections, 19 theorems, 137 equations, 1 figure)

This paper contains 33 sections, 19 theorems, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Novelty and significance.
  3. Overview.
  4. Geometry
  5. Formulation of the problem
  6. The fluid-structure interaction (FSI) problem.
  7. Variational formulation and the main results
  8. Formal a-priori estimates
  9. Traces. Function spaces. Variational formulation.
  10. Trace operator.
  11. Function spaces.
  12. Variational formulation of the equations.
  13. Main results
  14. Auxiliary tools
  15. Korn's inequality in moving Hölder domains
  16. ...and 18 more sections

Key Result

Lemma 2.1

Let $\eta \in H^{2}(\omega)$, $1<p<\infty$ and $1<r<p$. Then there exists a linear and continuous trace operator given by the formula whose continuity constant depends on $\Omega,r$ and an upper bound for $\left\Vert \eta\right\Vert _{H_{x}^{2}}$.

Figures (1)

  • Figure 1: The moving domains

Theorems & Definitions (40)

  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 2.1
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.1
  • Remark 2.4
  • Theorem 2.1
  • ...and 30 more