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Fluid-Structure interactions with Navier- and full-slip boundary conditions

Antonín Češík, Malte Kampschulte, Sebastian Schwarzacher

Abstract

We show the existence of weak solutions to the fluid-structure interaction problem of a largely deforming viscoelastic bulk solid with a viscous fluid governed by the incompressible Navier-Stokes equations. In contrast to previous works, the fluid is allowed to slip along the solid boundary; namely, the so called Navier-slip boundary conditions are considered. Such boundary conditions naturally involve the time-changing outer normal of the fluid domain. Hence, their dependence on the varying geometry is one degree higher than in the previously considered no-slip case, which makes it necessary to adjust the concept of weak coupled solutions. Two classes of test functions are introduced: test functions that are continuous over the fluid-solid domain, and fluid-only test functions with nonzero tangential component at the boundary. The weak equations are established until the point of contact, and moreover, compatibility with the strong formulation is shown.

Fluid-Structure interactions with Navier- and full-slip boundary conditions

Abstract

We show the existence of weak solutions to the fluid-structure interaction problem of a largely deforming viscoelastic bulk solid with a viscous fluid governed by the incompressible Navier-Stokes equations. In contrast to previous works, the fluid is allowed to slip along the solid boundary; namely, the so called Navier-slip boundary conditions are considered. Such boundary conditions naturally involve the time-changing outer normal of the fluid domain. Hence, their dependence on the varying geometry is one degree higher than in the previously considered no-slip case, which makes it necessary to adjust the concept of weak coupled solutions. Two classes of test functions are introduced: test functions that are continuous over the fluid-solid domain, and fluid-only test functions with nonzero tangential component at the boundary. The weak equations are established until the point of contact, and moreover, compatibility with the strong formulation is shown.
Paper Structure (28 sections, 27 theorems, 307 equations, 1 figure)

This paper contains 28 sections, 27 theorems, 307 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Assumptions on energy and dissipation
  3. The approximation
  4. Preliminaries
  5. Moving domains
  6. Transport theorem
  7. Spaces on a moving domain
  8. Convergence of domains
  9. Convergence of functions on different domains
  10. Approximation of test functions
  11. Eulerian to Lagrangian and back
  12. Extending the divergence-free domain
  13. Universal Bogovskii operator
  14. Approximating fluid-only test functions
  15. Strong and weak formulation
  16. ...and 13 more sections

Key Result

Theorem 1.1

The fluid-structure interaction problem eqn:solid-intro-eqn:init-cond-intro has a weak solution in the sense of Definition def:weak-solution-full, until the time of the first solid-solid collision.

Figures (1)

  • Figure 1: Geometrical depiction of the fluid-structure interaction problem.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 2.1: Transport theorem
  • proof
  • Remark
  • Proposition 2.2
  • proof
  • Lemma 2.3: Convergence of normal trace
  • proof
  • Lemma 2.4
  • proof
  • ...and 43 more