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On the long time behaviour of a system of several rigid bodies immersed in a viscous fluid

Marco Bravin, Eduard Feireisl, Arnab Roy, Arghir Zarnescu

TL;DR

The paper develops a dissipative weak solution framework for the fluid–structure interaction problem of multiple rigid bodies in a bounded 3D viscous domain, proving global existence for finite-energy data without smallness assumptions and allowing collisions. It extends Judakov’s weak formulation by enforcing a differential energy inequality, enabling analysis of long-time behavior. The main results show exponential decay of kinetic energy and rigid-body motions in the absence of forcing, and characterize energy behavior under gravitational forcing, with convergence of kinetic energy to zero and potential energy to a finite limit. The approach relies on careful approximation with highly viscous regularizations and a passage to the limit while preserving the differential energy balance, offering insights into the asymptotics of complex FSI systems beyond smooth strong solutions.

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 the long time behaviour of a system of several rigid bodies immersed in a viscous fluid

TL;DR

The paper develops a dissipative weak solution framework for the fluid–structure interaction problem of multiple rigid bodies in a bounded 3D viscous domain, proving global existence for finite-energy data without smallness assumptions and allowing collisions. It extends Judakov’s weak formulation by enforcing a differential energy inequality, enabling analysis of long-time behavior. The main results show exponential decay of kinetic energy and rigid-body motions in the absence of forcing, and characterize energy behavior under gravitational forcing, with convergence of kinetic energy to zero and potential energy to a finite limit. The approach relies on careful approximation with highly viscous regularizations and a passage to the limit while preserving the differential energy balance, offering insights into the asymptotics of complex FSI systems beyond smooth strong solutions.

Abstract

We consider several rigid bodies immersed in a viscous Newtonian fluid contained in a bounded domain in . 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.
Paper Structure (12 sections, 3 theorems, 45 equations)

This paper contains 12 sections, 3 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Fluid domain, state variables
  4. Mass conservation
  5. Momentum balance
  6. Energy balance
  7. Dissipative weak solutions
  8. Weak solutions to the fluid structure interaction problem revisited
  9. Approximate solutions with smooth data
  10. Long--time behaviour of dissipative solutions
  11. System without external forcing
  12. Gravitational forcing

Key Result

Theorem 2.1

Let $\Omega \subset R^3$ be a bounded domain. Let $(\mathcal{S}^i)_{i=1}^N$ be a family of connected compact sets satisfying rr1, rr2. Let the mass densities $(\varrho_{\mathcal{S}^i})_{i=1}^N$ be positive constants, Finally, let ${\bf g} \in L^\infty(\Omega; R^3)$ be a given volume force. Then the fluid--structure interaction problem admits a dissipative weak solution in $(0,\infty) \times \Omeg

Theorems & Definitions (6)

  • Definition 1.1: Dissipative weak solution
  • Theorem 2.1: Dissipative solution -- global existence
  • Remark 2.2
  • Theorem 3.1: Long--time behaviour without forcing
  • Remark 3.2
  • Theorem 3.3: Long--time behaviour, gravitational forcing