Stability of time-dependent motions for fluid-rigid ball interaction

TL;DR

This paper investigates the stability and decay of time-dependent motions of a rigid ball in a viscous fluid governed by Navier-Stokes equations, providing new insights into fluid-structure interaction stability.

Contribution

It introduces the first analysis of large-time behavior for solutions around nontrivial, possibly time-dependent states in fluid-rigid body interaction, establishing a novel stability theorem.

Findings

Derived $L^q$-$L^r$ decay estimates for linearized system

Applied estimates to nonlinear problem for decay properties

Established stability results for time-dependent motions of a rigid ball

Abstract

We aim at the stability of time-dependent motions, such as time-periodic ones, of a rigid body in a viscous fluid filling the exterior to it in 3D. The fluid motion obeys the incompressible Navier-Stokes system, whereas the motion of the body is governed by the balance for linear and angular momentum. Both motions are affected by each other at the boundary. Assuming that the rigid body is a ball, we adopt a monolithic approach to deduce $L^q$-$L^r$ decay estimates of solutions to a non-autonomous linearized system. We then apply those estimates to the full nonlinear initial value problem to find temporal decay properties of the disturbance. Although the shape of the body is not allowed to be arbitrary, the present contribution is the first attempt at analysis of the large time behavior of solutions around nontrivial basic states, that can be time-dependent, for the fluid-structure interaction problem and provides us with a stability theorem which is indeed new even for steady motions under the self-propelling condition or with wake structure.