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

Toshiaki Hishida

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.

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

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 - 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.
Paper Structure (22 sections, 23 theorems, 416 equations)

This paper contains 22 sections, 23 theorems, 416 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Nondimensional variables
  4. Stokes-structure operator
  5. Main results
  6. Basic motions
  7. Oseen-structure evolution operator
  8. Decomposition of $L^q_R(\mathbb R^3)$
  9. \ref{['resol1']} is equivalent to \ref{['st-str-resol']}
  10. Stokes-structure resolvent
  11. Generation of the evolution operator
  12. Smoothing estimates of the evolution operator
  13. Estimate of the pressure
  14. Adjoint evolution operator and backward problem
  15. Decay estimates of the evolution operator
  16. ...and 7 more sections

Key Result

Theorem 2.1

Suppose ass-1 and ass-2, then the operator family $\{L_+(t);\, t\in\mathbb R\}$ generates an evolution operator $\{T(t,s);\, -\infty<s\leq t<\infty\}$ on $X_q(\mathbb R^3)$ for every $q\in (1,\infty)$ with the following properties: Furthermore, if $\|U_b\|\leq \alpha_j$ being small enough, to be precise, see below about how small it is for each item $j=1,2,3,4$, then the evolution operator $T(t,s

Theorems & Definitions (53)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.2
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • ...and 43 more