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Equilibrium Configurations and their Uniqueness in a Fluid-Solid Interaction Problem

D. Bonheure, G. P. Galdi, C. Patriarca

TL;DR

The paper proves existence of equilibrium configurations for a Navier–Stokes fluid coupled to a rotating rigid body with elastic restoring forces in an exterior domain, under a uniform far-field flow. It reformulates the problem in a body-fixed frame on a fixed domain and uses invading-domain Schaefer fixed-point theory to obtain equilibria for any $\lambda>0$, complemented by a priori bounds and regularity results. Uniqueness is established for small $\lambda$ by exploiting an Oseen-type structure for the difference of two solutions and deriving $L^q$ bounds that yield a contraction. The results advance the mathematical understanding of fluid–structure interactions with rotation and provide a foundation for stability analysis in future work.

Abstract

We demonstrate existence in the ``large" and uniqueness in the ``small" of equilibrium configurations for the coupled system consisting of a Navier-Stokes fluid interacting with a rigid body subjected to spring forces and restoring moments. The driving mechanism is a uniform, given velocity field of the fluid at large spatial distances from the body. The main difficulty in the proof of the above properties arises from the fact that the body can rotate around a given axis, which produces a highly nonlinear problem.

Equilibrium Configurations and their Uniqueness in a Fluid-Solid Interaction Problem

TL;DR

The paper proves existence of equilibrium configurations for a Navier–Stokes fluid coupled to a rotating rigid body with elastic restoring forces in an exterior domain, under a uniform far-field flow. It reformulates the problem in a body-fixed frame on a fixed domain and uses invading-domain Schaefer fixed-point theory to obtain equilibria for any , complemented by a priori bounds and regularity results. Uniqueness is established for small by exploiting an Oseen-type structure for the difference of two solutions and deriving bounds that yield a contraction. The results advance the mathematical understanding of fluid–structure interactions with rotation and provide a foundation for stability analysis in future work.

Abstract

We demonstrate existence in the ``large" and uniqueness in the ``small" of equilibrium configurations for the coupled system consisting of a Navier-Stokes fluid interacting with a rigid body subjected to spring forces and restoring moments. The driving mechanism is a uniform, given velocity field of the fluid at large spatial distances from the body. The main difficulty in the proof of the above properties arises from the fact that the body can rotate around a given axis, which produces a highly nonlinear problem.
Paper Structure (8 sections, 9 theorems, 129 equations)

This paper contains 8 sections, 9 theorems, 129 equations.

Table of Contents

  1. Introduction
  2. Formulation of the Problem and Functional Setting
  3. Existence of Equilibria
  4. Resolution in Bounded Domains
  5. Proof of Proposition \ref{['prop:1']}
  6. Uniqueness of Equilibrium
  7. Bounds for the Solutions in terms of $\lambda$
  8. Proof of Theorem \ref{['th:unicita']}

Key Result

Theorem 3.1

For any $\lambda>0$, there is at least one solution $({\hbox{\boldmath $v$}} ,p,\hbox{\boldmath $\delta$},\theta)\in C^\infty(\Omega)\times C^\infty(\Omega)\times{\mathbb R}^3\times{\mathbb R}$ to (eq:7)-- (eq:8) that, in addition, satisfies for all $q\in (2,\infty]$, $r\in(\frac{4}{3},\infty]$, $s\in (\frac{3}{2},\infty]$, $\sigma\in (1,\infty)$.

Theorems & Definitions (19)

  • Theorem 3.1
  • Remark 3.1
  • Lemma 3.1
  • proof
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.1
  • proof : Proof of Theorem \ref{['th:1']}
  • Definition 3.2
  • Lemma 3.2
  • ...and 9 more