# On the Steady Motion of a Coupled System Solid-Liquid

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*Josef Bemelmans; Giovanni P. Galdi; Mads Kyed*

The authors study the unconstrained (free) motion of an elastic solid \(\mathcal B\) in a Navier-Stokes liquid \(\mathcal L\) occupying the whole space outside \(\mathcal B\), under the assumption that a constant body force \(\mathfrak b\) is acting on \(\mathcal B\). More specifically, the authors are interested in the steady motion of the coupled system \(\{\mathcal B,\mathcal L\}\), which means that there exists a frame with respect to which the relevant governing equations possess a time-independent solution. The authors prove the existence of such a frame, provided some smallness restrictions are imposed on the physical parameters, and the reference configuration of \(\mathcal B\) satisfies suitable geometric properties.

#### Table of Contents

# Table of Contents

## On the Steady Motion of a Coupled System Solid-Liquid

- Chapter 1. Introduction 18 free
- Chapter 2. Notation and Preliminaries 512 free
- Chapter 3. Steady Free Motion: Definition and Formulation of the Problem 1118
- Chapter 4. Main Result 1926
- Chapter 5. Approximating Problem in Bounded Domains 3946
- Chapter 6. Proof of Main Theorem 6370
- Chapter 7. Bodies with Symmetry 6976
- 7.1. Symmetry Function Spaces 6976
- 7.2. Main Theorem for Symmetric Bodies 7077
- 7.3. Stokes Problem for a Symmetric Body 7077
- 7.4. Reformulation of the Equations of Motion 7178
- 7.5. Compatibility Conditions 7380
- 7.6. Approximating Problem in Bounded Domains 7481
- 7.7. Fixed-Point Approach 7582
- 7.8. Validity of the Compatibility Conditions 7582
- 7.9. Solvability of the Fluid Equations 7683
- 7.10. Solvability of the Elasticity Equations 7784
- 7.11. Existence in a Bounded Domain 7986
- 7.12. Proof of Main Theorem for Symmetric Bodies 7986
- 7.13. Examples 8087

- Appendix A. Isolated Orientation 8390
- Bibliography 8794