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An iterative approach to a fluid-rigid body interaction problem

Charles M. Elliott, Thomas Sales

TL;DR

The paper develops a constructive, fixed-point framework to establish short-time existence of strong solutions to incompressible NS flow coupled with rigid-body motion in a 3D moving domain, under a small relative density condition. It combines an iteration over NS on evolving domains with a prescribed rigid-body motion and a separate ODE evolution for the rigid body, using a harmonic extension to drive domain evolution and the Piola transform to preserve divergence structure across mappings. Key results include energy estimates that close on short time intervals, a rigorous pushforward construction for the domain, and a contraction proof ensuring a unique fixed point corresponding to a strong FSI solution. The approach also yields a feasible numerical scheme via an evolving finite element method, with numerical experiments in 2D illustrating the necessity of the small-density assumption for convergence and stability.

Abstract

We study a novel approach for the existence of solutions to an incompressible fluid-rigid body interaction problem in three dimensions. Our approach introduces an iteration based on a sequence of related problems posed on domains with prescribed evolution. In particular we prove the short-time existence of strong solutions to a system coupling the incompressible Navier--Stokes equations to the ordinary differential equations governing the motion of a rigid body, with no slip boundary conditions on the boundary of the rigid body, provided that the relative density $\fracρ{ρ_B}$, is sufficiently small. We also discuss the use of our iterative approach in numerical methods for the moving boundary problem, and complement this with some numerical experiments in two dimensions which demonstrate the necessity of the smallness assumption on $\fracρ{ρ_B}$.

An iterative approach to a fluid-rigid body interaction problem

TL;DR

The paper develops a constructive, fixed-point framework to establish short-time existence of strong solutions to incompressible NS flow coupled with rigid-body motion in a 3D moving domain, under a small relative density condition. It combines an iteration over NS on evolving domains with a prescribed rigid-body motion and a separate ODE evolution for the rigid body, using a harmonic extension to drive domain evolution and the Piola transform to preserve divergence structure across mappings. Key results include energy estimates that close on short time intervals, a rigorous pushforward construction for the domain, and a contraction proof ensuring a unique fixed point corresponding to a strong FSI solution. The approach also yields a feasible numerical scheme via an evolving finite element method, with numerical experiments in 2D illustrating the necessity of the small-density assumption for convergence and stability.

Abstract

We study a novel approach for the existence of solutions to an incompressible fluid-rigid body interaction problem in three dimensions. Our approach introduces an iteration based on a sequence of related problems posed on domains with prescribed evolution. In particular we prove the short-time existence of strong solutions to a system coupling the incompressible Navier--Stokes equations to the ordinary differential equations governing the motion of a rigid body, with no slip boundary conditions on the boundary of the rigid body, provided that the relative density , is sufficiently small. We also discuss the use of our iterative approach in numerical methods for the moving boundary problem, and complement this with some numerical experiments in two dimensions which demonstrate the necessity of the smallness assumption on .
Paper Structure (24 sections, 21 theorems, 279 equations, 5 figures, 4 tables)

This paper contains 24 sections, 21 theorems, 279 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Some useful results
  5. Navier--Stokes equations on a domain with prescribed evolution
  6. Statement of the problem
  7. Transformation to homogeneous boundary conditions
  8. Energy estimates
  9. Construction of the velocity field
  10. The pushforward map
  11. Compatibility of function spaces
  12. The iterative argument
  13. Some preliminary calculations
  14. Pullbacks and the Piola transform
  15. The fixed point argument
  16. ...and 9 more sections

Key Result

Theorem 1.2

Let the relative density, $\frac{\rho}{\rho_B}$, be sufficiently smallThe smallness of $\frac{\rho}{\rho_B}$ is determined by the geometry of $\Omega(0)$ and $\mathrm{diam}(B(0))$ (see the proofs of Lemma lemma: non degeneracy2 and Lemma lemma: fsi contraction).. Then there exists some time $T$, dep of strong solutions solving eqn: FSI fluid, eqn: FSI rigid body in the sense of Definition defn: st

Figures (5)

  • Figure 1: A diagram showing a fluid-rigid body interaction problem.
  • Figure 2: Diagram showing the dependencies for the presented method. Notice that quantities on a diagonal line may be computed in parallel, e.g. $(k=1, n=2)$ and $(k=2, n=1)$.
  • Figure 3: Diagram showing the dependencies for the less memory-intensive method. Notice that the dependencies require this method to be computed in serial.
  • Figure 4: Plots of the magnitude of the fluid velocity with a rigid body with density $\rho_B = \frac{200}{\pi}$. Row corresponds to $t = 0.033$, $t = 0.066$, and $t = 0.1$ respectively, and each column corresponds to the $0$th, $2$nd and $4$th iteration respectively.
  • Figure 5: Plots of the magnitude of the fluid velocity, at time $t = 0.05$, over 9 iterations, with a rigid body with density $\rho_B = \frac{10}{\pi}$.

Theorems & Definitions (55)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: Helmholtz decomposition
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • ...and 45 more