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Representation formulas and far-field behavior of time-periodic incompressible viscous flow around a translating rigid body

Thomas Eiter, Ana Leonor Silvestre

Abstract

This paper is concerned with integral representations and asymptotic expansions of solutions to the time-periodic incompressible Navier-Stokes equations for fluid flow in the exterior of a rigid body that moves with constant velocity. Using the time-periodic Oseen fundamental solution, we derive representation formulas for solutions with suitable regularity. From these formulas, the decomposition of the velocity component of the fundamental solution into steady-state and purely periodic parts and their detailed decay rate in space, we deduce complete information on the asymptotic structure of the velocity and pressure fields.

Representation formulas and far-field behavior of time-periodic incompressible viscous flow around a translating rigid body

Abstract

This paper is concerned with integral representations and asymptotic expansions of solutions to the time-periodic incompressible Navier-Stokes equations for fluid flow in the exterior of a rigid body that moves with constant velocity. Using the time-periodic Oseen fundamental solution, we derive representation formulas for solutions with suitable regularity. From these formulas, the decomposition of the velocity component of the fundamental solution into steady-state and purely periodic parts and their detailed decay rate in space, we deduce complete information on the asymptotic structure of the velocity and pressure fields.
Paper Structure (11 sections, 18 theorems, 170 equations)

This paper contains 11 sections, 18 theorems, 170 equations.

Table of Contents

  1. Introduction
  2. Notations and fundamental solutions
  3. Notations
  4. Steady-state Stokes and Oseen fundamental solutions
  5. Time-periodic fundamental solutions
  6. Anisotropic convolution estimates
  7. Main results
  8. The linearized time-periodic problem
  9. Representation formulas for solutions
  10. Asymptotic expansion of the velocity and pressure fields
  11. Proofs of the main results

Key Result

Lemma 2.1

Let $g\in L^{\infty}({\mathbb R}^3)$ and $A\in[2,\infty)$, $B\in[0,\infty)$, $M>0$ such that Then there exists $C= C(A,B,\zeta)>0$ with the following properties:

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 26 more