An iterative approach to a fluid-rigid body interaction problem

TL;DR

This paper introduces an iterative method to prove short-time existence of strong solutions for a 3D fluid-rigid body interaction problem, coupling Navier-Stokes equations with rigid body dynamics, under small density ratio assumptions.

Contribution

The paper presents a novel iterative approach for establishing existence of solutions in fluid-rigid body interaction, applicable to moving boundary problems with numerical implications.

Findings

Proves short-time existence of strong solutions under small density ratio.

Develops an iterative scheme based on domain evolution for coupled fluid-structure problems.

Numerical experiments highlight the importance of the density ratio condition.

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 $\frac{\rho}{\rho_B}$, 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 $\frac{\rho}{\rho_B}$.