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Compressible fluids and elastic plates in 2D: A conditional no-contact theorem

Dominic Breit, Arnab Roy

Abstract

We consider the interaction of a compressible fluid with a flexible plate in two space dimensions. The fluid is described by the Navier--Stokes equations in a domain that is changing in accordance with the motion of the structure. The displacement of the latter evolves according to a beam equation. Both are coupled through kinematic boundary conditions and the balance of forces. We prove that for any weak solution to the coupled system, which satisfies certain additional regularity requirements, no contact occurs between the elastic wall and the bottom of the fluid cavity. This applies to both isentropic and heat-conducting fluids. As a special case of our general theory we extend the unconditional result from Grandmont and Hillairet (Arch. Ration. Mech. Anal. 220, 1283--1333, 2016) on incompressible fluids from visco-elastic to perfectly elastic plates.

Compressible fluids and elastic plates in 2D: A conditional no-contact theorem

Abstract

We consider the interaction of a compressible fluid with a flexible plate in two space dimensions. The fluid is described by the Navier--Stokes equations in a domain that is changing in accordance with the motion of the structure. The displacement of the latter evolves according to a beam equation. Both are coupled through kinematic boundary conditions and the balance of forces. We prove that for any weak solution to the coupled system, which satisfies certain additional regularity requirements, no contact occurs between the elastic wall and the bottom of the fluid cavity. This applies to both isentropic and heat-conducting fluids. As a special case of our general theory we extend the unconditional result from Grandmont and Hillairet (Arch. Ration. Mech. Anal. 220, 1283--1333, 2016) on incompressible fluids from visco-elastic to perfectly elastic plates.

Paper Structure

This paper contains 13 sections, 9 theorems, 96 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries & main results
  3. Conventions
  4. Classical function spaces
  5. Function spaces on variable domains
  6. The main result
  7. The distance estimate
  8. The stream function
  9. The main estimate
  10. Particular fluid models
  11. Isentropic compressible fluids
  12. Heat-conducting compressible fluids
  13. Incompressible fluids

Key Result

Theorem 1.1

Suppose that $(\eta,\varrho,\mathbf{u})$ is a weak solution to 2--interfaceCond which satisfies additionally for some $q>\frac{8}{5}$ Then there is no contact for $t<T$.

Figures (2)

  • Figure 1: Domain transformation in 2D.
  • Figure 2: Interaction with contact.

Theorems & Definitions (17)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2: Weak solution
  • Theorem 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • ...and 7 more